Function Spaces Fourth Conference on Function Spaces May 1419,2002 Southern Illrnois University at Edwardsville Krzysztof Jarosz Editor
Function Spaces Fourth Conference on Function Spaces May 1419,2002 Southern Illinois University at Edwardsville
Andy R. Magid
This volwne contains the proceedings of the Fourth Conference on Function Spaces, held May 1419, 2002, at Southern Illinois University at Edwardsville. 2000 Mathematics Subject Classification. Primary 32H02, 46E25, 46H05, 46JlO, 46J15, 46L07, 47AlO, 47B38, 47LlO, 54D05.
Library of Congress CataloginginPublication Data Conference on Function Space (4th: 2002 : Southern Illinois University at Edwardsville) Function spaces : Fourth Conference on Function Spaces, May 1419, 2002, Southern Illinois University at Edwardsville / Krzysztof Jarosz, editor. p. cm.  (Contemporary mathematics, ISSN 02714132 j 328) Includes bibliographical references. ISBN 0821832697 (softcover : alk. paper) 1. Function spacesCongresses. I. Jarosz, Krzysztof, 1953 II. Title. III. Contemporary mathematics (American Mathematical Society) j v. 328 QA323.C66 2002 515'.73dc21
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Components of resolvent sets and local spectral theory PIETRO AlENA AND FERNANDO VILLAFANE
JOHN R. AKEROYD
A CauchyGreen formula on the unit sphere in C 2 JOHN T. ANDERSON AND JOHN WERMER
A connected metric space that is not separably connected RICHARD M. ARON AND MANUEL MAESTRE
Weighted Chebyshev centres and intersection properties of balls in Banach spaces PRADIPTA BANDYOPADHYAY AND S. DUTTA
Complete isometries  an illustration of noncommutative functional analysis DAVID P. BLECHER AND DAMON M. HAY
Some recent trends and advances in certain lattice ordered algebras KARIM BOULABIAR, GERARD BUSKES, AND ABDELMAJID TRIKI
An extension of a theorem of Wermer, Bernard, Sidney and Hatori to algebras of functions on locally compact spaces 135
The unique decomposition property and the BanachStone theorem AUDREY CURNOCK, JOHN HOWROYD, AND NGAICHING WONG
Some more examples of subsets of Co and L1 [0, 1] failing the fixed point property P. N. DOWLING, C. J. LENNARD, AND B. TURETT
Homotopic composition operators on HOC) (Bn) PAMELA GORKIN, RAYMOND MORTINI, AND DANIEL SUAREZ
Characterization of conditional expectation in terms of positive projections J. J. GROBLER AND M. DE KOCK
Characterizations and automatic linearity for ring homomorphisms on algebras of functions OSAMU HATORI, TAKASHI ISHII, TAKESHI MIURA, AND SINEI TAKAHASI
Weak *extreme points of injective tensor product spaces KRZYSZTOF JAROSZ AND T. S. S. R. K. RAO
Determining sets and fixed points for holomorphic endomorphisms KANGTAE KIM AND STEVEN G. KRANTZ
Localization in the spectral theory of operators on Banach spaces T. L. MILLER, V. G. MILLER, AND M. M. NEUMANN
Abstract harmonic analysis, homological algebra, and operator spaces VOLKER RUNDE
Uniform algebras generated by unimodular functions STUART J. SIDNEY
Analytic functions on compact groups and their applications to almost periodic functions THOMAS TONEV AND S. A. GRIGORYAN
Preface The Fourth Conference on Function Spaces was held at Southern Illinois Universityat Edwardsville from May 14 to May 19, 2002. It was attended by over 100 participants from 25 countries. The lectures covered a broad range of topics, including spaces and algebras of analytic functions of one and of many variables (and operators on such spaces), LPspaces, spaces of Banachvalued functions, isometries of function spaces, geometry of Banach spaces, and other related subjects. The main purpose of the conference was to bring together mathematicians interested in various problems within the general area of function spaces and to allow a free discussion and exchange of ideas with people working on exactly the same problems as well as with people working on related questions. Hence, most of the lectures, and therefore the papers in this volume, have been directed to nonexperts. A number of articles contain an exposition of known results (known to experts) and open problems; other articles contain new discoveries that are presented in a way that should be accessible also to mathematicians working in different areas of function spaces. The conference was the fourth in a sequence of conferences on function spaces at SlUE, with the first held in the spring of 1990, the second in the spring of 1994, and the third one in the spring of 1998. The proceedings of the first two conferences were published with Marcel Dekker in the Lecture Notes in Pure and Applied Mathematics series (#136 and #172); the proceedings of the third conference were published by the AMS in the Contemporary Mathematics series (#232). The abstracts, the schedule of the talks, and other information, as well as the pictures of the participants, are available on the conference Web page at http://www.siue.edu/MATH/conference/. The conference was sponsored by grants from Southern Illinois University and from the National Science Foundation. The editor would like to thank everyone who contributed to the proceedings: the authors, the referees, the sponsoring institutions, and the American Mathematical Society. The editor would also like to express very special thanks to his wife, Dorota, for her active professional help during all of the stages of the organization  without her help the conference and the proceedings would not have been possible. Krzysztof Jarosz
Components of resolvent sets and local spectral theory Pietro Aiena and Fernando Villafane ABSTRACT. In this paper we shall study the components of various resolvent sets associated with some spectra originating from Fredholm theory. In particular, we obtain a classification of these components by using, in the case of operators of Kato type, the equivalences between the single valued extension property at a point and some kerneltype and range type conditions established in [2], [6], [3] and [5]. We also show that certain subspace valued mappings coincide on the components of the Kato type resolvent and give a precise description of the operators having empty Kato type spectrum.
1. Single valued extension property
Throughout this paper, T is assumed to be a bounded linear operator on a complex Banach space X and L( X) will denote the algebra of all bounded linear operators on L(X). If x E X, the local resolvent set of T at x E X, denoted by PT(X), is defined as the union of all open subsets U of C such that there is an analytic function f : U  X which satisfies the equation (1.1)
x for all >.. E U .
The local spectrum aT (x) of T at x is defined by aT (x) := C \ PT (x) and obviously aT(x) ~ a(T), where a(T) denotes the spectrum of T. The operator T E L(X) is said to have the singlevalued extension property at >"0 E C ( SVEP at >"0 for brevity) if, for every neighborhood U of >"0, the only analytic function f : U  X which satisfies the equation (>.1  T)f(>..) = 0 for all >.. E U is the function f == O. The operator T E L(X) is said to have SVEP if T has SVEP at every point>" E C. Clearly, if T has SVEP at >"0, then the analytic solution of (1.1) in a neighborhood U of >"0 is uniquely determined. The SVEP was first introduced by Dunford [8], [9] and has later received a systematic treatment in DunfordSchwartz [10]. The SVEP at a point was first introduced by Finch [11] and successively investigated by several authors, see [20], [28], [2], [6], [3] and [5] . The basic role of SVEP arises in local spectral theory, since every operator which satisfies the socalled Bishop's property (13) enjoys this 1991 Mathematics Subject Classification. Primary 47AlO, 47A11. Secondary 47A53, 47A55. Key words and phrases. Single valued extension property, semiregular operators, Kato decomposition property . The research was supported by the International Cooperation Project between the University of Palermo (Italy) and the University of Barquisimeto.
property, see [18] for definition and results. Recall that an operator T E L(X) on a Banach space X is said to be decomposable if for every open cover {UI , U2 } of C there exist Tinvariant closed linear subspaces Xl and X 2 of X for which X = Xl + X 2 , a(T IXd ~ Ul and a(T IX2 ) ~ U2 • The class of decomposable operators contains, for instance, all normal operators, all spectral operators, all operators with a nonanalytic functional calculus and all compact operators, or more generally all operators with a totally disconnected spectrum. Note that T is decomposable if and only if T and its dual T* have property ((3), see Theorem 2.5.19 [18]. Consequently, if T is decomposable, then both T and T* have SVEP. For an arbitrary subset F of C let XT(F) be the local spectral subspace associated with F, defined by XT(F) := {x EX: aT(x) ~ F}. If F is a closed subset of C, let XT(F) be the glocal spectral subspace associated with F, defined as the set of all x E X for which there exists an analytic function f : C \ F + X which satisfies (>.J  T)f(A) = x for all A E C \ F. Clearly, XT(F) and XT(F) are (not necessarily closed) linear subspaces of X with XT(F) ~ XT(F) for all closed sets F ~ C. Note that, by Proposition 3.3.2 of [18], the identity XT(F) = XT(F) holds for all closed sets F ~ C precisely when Thas SVEP, and this is the case if and only if X T (0) = {O}, see Proposition 1.2.16 of [18]. The SVEP at a point Ao may be characterized in a similar way: T has SVEP at Ao if and only if ker (AoI  T) n X T (0) = {O}, cf. [1, Theorem 1.9]. Two important subspaces in Fredholm theory are the hyperrange of T, defined by TOO(X) = n:'=l Tn(X), and the hyperkernel of T defined by N°O(T) = U:'=l ker Tn. Recall that the ascent of an operator T is the smallest nonnegative integer p := p(T) such that ker TP = ker TP+l. If such integer does not exist, we put p(T) = 00. Obviously, if T has finite ascent p then N°o (T) = ker TP. Analogously, the descent q := q(T) of an operator T is the smallest nonnegative integer q such that Tq(X) = Tq+l(X). If such integer does not exist we put q(T) = 00. Also, if T has finite descent q then TOO(X) = Tq(X). It is wellknown that if p(T) and q(T) are both finite then p(T) = q(T), see [15, Proposition 38.3]. Furthermore, p(AoI  T) = q(AoI  T) < 00 if and only if Ao is a pole of the resolvent R(A, T) := (>.J  T)l, [15, Proposition 50.23]. Associated with T E L(X) there is another linear subspace of X, the quasinilpotent part of T defined as
Ho(T) := {x EX: lim IITnxll l / n = O}. n>oo
Evidently, N°O(T) ~ Ho(T). Moreover, Ho(T) = X if and only if T is quasinilpotent, i.e. a(T) = {O}, see [20, Theorem 1.5]. The following decomposition property studied by Mbekhta [20], [19], Mbekhta and Ouahab [21], has its origin in the classical treatment of perturbation theory due to Kato [17], who showed an important decomposition for semiFredholm operators: DEFINITION 1.1. An operator T E L(X) is called semiregular if T(X) is closed and N°O(T) ~ TOO(X). An operator T E L(X) is said to be of Kato type if there exists a pair of Tinvariant closed subspaces (M, N) such that X = M EB N, the restriction T 1M is semiregular and T IN is nilpotent. The pair (M, N) is called a generalized Kato decomposition ( GKD, for brevity) for T.
If, additionally, in the definition above we assume that N is finitedimensional then T is said to be essentially semiregular, see Rakocevic [25] and Miiller [24]. By Proposition 3.1.6 of [18], T is semiregular if and only if T* is semiregular. Analogously, by Corollary 3.4 of [24], T is. essentially semiregular if and only if T* is essentially semiregular. Moreover, if T is essentially semiregular then Tn is essentially semiregular for all n E N, again by Corollary 3.4 of [24]. It should be noted that the range of an essentially semiregular operator is always closed. In fact, if (!vI, N) is a GKD for T with N finitedimensional, then T(X) is the direct sum of T(!vI), which is closed because T I !vI is semiregular, and of T(N), which is finitedimensional.
Two very important class of essentially semiregular operators is given by the class of upper semiFredholm operators, defined by ~+(X) := {T E L(X) :
see Kato [17, Theorem 4] or West [31]. The class of Fredholm operators is defined by ~(X) := ~+(X)n~_(X). Note that a semiFredholm operator T is semiregular if and only if its jump j(T) is zero, see ([31, Proposition 2.2]. Moreover, if T is of Kato type, and in particular if T is semiregular, then TOO(X) is closed with T(TOO(X)) = TOO(X), see Theorem 2.3 and Theorem 2.4 of [2], and TOO(X) coincides with the analytical core K(T) := {x E X: there exist a constant c> 0 and a sequence of elements Xn E X such that Xo = x, TX n = Xnl, and Ilxnll ::; cnllxll for all n EN}. It should be noted that both subspaces K(T) and Ho(T) admit a local spectral characterization. In fact, (1.2)
= {x EX: 0 f/. O"T(X)},
see Mbekhta [20], Vrbova [30] and also Propositions 3.3.7 and 3.3.13 of [18], and Ho(T) = XT({O}, see Propositions 3.3.7 and 3.3.13 of [18]. Therefore, if T has SVEP, then Ho(T) = X T ( {O}). In the following lemma by A.L and its proof we denote the annihilator of a subset A ~ X, and by .L B the preannihilator of a subset B ~ X*. LEMMA
1.2. For every T E L(X), the following statements hold
(i) Ho(T) ~.L K(T*) and K(T) ~.L Ho(T*). (ii) If T is a Kato type operator and the pair (!vI, N) is a GKD for T then
Ho(T) = Ho(T I !vI) EB Ho(T I N) = Ho(T I !vI) EB N.
K(T).
PROOF. (i) See Proposition 4.1 of [5]. (ii) The proof of the equality K(T) = K(T I M) may be found in [1]. The second equality in (1.3) is clear, since the nilpotency of TIN implies Ho(T I N) = N. The inclusion Ho(T I M) + Ho(T I N) £;;; Ho(T) is evident. To show the opposite inclusion, consider an arbitrary element x E Ho(T) and set x = y + z, with y E M, zEN. Since TIN is quasinilpotent then N = Ho(T) £;;; Ho(T). Therefore y = x  z E Ho(T) n M = Ho(T I M) and consequently Ho(T) £;;; Ho(T I M) + Ho(T IN). (iii) Assume that T is essentially semiregular and hence also T* essentially semiregular. Then T*n is essentially semiregular for all n E N, so that T*n(X*) is closed for all n E N. From part (i) we know that N°O(T) £;;; Ho(T) £;;;1. K(T*), so that, to show the first two equalities of (1.4), we need only to prove the inclusion 1. K(T*) £;;; N°O(T). For every T E L(X) and every n E N we have ker Tn £;;; N°O(T) and hence N°O(T)1. £;;; ker Tn1. = T*n(X*), because the last subspace is closed for :.:::::,1.
all n E N. From this we easily obtain that N°O(T) £;;; T*OO(X*) = K(T*), where the last equality holds since T* is essentially semiregular and hence of Kato type. Consequently, 1. K(T*) £;;; N°O(T). Thus the first two equalities of (1.4) are proved. The equality K(T) =1. Ho(T*) is proved in a similar way. (iv) The semiregularity of T entails that N°O(T) £;;; TOO(X) = K(T), so that, by part (ii), Ho(T) = Noo(T) £;;; K(T) = K(T) , and this concludes the proof. 0 We have already observed that an operator T E L(X) has SVEP at >'0 precisely when ker (>'01  T) n K(>'ol  T) = {a}. From the inclusion ker (>'01  T) £;;; N°O(>'ol  T) £;;; Ho(>'ol  T) it then follows that the condition Ho(>'ol  T) n K(>'ol  T) = {a} implies that T has SVEP at >'0' Example 2.5 of [5] shows that SVEP for T at a point does not necessarily imply that Ho(>'ol  T) n K(>'ol  T) = {a}. In [2] it has been shown that also the condition N°O(>'ol  T) n (>'01  T)OO(X) = {a} implies SVEP at >'0 for T. The next result shows that these implications are actually equivalences in the case that >'01  T is of Kato type. THEOREM 1.3. If >'01  T is of Kato type then the following properties are equivalent: (i) T has the SVEP at >'0; (ii) Ho(>'ol  T) n K(>'ol  T) = {a}; (iii) Ho(>'ol  T) is closed; (iv) >'01  T has finite ascent; (v) N°O(>'ol  T) n (>'01  T)OO(X) = {a}. Furthermore, if >'01  T E L(X) is essentially semiregular, the assertions (i)(v) are equivalent to the following conditions: (vi) Ho(>'ol  T) is finitedimensional. (vii) N°O(>'ol  T*) + (>'01  T*)(X*) is weak *dense in X*; (viii) Ho(>'ol  T*) + (>'01  T*)(X*) is weak *dense in X*; (ix) Ho(>'ol  T*) + K(>'ol  T*) is weak *dense in X*. In this case >'01  T E iP+(X).
PROOF. The equivalence of (i), (ii), (iii) and (iv) has been established in [3, Theorem 2.6 and Corollary 2.7]. The equivalence (i) {:} (v) has been proved in Theorem 2.6 of [2], see also Theorem 2.1 of the present paper. Assume now that AoI  T is essentially semiregular. We may assume that AO = O. (i) {:} (vi) Obviously, if Ho(T) is finitedimensional then Ho(T) is closed, so T has SVEP at 0, by the equivalence (i) {:} (iii). Conversely, if T has SVEP at 0 then also TIM has SVEP at 0, since the local SVEP is inherited by the restrictions to closed invariant subspaces. The semiregularity of TIM then implies that TIM is injective, see Theorem 2.14 of [1] and therefore N°O(T) = {O}. By part (iii) of Lemma 1.2 we then conclude that Ho(T I M) = N°O(T 1M) = {O}. From part (ii) of Lemma 1.2 it follows that Ho(T) = {O} EB N = N is finitedimensional. Finally, if T is essentially semiregular then also Tn is essentially semiregular and therefore, the ranges Tn(x) are closed for all n E N. From Theorem 4.3 of [5] it then follows that the condition (i) is equivalent to each one of the conditions (vii), (viii) and (ix). It remains to establish that (i) implies that AoI  T E ~+(X). Clearly, if Ho(AoI  T) is finite dimensional then also its subspace ker (AoI  T) is finitedimensional. Since (AoI T)(X) is closed we then conclude that AoI T E ~+(X).
The next Theorem 2.1 will show that, if AoI  T of Kato type, then Ho(AoI T) n K(AoI  T) = N°O(AoI  T) n (AoI  T)OO(X). The following characterizations of SVEP for the dual T* are dual, in a sense, to those given in Theorem 1.3. THEOREM 1.4. Suppose that AoI  T is of Kato type. Then the following statements are equivalent: (i) T* has SVEP at AO; (ii) X = Ho(AoI  T) + K(AoI  T); (iii) AoI  T has finite descent; (iv) X = N°O(AoI  T) + (AoI  T)OO(X); (v) Ho(AoI  T) + K(AoI  T) is normdense in X; (vi) N°O(AoI  T) + (AoI  T)OO(X) is normdense in X; Furthermore? if AoI  T E L(X) is a essentially semiregular then the assertions (i)(vi) are equivalent to the following conditions: (vii) K(AoI  T) is finitecodimensional; (viii) N°o (AoI  T) + (AoI  T) (X) is normdense in X; (ix) Ho(AoI  T) + (AoI  T)(X) is normdense in X. In this case AoI  T E ~_(T). PROOF. Also here we may assume that AO = 0 and T is of Kato type. The equivalence of (i), (ii) and (iii) has been established in Theorem 2.9 of [3]. The equivalence of (i) and (iv) has been proved in Theorem 2.9 of [2], see also Theorem 2.1 of the present paper. Clearly, (ii) ::::} (v), (iv) ::::} (vi). The implications (v) ::::} (i) and (vi) ::::} (i) have been proved in Corollary 4.2 of [5], so that the statements (i)(vi) are equivalent. Now, assume that T is essentially semiregular. Then Tn(x) is closed for
all n E N, so that, by Theorem 4.3 of [5], the statements (i), (viii) and (ix) are equivalent. To conclude the proof note first that if (M, N) is a GKD for T then the pair (Nl.,Ml.) is a GKD for T*. Now, if T* has SVEP at 0, then, as observed in the proof Theorem 1.3, T* I Nl. is injective and therefore, see Lemma 2.8 of [2], TIM is surjective. Therefore K(T) = K(T I M) = M is finitecodimensional, so that the implication (i) => (vii) is proved. Conversely, suppose that the analytical core K(T) is finiteco dimensional. From K(T) = TOO(X) ~ Tn(x) we deduce that q(T) < 00, so that (vii) implies (iii), and the proof of the equivalences is complete . Finally, from the inclusion K(>"ol T) ~ (>"01 T)(X) we infer that, if K(>"o/T) finiteco dimensional, then also (>"01  T)(X) is finitecodimensional, so that >"01  T E
1.5. If >"01  T is of Kato type, then the following equivalences hold:
(i) T has the SVEP at >"0 precisely when l7ap(T) does not cluster at >"0, [6, Theorem 2.2]; (ii) T* has the SVEP at >"0 precisely when l7su(T) does not cluster at >"0,' [6, Theorem 2.5].
2. Components In this section we shall take a closer look at the components of some resolvent sets associated with the various spectra originating from Fredholm theory. In particular, we shall obtain a classification of these components, by using the constancy of some mappings and the equivalences between the SVEP at a point and the kerneltype and range type conditions, established in the previous section. For an operator T E L(X), we consider the following parts of the ordinary spectrum: the Kato spectrum l7k(T) := {>.. E C : >..I  T is not semiregular},
and the essential Kato spectrum l7ke(T) := {>.. E C : >..I  T is not essentially semiregular}.
Moreover, we define l7kt(T) := {>.. E C : >..I  T is not of Kato type}.
COMPONENTS OF RESOLVENT SETS AND LOCAL SPECTRAL THEORY
7
It is known that the three sets O'k(T), O'kt(T) and O'ke(T) are closed, for the first set see [18, Proposition 3.1.9], for the other two sets see [4, Corollary 1]. Moreover, O'k(T) and O'ke(T) are nonempty, since the first spectrum contains the boundary of O'(T), see [18, Proposition 3.1.6], while the second spectrum contains the boundary of the Fredholm spectrum O'f(T) := p, E C : >.J  T ~ cJ>(X)}, see [24, Theorem 3.8]. Next we shall show that O'kt(T) is nonempty precisely when O'(T) is not a finite set of poles. Let Pk(T) := C \ O'k(T), Pkt(T) := C \ O'kt(T) and Pke(T) := C \ O'ke(T) be the resolvents associated with these spectra. The sets Pk(T), Pkt(T) and Pke(T) are open subsets of C, so they may be decomposed in connected disjoint open nonempty components. Clearly, (2.1) Note that for every T E L(X) we have Pk(T) In [29]
= Pk(T*)
and Pke(T)
= Pke(T*).
6 Searcoid and West showed the constancy of the mappings
on the components of the semiFredholm resolvent Psf(T) := C \ O'sf(T), where O'sf(T) is the semiFredholm spectrum defined by
O'sf(T) := {A E C: >.J  T
~
cJ>+(X) U cJ>_(X)}.
From the Kato decomposition for semiFredholm operators we easily obtain the following inclusions (2.3) The work of 6 Searcoid and West [29] extended previous results established by Homer [16], by Goldmann and Kraekovskii [13], [14], and by Saphar [26], which have established the constancy of the functions
on a component of the semiFredholm resolvent Psf(T), except for the discrete subset of points for which AI  T is not semiregular. In the same vein, Forster [12] showed that the mappings
are constant as A ranges through a component of the Kato resolvent Pk(T). The constancy of these mappings has also been studied by Mbekhta and Ouahab [22], which showed the constancy of the mappings (2.4)
A ~ Ho(>.J  T)
+ K(>.J 
T),
A ~ Ho(>.J  T) n K(>.J  T)
on the components of Pkt(T). The next result shows that the mappings (2.2) and (2.4) coincide, respectively, on the components of Pkt(T), so that the Mbekhta and Ouahab result extends the previous result of 6 Searcoid and West. 2.1. Let >.J  T be of Kato type. Then (i) N°O(>.J  T) + (>.J  T)OO(X) = Ho(>.J  T) + K(>.J  T).
THEOREM
(ii) N°o(>.J  T)
n (>.J 
T)OO(X) = Ho(>.J  T)
n K(>.J 
T).
PIETRO AlENA AND FERNANDO VILLAFANE
8
PROOF. (i) Throughout this proof we may take>. = O. Let (M, N) be a GKD for T such that (T I N)d = 0 for some integer dEN. By part (ii) of Lemma 1.2 we know that K(T) = K(T I M) = K(T) n M. Moreover, by part (iv) of Lemma 1.2, the semiregularity of TIM implies that Ho(T I M) ~ K(T I M) = K(T). From this we obtain
n K(T) = Ho(T) n (K(T) n M) = (Ho(T) n M) n K(T) = Ho(T I M) n K(T) = Ho(T 1M), and therefore Ho(T) n K(T) = Ho(T 1M). Ho(T)
We claim that Ho(T) + K(T) = NEB K(T). From N ~ ker Td ~ Ho(T) we obtain that NEB K(T) ~ Ho(T) + K(T). Conversely, from part (ii) of Lemma 1.2 we have
Ho(T) = NEB Ho(T I M) = NEB (Ho(T) n K(T))
~
NEB K(T),
so that
Ho(T) + K(T)
~
(N EB K(T)) + K(T)
~
NEB K(T),
so our claim is proved. Since K(T) = TOO(X) for every operator of Kato type, we obtain from the inclusion N ~ ker Td ~ N°O(T), that
Ho(T)
+ K(T) =
so the equality N°O(T)
NEB K(T)
+ TOO (X)
~
N°O(T)
= Ho(T)
+ TOO (X)
~
Ho(T)
+ K(T),
+ K(T) is proved.
(ii) Suppose again that>. = O. Let (M, N) be a GKD for T such that, for some dEN, we have (T IN)d = O. Then ker Tn = ker (T IM)n for every natural n :2: d. Since ker Tn ~ ker Tn+! for all n E N we then have
N°O(T) =
00
00
n2:d
n2:d
U ker Tn = U ker(T I M)n = N°O(T 1M).
The semiregularity of TIM then implies, by part (iii) of Lemma 1.2, that (2.5)
N°O(T) = N°O(T I M) = Ho(T I M) = Ho(T)
n M.
Next we show that the equality Ho(T) n M = Ho(T) n M holds. The inclusion Ho(T) n M ~ Ho(T) n M is evident. Conversely, suppose that x E Ho(T) n M. Then there is a sequence (xn) C Ho(T) such that Xn + x as n + 00. Let P be the projection of X onto M along N. Then PX n + Px = x and PX n E Ho(T) n M. Therefore x E Ho (T) n M. Finally, from (2.5) and taking into account that K(T) n M = K(T) = TOO(X), we then obtain
N°O(T)
n TOO (X) = Ho(T) n M n K(T) = Ho(T) n (M n K(T)) = Ho(T) n K(T),
so the proof is complete.
D
From the constancy of the mappings>. + Ho(>.J  T)nK(>.J T), or, which is the same, of the mappings>. + Noo(>.J  T) n (>.J  T)OO(X), on the components of Pkt(T) and the results established in the previous section we now obtain the following classification.
COMPONENTS OF RESOLVENT SETS AND LOCAL SPECTRAL THEORY
9
THEOREM 2.2. LetT E L(X) and 0. a component ofpkt(T). Then the following alternative holds: either (i) T has SVEP for every point ofn. In this case p(>.J  T) < 00 for all A E n. Moreover, aap(T) does not have limit points in 0.; every point of 0., except possibly for at most countably many isolated points, is not an eigenvalue of T.
or (ii) T has SVEP at no point of n. In this case p(>.J  T) = Every point of 0. is an eigenvalue of T, PROOF. (i) Suppose that T has SVEP at AO E Ho(>.J  T) is closed and
n.
00
for all A E
n.
Then, by Theorem 1.3,
n K(AoI  T) = HO(AoI  T) n K(AoI  T) = {O}. Since the mapping A + Ho(>.J  T) n K(>.J  T) is constant on the component 0., then Ho(>.J  T) n K(>.J  T) = {O} for all A E 0. and this implies, again by HO(AoI  T)
Theorem 1.3, that T has SVEP at every A E n. This is equivalent, also by Theorem 1.3, to saying that p(AI  T) < 00 for all A E n. Moreover, from Theorem 1.5, aap(T) does not cluster in 0. and, consequently, every point of 0. is not an eigenvalue of T, except a subset of 0. which consists of at most count ably many isolated points.
o
(ii) This is clear, again by Theorem 1.3.
Recall that A E C is said to be a deficiency value for if >.J  T is not surjective. THEOREM 2.3. Let T E L(X) and 0. a component of Pkt(T). Then the following alternative holds: either (i) T* has SVEP for every point of n. In this case q(>.J  T) < 00 for all A E n. Moreover, asu(T) does not have limit points in 0.; every point ofn, except possibly for at most countably many isolated points, is not a deficiency value of T.
or (ii) T* has the SVEP at no point of n. In this case q(>.J  T) = A E 0. and every A E 0. is a deficiency value of T.
00
for all
PROOF. Proceed as in the proof of Theorem 2.2, combining the constancy on the components of Pkt(T) of the mapping A E 0. + K(>.J  T) + Ho(>.J  T) (or, equivalently, the constancy of the mapping A E 0. + N°O(>.J T) +(>.J T)OO(X)), with Theorem 1.4 and Theorem 1.5. 0 The previous results lead to a precise description of the operators whose Kato type spectrum akt(T) is empty. Most of the results of the following theorem may be found in Mbekhta [23] in the context of operators on Hilbert spaces. However, our proofs, involving local spectral theory, are considerably simpler and are established in the more general context of operators acting on Banach spaces. Recall first that T E L(X) is algebraic if there exists a nontrivial polynomial h such that h(T) = O. THEOREM 2.4. For an operator T E L(X) the following statements are equivalent: (i) akt(T) is empty;
PIETRO AlENA AND FERNANDO VILLAFANE
10
(ii) AI  T has finite descent for every A E C; (iii) AI  T has finite descent for every A E 8a(T), where 8a(T) is the topological boundary of a(T); (iv) a(T) is a finite set of poles of R(A, T); (v) T is algebraic.
*
PROOF. (i) (ii) Suppose that akt(T) = 0. Then Pkt(T) has an unique component n = C and therefore, by Theorem 2.2, T has SVEP at every point of C, since T has SVEP at the point ofthe resolvent p(T). On the other hand, if >.1 T is of Kato type, then also Al*  T* is of Kato type, see [4, Proposition 1). Therefore, C = Pkt(T) = Pkt(T*) and consequently, by Theorem 2.3, also T* has SVEP. Since >.1  T is of Kato type, by Theorem 1.4, we then conclude that q(>.1  T) < 00 for every A E C.
(ii)* (iii) Obvious. (iii) * (iv) Since T has SVEP at every A E 8a(T) then the condition q(>.1 T) < 00 entails that every A E 8a(T) is a pole of R(A, T), see Corollary 1 of [27), and hence an isolated point of a(T). Clearly, this implies that a(T) = 8a(T), so a(T) is a finite set of poles. (iv) (i) It suffices to prove that >.1  T is of Kato type for all A E a(T). Suppose that a(T) is a finite set of poles of R(A, T). If A E a(T), let P be the spectral projection associated with the singleton {A}. Then X = M EB N, where M := K(>.1  T) = ker P and N := Ho(>.1  T) = P(X), see the proof of Theorem 1.6 of [19) or also Theorem 1 of [27). Since A is a pole of R(A, T), by Proposition 50.2 of [15), >.1  T has positive finite ascent and descent, and if p := p(Aol  T) = q(>.1  T), then N = ker (>.1  T)P. From the classical Riesz functional calculus we know that a(T I M) = a(T) \ {A}, [15, Theorem 49.1), so that (>.1  T) I M is bijective, while (>.1  T I N)P = O. Therefore >.1  T is of Kato type for every A E C. (iv) (v) Assume that a(T) is a finite set of poles {Al,··· ,An}, where for every i = 1,· .. ,n with Pi we denote the order of Ai. Let h(A) := (AI  A)Pl ... (An  A)Pn. Then, see Lemma 3.1.15 of [18),
*
*
h(T)(X)
=
n(Ail  T)Pi (X) = nK(Ai1  T), n
n
i=l
i=l
where the last equality follows since T has SVEP and Ail  T is of Kato type, see Theorem 2.9 of [3). But the last intersection is {O}, since, by the local spectral characterization of the analytical core (1.2), if x E K(Ai1  T) n K(Ajl  T), with Ai =1= Aj, then aT(x) ~ {Ai} n {Aj} = 0 and hence x = 0, since T has SVEP. Therefore h(T) = O. (v) (i) As in the proof of (iv) (i) it suffices to show that AI  T is of Kato type for all A E a(T). Let h be a polynomial such that h(T) = O. From the spectral mapping theorem we easily deduce that a(T) is a finite set {AI,··· ,An}. The points AI,··· ,An are zeros of finite multiplicities of h, say k 1 ,··· ,kn' respectively, so that h(A) = (AI  A)kl ... (An  A)kn and hence
*
*
n
X = ker h(T) = $ker (Ail  T)ki, i=l
COMPONENTS OF RESOLVENT SETS AND LOCAL SPECTRAL THEORY
11
see Lemma 3.1.15 of [18]. Now, suppose that A = Ai for some j and define ho(A) :=
II(Ai 
A)ki.
i"lj
We have M := ker ho(T) =
EB ker (Ail 
T)ki
i"lj
and if N := ker (Ail  T)kj, then X = M EEl Nand M, N are invariant under Ail  T. From the inclusion ker (Ail  T) ~ ker (Ail  T)k j = N, we infer that the restriction of Ail  T on M is injective. It is easily seen that (Ail  T)(ker (AJ  T)ki) = ker (AJ  T)ki,
i =I j,
so that (Ai 1 T)( M) = M. Hence the restriction of Ai 1 T on M is also surjective and therefore bijective. Obviously, (Ajl  T) I N)k j = 0, so that Ajl  T is of Kato type, as desired. 0 A bounded operator on a Banach space X is said to satisfy a polynomial growth condition, if there exists a K > 0, a 8 > for which
°
IIexp(iAT)II ::; K(1
+ IAleS)
for all A E JR,
Examples of operators which satisfy a polynomial growth condition are hermitian operators on Hilbert spaces, nilpotent and projection operators, algebraic operators with real spectra, see Barnes [1]. In Laursen and Neumann [18, Theorem 1.5.19] it is shown that the class P(X) of operators which satisfy a polynomial growth condition coincides with the class of all generalized scalar operators having real spectra. As noted in Barnes [1], if T E P(X) and Aol  T has closed range for some Ao E C then q(Aol  T) is finite. From Theorem 2.4 it follows that, if T E P(X), then the condition (AI  T)(X) closed for all A E C implies that O"kt(T) = 0. Other classes of operators for which O"kt(T) = 0 may be found in [23]. The classification of the components of Pes(T) may be easily obtained from Theorem 2.2 and Theorem 2.3, once it has been observed, that the two sets Pes (T) and Pkt(T) may be different only for a denumerable set, see for instance Corollary 1 of [4]. We now look at the components of PSf(T). Recall that for a semiFredholm operator T E
°
o
PIETRO AlENA AND FERNANDO VILLAFANE
12
The eigenvalues do not have a limit point in n and every point of value. (iii) T* has SVEP at the points of n, while T fails to have SVEP at the points of n. In this case we have ind (AI  T) > 0, p(AI  T) = 00 and q(AI  T) < 00 for every A E n. The deficiency values do not have a limit point in n, while every point of n is an eigenvalue. (iv) Neither T or T* has SVEP at the points of n. In this case we have p(AI  T) = q(AI  T) = 00 for every A E n. The index may assume every value in Z; all the points of n are eigenvalues and deficiency values. for every A E
n.
n is a deficiency
PROOF. The case (i) is clear from the results established in the previous section, Theorem 2.2 and Theorem 2.3. The index ind (AI  T) = 0 by Proposition 38.6 of [15J. In the case (ii) the condition p(AI  T) < 00 implies that AI  T has index less or equal to 0, while the condition q(AI  T) = 00 excludes that ind (AI  T) = 0, see Proposition 38.5 of [15J. A similar argument shows in the case (iii) that ind (AI  T) > O. The statements of (iv) are clear. 0 The following corollary establishes that a very simple classification of the components of semiFredholm resolvent may be obtained in the case that T, or T* has SVEP. Recall that the case that both T and T* have SVEP applies in particular to the decomposable operators. COROLLARY 2.6. Let T E L(X) and
n
any component of PSf(T). If T has
SVEP then only the case (i) and (ii) of Theorem 2.5 are possible, while if T* has SVEP only the case (i) and (iii) are possible. Finally, if both T and T* have SVEP then only the case (i) is possible.
In the next result we consider the components of Pk(T), which is the smallest of the resolvent sets that we have considered. THEOREM 2.7. Let T E L(X) and n any component of Pk(T). Then one of the following possibilities occurs: (i) Both T and T* have SVEP at every point of n. In this case we have n ~ p(T). (ii) T has SVEP at the points of n, while T* fails to have SVEP at every point of n. In this case we have n n aap(T) = 0 and n ~ asu(T). (iii) T* has SVEP at the points of n, while T* fails to have SVEP at the points of n. In this case we have n n asu(T) = 0 and n ~ aap(T). (iv) Neither T or T* have SVEP at the points of n. In this case we have n ~ aap(T) n asu(T). PROOF. (i) Let Ao E n. The subspaces M:= X and N := {O} give a GKD for T and the subspaces ..L N = X and ..L M = {O} give a GKD for T*. As observed in the proof of Theorem 1.3 and Theorem 1.4 if T has SVEP at Ao then Ho(AoI  T) = N = {O} and if T* has SVEP at Ao then K(AoI  T) = M = X. Therefore Ao E p(T). (ii) In this case Ho(AI  T) = {O} and (AI  T)(X) is closed for every A E n. If A ~ asu(T) then A E p(T) = p(T*) and this is impossible, since T* does not have SVEP at A.
COMPONENTS OF RESOLVENT SETS AND LOCAL SPECTRAL THEORY
13
(iii) In this case K(M  T) = X for every A E n, so A f/. O'su(T). If A f/. O'ap(T) then A E p(T) and this is impossible, since T does not have SVEP at the point A. (iv) Use the same arguments as in part (ii) and (iii).
o References [IJ P. Aiena, o. Monsalve Operators which do not have the single valued extension property. J. Math. Anal. Appl. 250, (2000),435448. [2J P. Aiena, O. Monsalve The single valued extension property and the generalized Kato decomposition property. Acta Sci. Math. (Szeged) 67, (2001), 461477. [3J P. Aiena, M. L. Colasante, M. Gonzalez Operators which have a closed quasinilpotent part, Proc. Amer. Math. Soc. 130, (2002), 27012710. [4J P. Aiena, M. Mbekhta Characterization of some classes of operators by means of the Kato decomposition. (1996), Boll. Un. Mat. It. 10A, 60921. [5J P. Aiena, T. L. Miller, M. M. Neumann On a localized single valued extension property, (2001), to appear on Proc. Royal Irish Acad. [6J P. Aiena, E. Rosas The single valued extension property at the points of the approximate point spectrum. to appear on J. Math. Anal. Appl. [7J B. A. Barnes Operators which satisfy polynomial growth conditions. Pacific. J. Math. 138 (1989), 20919. [8J N. Dunford Spectral theory I. Resolution of the identity. Pacific J. Math. 2 (1952), 559614. [9J N. Dunford Spectral operators. Pacific J. Math. 4 (1954), 321354. [lOJ N. Dunford, J. T. Schwartz Linear operators, Part Ill. (1971), Wiley, New York. [I1J J. K. Finch The single valued extension property on a Banach space Pacific J. Math. 58 (1975), 6169. [12J K. H. Forster Uber die Invarianz eineger Riiume, die zum Operator T  >'A gehoren. Arch. Math. 17 (1966), 5664. [13J M. A. Goldman, S. N. Kraekovskii Invariance of certain subspaces associated with A  >'1. Soviet Math. Doklady 5, (1964), 1024. [14J M. A. Goldman, S. N. Krackovskii Behaviour of the space of zero elements with finitedimensional salient on the zero kernel under perturbations of the operator. Soviet Nath. Doklady 16, (1975), 3703. [15J H. Heuser Functional Analysis (1982), Marcel Dekker, New York. [16J R. H. Homer Regular extensions and the solvability of operator equations. Proc. Amer. Math. Soc. 12 (1961), 41518. [17J T. Kato Perturbation theory for nullity, deficiency and other quantities of linear operators. J. Anal. Math. 6 (1958), 261322. [18J K. B. Laursen, M. M. Neumann Introduction to local spectral theory, Clarendon Press, Oxford 2000. [19J M. Mbekhta Sur I 'unicite de la decomposition de Kato generalisee. Acta Sci. Math. (Szeged) 54 (1990), 36777. [20J M. Mbekhta Sur la theorie spectrale locale et limite des nilpotents. Proc. Amer. Math. Soc. 110 (1990), 621631. [21J M. Mbekhta, A. Ouahab Operateur sregulier dans un espace de Banach et theorie spectrale. Acta Sci. Math. (Szeged) 59 (1994), 52543. [22J M. Mbekhta, A. Ouahab Perturbation des operateurs sreguliers . Topics in operator theory, operator algebras and applications, Timisoara (1994), Rom. Acad. Bucharest, 239249. [23J M. Mbekhta Ascent, descent et spectre essential quasiFredholm., Rendiconti Circ. Mat. Palermo (2), 46, (1997), 175196. [24J V. Miiller On the regular spectrum., J. Operator Theory 31 (1994), 363380. [25J V. Rakocevic Generalized spectrum and commuting compact perturbation. Proc. Edinburgh Math. Soc. 36 (2), (1993), 197209. [26J P. Saphar Contribution a l'etude des applications lineaires dans un espace de Banach. Bull. Soc. Math. France 92 (1964), 36384. [27J C. Schmoeger On isolated points of the spectrum of a bounded operator. Proc. Amer. Math. Soc. 117 (1993), 71519.
14
PIETRO AlENA AND FERNANDO VILLAFANE
[28] C. Schmoeger (1995). SemiFredholm opemtors and local spectml theory., Demonstratio Math. 4, 9971004. [29] M. 6 Searc6id , T. T. West Continuity of the genemlized kernel and mnge for semiFredholm opemtors. Math. Proc. Camb. Phil. Soc. 105, (1989), 513522. [30] P. Vrbova On local spectml properties of opemtors in Banach spaces. Czechoslovak Math. J. 23(98) (1973a), 48392. [31] T. T. West A RieszSchauder theorem for semiFredholm opemtors. Proc. Roy. Irish. Acad. 87 A, N.2, (1987), 137146. DIPARTMENTO DI MATEMATICA ED ApPLICAZIONI, VIALE DELLE SCIENZE, UNIVERSITA DI PALERMO, 90128 PAI,ERMO, ITALY Email address:paienafDmbox.unipa.it DEPARTAMENTO DE :tViATEMATICAS, FACULTAD DE CIENCIAS, UNIVERSIDAD UCLA DE BARQUISIMETO (VENEZUELA) Email address:fvillafa«luicm.ucla.edu.ve
Contemporary Mathematics Volume 328, 2003
The FejerRiesz Inequality and the Index of the Shift John R. Akeroyd ABSTRACT. In this brief article we consider a result that can be characterized as the converse of the FejerRiesz inequality. This result has bearing on theory concerning the index of the shift.
Let I" be a finite, positive Borel measure with support in {z : Izl ::; I} (~ := {z : Izl < I}) and let p2(1") denote the closure of the polynomials in L2(1")' We assume throughout that p2(1") is irreducible (i.e., it contains no nontrivial characteristic functions). From this it follows that: a) I"lalIJl« m (normalized Lebesgue measure on 8~), and b) for any w in ~, f f+ f(w) defines a bounded linear functional for polynomials f with respect to the L2(1") norm, that is bounded independent of w in any compact subset of~; cf. [15], Theorem 5.8. In other words, ~ = abpe(p2(1"))  the collection of analytic bounded point evaluations for P2(1")' In the case that 1"(8~) > 0 and I"llIJl is radially weighted area measure, there is much in the literature concerning which weights have the property that p2(1") is irreducible; for instance, cf. [9] and [10]. Returning to our general setting, notice that multiplication by the independent variable z is a bounded operator on p2 (1"). We call this operator the shift (on p2 (1")) and denote it by M z, suppressing reference to 1". Let Lat(Mz ) denote the collection of closed invariant subspaces for the shift (on p2(1"))' If {O} "I M E Lat(Mz), then, since 0 E abpe(p2(1")), zM is a closed subspace of M and in fact dim(M e zM) 2: 1. In the case that 1"(8~) = 0, C. Apostol, H. Bercovici, C. Foias and C. Pearcy have shown that for any natural number n, and for n = 00, there exists Min Lat(Mz ) such that dim(MezM) = n; cf. [3], and for related work see [7]. This result is an indication of how very large Lat(Mz ) is in the case that 1"(8~) = O. In fact, it is large enough to "model" the general invariant subspace problem for bounded operators on a Hilbert space; again, cf. [3] and [7]. A classical example that falls under this heading (1"(8~) = 0) is the Bergman space L~(~), which equals p2(1") when I" = A  area measure on ~. At the other extreme, if I" = m, then P2(1") represents the Hardy space H2(~) and so, by Benrling's Theorem, dim(M e zM) = 1 for all nontrivial members M 1991 Mathematics Subject Classification. Primary 47 A53, 47B20, 47B38; Secondary 30ElO, 46E15. © 15
2003 American Mathematical Society
16
JOHN R. AKEROYD
of Lat(Mz ). It has been conjectured that for any measure I" with mass on the unit circle (i.e., 1"(8lI))) > 0), the outcome mimics that of Hardy space case and dim(M8 zM) = 1 whenever {OJ =I M E Lat(Mz ); cf. [5]. There are a number of results in the literature that support this conjecture. The first of these is found in a paper ofR. Olin and J. Thomson (cf. [12]) who show that it holds whenever I" that has a socalled "outer hole" in its support. Subsequently (in [11]), L. Miller shows that it also holds in the case that I" = A + mi.,!' where 'Y is some nontrivial sub arc of 8lI)). In [17], L. Yang extends this result of L. Miller to the case: I" = A + mlE, where E is any compact subset of 8lI)) of positive Lebesgue measure that satisfies the Carleson condition 1
Ln m(In)log(m(In)) <
00;
{In} are the intervals that are complementary to E in 8lI)). And then (in [16]) J. Thomson and L. Yang obtain this extension of L. Yang in the more general context of the shift on pt(I"), for 1 < t < 00. The conjecture has recently been established for any measure I" for which there is a nontrivial subarc 'Y of 8lI)) such that
1
10g ( 1;; )dm >
00,
with no special assumption made concerning 1"10; cf. [2]. In [2], the author makes use ofresults in an earlier paper (cf. [1]) that are intricately related to the seminal work of R. Olin and J. Thomson in [12]. Specifically, in [1] the author defines what it means for I" to be strongly inscribed and shows that if I" is such, then indeed dim(M 8 zM) = 1 for each nontrivial member M of Lat(Mz ). To be explicit, I" is said to be strongly inscribed if there is a Jordan subregion W of lI)) with rectifiable boundary (we let Ww denote harmonic measure on 8W for evaluation at some point in W) with the properties: i) ww(8lI))) > 0, and ii) there is a nonnegative function h in LOO(ww) such that log(h) E L1 (ww)
and
law Ifl 2 w: ; f Ifl2dl" hdw
for all polynomials
f.
Since 8W is rectifiable, (i) is equivalent to: m((8W)n(8lI)))) > O. And this definition is not truly altered if we drop the requirement that W has rectifiable boundary, because if W were any simply connected subregion of lI)) that satisfies (i) and (ii), then we could find a Jordan subregion V of W, where V has rectifiable boundary and V itself satisfies (i) and (ii); cf. [14], Proposition 6.23. It is still an open question as to whether or not our general assumptions concerning 1", along with the hypothesis that 1"(8lI))) > 0, together imply that I" is strongly inscribed. In this brief article we discuss what amounts to the converse of the FejerRiesz inequality and find that this converse has close ties to the definition of "strongly inscribed" . The FejerRiesz inequality, whose statement follows, falls under the general heading of results concerning Carleson measures. For a proof, see [6], Theorem 3.13.
THE FEJERRIESZ INEQUALITY AND THE INDEX OF THE SHIFT THEOREM 1 (FejerRiesz Inequality). If f E HP(][))) (0
ill
17
< p < 00), then
If(tei'l')lPdt ::; 4121r If(eill)IPdO
for 0 ::; 'P < 27f. The constant
!
is best possible.
With z fixed in ][)), ( f+ Pz (() := 1~=~J~2 is the Poisson kernel on 8][)) for evaluation at z. It is wellknown that falf» Pz(()dm(() = 1 independent of z, and that if hE L1(m), then (by Fatou's Theorem)
( Pz(()h(()dm(() ~ h(~) lalf» as z nontangentially approaches ~ for ma.a. ~ in 8][)). One may consult [6] and [8] as good references for these results. We begin with a rather straightforward observation whose proof appears in [4]; see the proof of Lemma 3.1 in this reference. LEMMA 2. Let rJ be a finite, positive Borel measure with support in ll} such that rJ(8][))) = O. Then lim r+1
for ma.a.
~
1  r2
1m
If» 11r~wI2
drJ(w) = 0
in 8][)).
Our next result can be viewed as the converse of the FejerRiesz inequality. THEOREM 3. Let v be a finite, positive Borel measure with support in ll} such that vlalf» « m and][)) = abpe(p2(v)). For c > 0, let Bv(c) be the set of all ~ in 8][)) such that
11 If(t~Wdt J ::; c·
Ifl2dv
for all polynomials f. Then Bv (c) is a closed subset of 8][)) and :~ ;::: ~ (a. e. m) on Bv(c). Furthermore, if E is a Lebesgue measurable subset of Bv(c) and m(E) > 0, then XE ¢ p 2(v). Proof. That Bv(c) is closed is an immediate consequence of the fact that any polynomial is uniformly continuous on ll}. Now if 0 < r < 1 and I~I = 1, then
v'f=T2
g(w) := =1r~w
is analytic in a region containing ll} and so, by Runge's Theorem, is the uniform limit (on ll}) of polynomials. Therefore we can apply our hypothesis to get that
11 Ig(t~)12dt J ::; c·
which yields:
1 1
l+r=
o
1  r2
(1
 rt
)2 dt ::;c.
Igl 2dv,
JI
1  r2
1 r~wl
2 dv (W),
for 0 < r < 1 and any ~ in Bp.(c). Letting r + 1 and applying Lemma 2, we find that :~ ;::: ~ (a.e. m) on Bv(c). To finish the proof of this theorem, let E be a Lebesgue measurable subset of Bv(c) such that m(E) > 0, and suppose
JOHN R. AKEROYD
18
that XE E P2(v); we look for a contradiction. Now h := (1  XE) E p 2(v) (since XE E p2(v)), and in fact, hg E P2(v) whenever 0 < r < 1 and ~ E E. Arguing as before, with hg now in the place of g, we obtain: ~ ~ h· :::. = 0 a.e. rn on E clearly a contradiction.D It turns out that if f.L is strongly inscribed, then in fact there exists c > 0 such that rn(BJ.L(c)) > o. En route to this result (Theorem 5, below), we make the following observation. PROPOSITION 4. Let f.L be a finite, positive Borel measure with support in iID such that p2 (f.L) is irreducible. Then the following are equivalent. 1) f.L is strongly inscribed. 2) There is a Jordan subregion V ofH}, where 8V is rectifiable and rn((8V) n (8H})) > 0, and there is a positive constant M, such that
lav Ifl 2dw v
~ M·
f
Ifl2df.L
for all polynomials f. Proof. We first assume (1), and so by definition there is a Jordan subregion W of H} and a nonnegative function h in LOO(ww) that satisfy certain requirements. One of these requirements, namely that log( h) EL I (ww ), guarantees the existence of a bounded analytic function g in W such that g 0 'P is an outer function ('P is a conformal mapping from H) onto W) and Igl has "boundary values" equal to h (a.e. ww). Applying Proposition 2.2 of [1], we can find a Jordan subregion V of W, where V has rectifiable boundary, wv(8H}) > 0 and Igl ~ e > 0 on V. So by the subharmonicity of Ifl 21g1 in W, (2) holds, with M := ~. That (2) implies (1) is immediate, and our proof is complete.D THEOREM 5. Let f.L be a finite, positive Borel measure with support in iID such that p2 (f.L) is irreducible. If f.L is strongly inscribed, then there exists c > 0 such that rn(BJ.L(c)) > o.
Proof. Assuming that f.L is strongly inscribed, Proposition 4 provides a Jordan subregion V of H} with the properties listed in (2). Let 'P be a conformal mapping from H} onetoone and onto V, and let 1/J = 'P 1. Since Wv (8H}) > 0, we can find (cf. [14], Theorem 6.8 and Theorem 3.7) a closed subset E of (8V) n (8H}), where each point in E is a point of tangency of 8V with 8H}, such that:
i) rn(E) > 0, ii) for any ~ in E and any Stolz angle ~ whose closure is contained in VU{O, there is a constant M > 1 such that
~ ~ 11/J'(z)l, 11/J(Z; =t~) I ~ M for all Z in ~, and iii) if ~ E E and 'Y is a smooth arc in V U {O having nontangential approach in V to ~, then 1/J("() is smooth and has nontangential approach in H} to 1/J(~).
Now choose ~ in E. Since ~ is a point of tangency of 8V with 8H}, there exists s, o < s < 1, such that t~ E V whenever s ~ t < 1. Let'Y be the smooth curve in
THE FEJERRIESZ INEQUALITY AND THE INDEX OF THE SHIFT ID>U{'l/J(~)}
constants
19
defined by ')'(t) = 'l/J(t~), s::; t::; 1. Then, by (i)  (iii), there are positive (k = 1,2,3) independent of to in [s, 1) such that
Ck
lengthb([to, l])) 1  b(to)1
It: W(t~)ldt
=
1 1'l/J(to~)1 I~
<
C1 .
 to~1
1  I'l/J(to~) I
I~  to~1 C2· I'l/J(O  'l/J(to~)1 < C3·
<
From this it follows that arclength measure on ')'([s, 1)) is a Carleson measure for H2(1D». And so there are positive constants Ck (k = 3,4) such that, for any polynomial f,
11 If(t~)12dt
=
llU 0 cp)(wWlcp'(w)ldlwl
<
C3
·llU 0 cp)(wWdlwl
< C4· [ IU 0 cpWdm
laI)
C4· [
lav
IfI 2ru.vv;
by Harnack's inequality, we may assume that cp(O) is the point in V of evaluation for wv. Once again recalling (2) (of Proposition 4), we can now find a positive constant C5 such that
11 If(t~Wdt
::;
C5 •
J
Ifl2djj
for all polynomials f. Since ID> = abpe(p2(jj)), we may apply Lemma 2.6 of [13] and find another positive constant C6 such that, for all polynomials f,
1If(t~Wdt J s
::;
C6 •
Ifl 2djj.
Consequently, ~ E B/L(c) , for C := C5 + C6. Thus we have shown that E ~ U~=1 B/L(n). Since m(E) > 0, we can assert that m(B/L(n)) > 0 for some integer n, which completes the proof.D QUESTION 6. Does the converse of Theorem 5 hold? That is, if jj is a finite, positive Borel measure with support in iij such that p 2 (jj) is irreducible, and if m(B/L(c)) > 0 for some positive constant c, then is jj strongly inscribed? Indeed, can we even assert that dim(M 8 zM) = 1 for each nontrivial, closed invariant subspace M for the shift on p2(jj)? We conclude this article with a rather anemic response to Question 6 that supports an affirmative answer.
20
JOHN R. AKEROYD
REMARK 7. There are other more general forms of the FejerRiesz inequality, where the integral on the left is taken over chords of the unit circle and not just over diameters. A converse to this FejerRiesz inequality (for chords) can be established, and involves integrals over segments that have nontangential approach in lI} to certain points in alI}. Thus, analogues of B/I(c) can be defined, where the integral on the left is taken over segments in various Stolz angles. All of this leads to a counterpart of Theorem 5, whose converse appears to be manageable. To this author it seems most likely that if m(B/I(c)) > 0, then there is a sizeable subset E of B/I(c) that is contained in these collections that are analogous to B/I(c), and thus the converse of Theorem 5 is likely a consequence of its counterpart in the context of the FejerRiesz inequality for chords. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
[10] [11] [12] [13] [14] [15] [16] [17]
J. Akeroyd, Another look at some index theorems for the shift, Indiana Univ. Math. J., 50 (2001),705718. J. Akeroyd, A note concerning the index of the shift, Proc. Amer. Math. Soc., Vol. 130, No. 11 (2002), 33493354. C. Apostol, H. Bercovici, C. Foias, C. Pearcy, Invariant subspaces, dilation theory, and the structure of the predual of a dual algebra, I, J. Functional Analysis, 63 (1985),369404. A. Aleman, S. Richter, C. Sundberg, The majorization function and the index of invariant subspaces in the Bergman spaces, J. Analyse Math., 86 (2002), 139182. J. B. Conway, L. Yang, Some open problems in the theory of subnormal operators, Holomorphic spaces, Cambridge University Press, 33 (1998), 201209. P. L. Duren, Theory of HP Spaces, Academic Press, New York, 1970. H. Hedenmalm, S. Richter, K. Seip, Interpolating sequences and invariant subspaces of given index in the Bergman spaces, J. Reine Angew. Math., 477 (1996), 1330. K. Hoffman, Banach Spaces of Analytic Functions, PrenticeHall, Englewood Cliffs, N.J., 1962. S. Hruscev, The problem of simultaneous approximation and removal of singularities of Cauchy type integrals, Trudy Mat. Inst. Steklov 130 (1978), 124195; English transl., Proc. Steklov Inst. Math. 130 (1979), no. 4, 133203. T. L. Kriete, B. D. MacCluer, Meansquare approximation by polynomials on the unit disk, Trans. Amer. Math. Soc., vol. 322, no. 1 (1990), 134. T. L. Miller, Some subnormal operators not in A2, J. Functional Analysis, 82 (1989), 296302. R. F. Olin, J. E. Thomson, Some index theorems for subnormal operators, J. Operator Theory, 3 (1980), 115142. R. F. Olin, L. Yang, A subnormal operator and its dual, Canad. J. Math., 48 (1996), 381396. Ch. Pommerenke, Boundary Behaviour of Conformal Maps, SpringerVerlag, BerlinHeidelberg, 1992. J. E. Thomson, Approximation in the mean by polynomials, Ann. Math., 133 (1991), 477507. J. E. Thomson, L. Yang, Invariant subspaces with the codimension one property in Lt(J.I), Indiana Univ. Math. J., vol. 44, no. 4 (1995),11631173. L. Yang, Invariant subspaces of the Bergman space and some subnormal operators in Al\A2, Mich. Math. J., 42 (1995), 301310. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ARKANSAS, FAYETTEVILLE, ARKANSAS
Email address: jakeroydlDcomp.uark.edu
72701
Contemporary Mathematics Volume 328, 2003
A CauchyGreen Formula on the Unit Sphere in C 2 John T. Anderson and John Wermer In 1977 G. Henkin introduced an integral formula for solving where tL is a measure, on the boundary of a smooth strictly convex domain. This result is closely related to a "CauchyGreen" formula on the sphere (see Chen and Shaw [3]). We give a direct elementary proof of the CauchyGreen Theorem on the unit sphere and derive Henkin's solution of the 8b equation from this. We also give an application to an approximation result. ABSTRACT.
8bf
= tL
1. Introduction
Let n be a domain in the plane, with smooth boundary r. CauchyGreen formula states that for any c/> E C 1 (0') and zEn, (1.1 )
c/>( z) = _1 27l'i
r c/>( () d( _ _ lnr o~ d( 1
lr (  Z
27l'i
o(
1\
(
The classical
d( Z
Note that the first term on the right of (1.1) is a holomorphic function 4> of Z in the domain n. In fact, 4> extends continuously to 0', and hence defines an element of the algebra A(O') consisting of functions holomorphic in n and continuous on 0'. Of course, if c/> E A(O'), (1.1) reduces to the Cauchy integral formula and 4> = c/>. The representation (1.1) has many applications in complex analysis. In the theory of approximation of continuous functions on a compact set K c C by rational functions with poles off K, one is led by considerations of duality to examine measures supported on K. The Cauchy transform of such a measure J.l is defined by
p,(Z) =
r
dJ.l(() lK (z The integral defining [J, converges absolutely for almost all z E C. Using (1.1), one can easily show that for any smooth compactly supported function C/>, (1.2)
r
r
c/>(z) dJ.l(z) = ~ °o~ p,(z) di 1\ dz lK 27l'zlc z That is, [J, satisfies the equation (1.3)
(1.4)
0[J,
oi =7l'J.l
1991 Mathematics Subject Classification. Primary 32A25, Secondary 32E30.
© 21
2003 American Mathematical Society
22
JOHN T. ANDERSON AND JOHN WERMER
in the sense of distributions, and hence defines a holomorphic function on e \ K. The Cauchy transform is a key tool in rational approximation theory in the plane. We have been motivated by problems of rational approximation for subsets of the boundary S of the unit ball in e 2 . It is posible to do a kind of function theory on S analogous to the theory of analytic functions in the plane. The operator a / az is replaced by the tangential CauchyRiemann operator
x
(1.5)
a
= Z2
aZl

a
Zl.
aZ2
x is welldefined 011 C l (S) and for any relatively open subset n of S, annihilates the restrictions to n of functions holomorphic in a neighborhood of n in e 2 • The solutions to X ¢ = 0 on n are known as CR functions on n. A good general reference for the theory of CR functions is the book [2]. One would like an analogue of the Cauchy transform for measures on S. Given a measure JL on S, G. Henkin in 1977 [4] constructed a function KJ1.' summable with respect to threedimensional Hausdorff measure da on S, satisfying 
(1.6)
2
abKJ1. = 27r JL
in the sense of distributions, i.e.,
(1.8)
[ ¢(z) dJL(z) =
~
[ KJ1. X¢ da(z) 27r for all smooth ¢, provided that JL satisfies the necessary condition that P dJL = 0 for all polynomials P. Note that (1.7) implies that KJ1. is a CR function (in the sense of distributions) off the support of JL. In attempting to use and understand Henkin's construction in the study of rational approximation on subsets of S, we were led to the analogue of the CauchyGreen formula (1.1) that we present below. It plays the same role with respect to Henkin's formula (1.6) as the classical CauchyGreen formula on the plane does to equation (1.4). The resulting formula, which is contained in our Theorems 2.1 and 3.1 below, is not new. It is given in a more general setting in Chen and Shaw ([3], see the remarks following Corollary 11.3.5) as a consequence of the theory of Henkin for solving the 8b equation on the boundary of a strictly convex domain in en. Our approach to establishing this CauchyGreen formula on the sphere in e 2 is direct and elementary, and leads immediately to the property (1.6) of Henkin's transform K J1.' Let A(B) denote the algebra of functions holomorphic in the open unit ball B of e 2 and continuous on its closure. We seek a kernel H((, z), defined for ((, z) E S x S, such that for all ¢ E Cl(S), there exists E A(B) with
(1.7)
is
¢(z)
= (z) + c
Is
is
Is
H((, z) 8¢(() 1\ w(()
for all z E S, where w(() = d(l 1\ d(2, 8¢ = (a¢/azddz 1 + (a¢/az2)dz2, and c is a universal constant. We call (1.8) a "CauchyGreen formula for S". We will demand that H have the following properties:
A CAUCHYGREEN FORMULA ON THE UNIT SPHERE IN C 2
23
a: H((, z) is continuous on S x S \ {z = 0; b: For all unitary transformations U of determinant 1, H (U (, U z) = H ((, z); c: jH((, el)j da(() < 00, where el = (1,0), and da is threedimensional Hausdorff measure 1 on S. Properties (b) and (c) together with the unitary invariance of da imply that H is uniformly summable with respect to da, Le., there exists a constant C so that
Is
is
(1.9)
jH((, z)j da(() ::; C, Vz
They also imply that the integral
K(z) ==
(1.10)
is
ES
H((, z) a¢(() /\ w(()
appearing in (1.8) is finite for all z E S, since a¢ /\ w is absolutely continuous with respect to da. A routine calculation gives
a¢ /\ w = 2(X¢) da
(1.11)
on S, where X is the operator in (1.5), for smooth ¢. We can say more about K: LEMMA
1.1. If H satisfies properties (a), (b) and (c), then K is continuous on
S. PROOF. Fix z E S. For f > 0, put S«z) = S \ {jz  (j ::; f} and S~ S n {jz  (j ::; fl. Let
K«z)
=
r
=
H((, z) a¢(() /\ w(()
ls.(z)
Then K< is continuous on S, by property (a) of H. For all z E S, by (1.11),
I
jK(z)  K,(z)j =
r
ls~(z)
r
H((, z) a¢(() /\ W(()I ::; M
jH((, z)jda(()
ls~(z) Let el = (1,0) and choose a
where M is a constant independent of z and f. unitary transformation U of C 2 with Uel = Z; then U(S~(el)) = S~(z). Then using property (b),
r
ls~(z) Since
jH((, z)j da(()
=
r
lS~(e,)
jH(Ury, Uel)j da(Ury)
=
r
jH(ry, el)j da(ry)
lS:(e')
Is jH(ry, edjda(ry) is finite by assumption (c), lim <+0
It follows that K,
t
1rS~(e,) jH(ry, el)jda(ry) = 0
K uniformly on S, and so K is continuous, as claimed.
We say that a measure /L on S is orthogonal to polynomials if
(1.12)
is
Pd/L = 0, V holomorphic polynomials P
Given any measure /L on S, define
(1.13)
K Il (() =
Ida is not normalized; oo(S) =
is
21T2.
H((, z) d/L(z), (E S
JOHN T. ANDERSON AND JOHN WERMER
24 LEMMA
only
1.2. A kernel H((, z) satisfying (a), (b) and (c) satisfies (1.8) if and
if for each measure J.L on 8 orthogonal to polynomials
Is
(1.14) for all ¢
E
¢ dJ.L = cis K,.. 8¢ A w
C1(8).
PROOF. Suppose first that H((, z) satisfies (a), (b), (c) and (1.8). Let J.L be a measure on 8 orthogonal to polynomials. Fix ¢ E C 1 (8). and let
Is
Is
¢(z)dJ.L(z)
=
Is (c Is Is (c Is Is
c
H((,z) 8¢(() AW(()) dJ.L(z) H((, Z)dJ.L(Z)) 8¢(() A w(()
K,..(() 8¢(() A w(()
so that (1.14) holds. The application of Fubini's theorem is justified by (1.9). Next, suppose that (1.14) holds, for H satisfying (a), (b) and (c). Choose a measure J.L on 8 orthogonal to polynomials. Fix a function ¢ E C 1 (8), and define
Is
H((, z) 8¢(() 1\ w(()
By Lemma 1.1,
Is
Is Is
¢(Z)dJL(Z)  c
Is (Is
H((, Z)dJ.L(Z)) 8¢ A w(()
¢(z)dJ.L(z)  cis K,..(() 8¢(() A w(()
o by (1.14). Since this holds for all J.L orthogonal to polynomials,
( 1.15)
H(( z) =
,

(1 22  (2 21
(, Z
E8
11
Zl(l + Z2(2, and proved the formula (1.14) using this kernel. It is easy to check that H satisfies properties (a), (b) and (c) above. Formula (1.14) on 8 is actually very special case of a class of general integral formulae on smooth convex domains established in [4]. In her thesis [5], H.P. Lee gave an elementary proof of Henkin's formula for 8j the paper [8] of Varopoulous also contains an exposition of Henkin's results on the sphere. For applications of Henkin's formula to rational approximation, see the paper [6] of Lee and Wermer.
In this paper, we shall (1) give a direct proof of (1.8), using Henkin's kernel (1.15)j (2) give a formula for
25
A CAUCHYGREEN FORMULA ON THE UNIT SPHERE IN C 2
1.1. Acknowledgment. The first author wishes to thank Joseph Cima for helpful conversations on the results in section 3.
2. A CauchyGreen Formula using Henkin's Kernel With H as in (1.15) and ¢ E C 1 (S) as in section 1 put
K(z) = For a E int(6), put r = the z2plane. LEMMA
2.1. Fix a
E
Is
H((, z) 8¢(() 1\ w(()
Jl lal 2 and denote by "fa the circle Z2 =
rT, ITI = 1 in
6. For n = 0,1,2, ... we have, putting z = (a, Z2),
(2.1) PROOF.
We denote the inner integral by J((). Multiplying both numerator and denominator of the integrand by T, we get
Let and Note that
T1 T2
r(2
T2
=
1 a1 (
= 1. We have
Ir(21 2 11  a(11 2
= =
(1laI 2)(11(112) 11  a(11 2 1  lal 2  1(11 2 + la1 211(112  1  la1 21(112 + a(l (laI 2 + 1(11 2  a(l  a(t)
Ia 
(11 2
+ a(l
26
JOHN T. ANDERSON AND JOHN WERMER
Thus
(2.2) Let Se be the part of S lying over the region
al
{I(I 
~
f} n {I(II ::; I}
in the (Iplane. Let TE denote the boundary of SE' We claim that
(2.3)
la 1
K(Z)Z2 dZ 2 = 
!~~ [l,4>(()I(()W(()]
To establish the claim, note that
K(Z)Z2 dZ 2
'Ya
lim [ 84> /\ w . I <+0
=
lim [ d(4) wI) e+O
since I is holomorphic on Se for equals
f
Js,
Js,
> O. By Stokes' Theorem, the latter integral  [ 4> wI
JT,
proving the claim. Note that T, is the torus (1 = a + fe ifJ , (2 = JI 1(II 2ei .p, 0::; 0, 'IjJ ::; 27l'.
A CAUCHYGREEN FORMULA ON THE UNIT SPHERE IN C 2
27
On T< we have the following relations:
¢(() = ¢(a, rei1/J) + O(f);
d(1 = ife i9 dB, d(~ = ife i9 dB; d(2 = (1 d(;.  (1 d(1 ei1/J + iJI 1(11 2ei 1/Jd'lj; = i re i1/Jd'lj; + O(f); 2JII(11 2 1 1 (1  a fe i9 · Using this information together with (2.2) and (2.3) we obtain
1K(z)z~ "fa
For fixed
f,
= lim [27ri [ ¢(()r2n+2 (
Jr,
<>0
1
a(~1 )
+1 (( 1
n
1 
a) d(11\. d(2]
we rewrite the expression in brackets as 27ri [ ¢(a, rei1/J)rnein1/JidB I\. irei'IjJd'lj; + O(f)
Jr.
1K(z)z~
dZ2 =
"fa
This completes the proof of (2.1) and Lemma 2.1.
o
Next, we define an operator T on C 1 (8) as follows:
(2.4)
for z E 8, ¢ E C 1(8)
(T¢)(z) = 47r 2 ¢(z)  K(z),
Letting X denote the tangential CauchyRiemann operator on 8 as in section 1, using (1.11) we can write
T¢ = 47r 2 ¢ 
is
H((, z) (X¢)(()dcr(()
LEMMA 2.2. Fix ¢ E C 1 (8). Let L be a complex line in C2. Then the restriction ofT(¢) to L n 8 extends analytically to L n B. PROOF.
(2.5)
Lemma 2.1 gives us, for each a
1(T¢)(z)z~
E
int(h.), that
dZ 2 = 0, n = 0,1,2, ...
"fa
Note that "fa = La n 8, where La is the line {Z1 = a}. Then (2.5) implies that T¢ extends analytically to the disk La n B. Using the unitary invariance of H,cr, and X, it is not hard to check that for all ¢ E C 1 (8),
(2.6)
(T¢)oU=T(¢oU)
Fix a complex line L. Let N denote the complex line passing through the origin which is orthogonal to L, and let zO denote the intersection point N n L. Write L = {zO + (t I t E C} for some unit vector (. If U is a unitary transformation with Ue2 = (, where e2 = (0,1) then U maps the line {Z2 = O} to N, and maps some
JOHN
28
T.
ANDERSON AND JOHN WERMER
point (a,O) to zOo Then U((a,O) + t(O, 1)) = zO + t(, for all t E C. So U maps the line La to L and maps the disk La n B to L n B. By (2.6), T¢ I Lns extends analytically to the disk L n B if and only if (T¢) 0 U 1La ns extends to La n B. This last is true by (2.5), as we have noted earlier, and so the proof is complete. 0 By Lemma 1.1, since H satisfies properties (a), (b) and (c) of section 1, K and thus T¢ are continuous on S. By Lemma 2.2, T¢ has the "onedimensional extension property" as defined by Stout in [7], p. 105. A theorem of Agranovskii and Val'skii [1] then gives that T¢ lies in the ball algebra A(B). Putting cP = T(¢), we have arrived at THEOREM 2.3. Let ¢ E C 1 (S). Then there exists cP E A(B) such that 47r 2 ¢(z) = cP(z)
+
is
H((, z) 8¢(() /\ w(()
where H is Henkin's kernel
3. The CauchyGreen formula and the Cauchy transform In this section we identify the ball algebra function cP appearing in Theorem 2.3 as a certain principal value of the Cauchy transform of ¢. The Cauchy kernel for B is 1 C(z,() = (1 < z,( »2 For z E S we set N. (z) = {( E S : I < (, z > I > 1  f} and we denote the boundary of N. (z) by r f (z). THEOREM 3.1. Fix ¢ E C 1 (S). If cP is as in Theorem 2.3, then for z E S,
cP(z) = 2 lim
f
¢(()C(z, () du(()
.+0 JS\N.(z)
REMARK 3.2. Since C(z,·) Theorem 3.1 exists.
rt. Ll(du),
it is not immediate that the limit in
PROOF. As in sections 1 and 2, set
f H((, z) 8¢(() /\ w(() = JS
K(z) = For
f
f+O
f H((, z) 8¢(() /\ w(() JS\N, (z)
f
d[H((, z)¢(() /\ w(()]
lim
> 0 fixed,
f
H((, z) 8¢(() /\ w(()
=
JS\N,(z)
JS\N.(z)
 f
[8H((, z)]/\ ¢(() /\ w(()
JS\N.(z)
=
f
H((, z) 8¢(() /\ w(()
Jr,(z)
2
f JS\N.(z)
(XH)((,z) ¢(() du(()
A CAUCHYGREEN FORMULA ON THE UNIT SPHERE IN C 2
29
by Stokes' theorem, if r e(z) is oriented as the boundary of S \ Ne (z). We have also used equation (1.11) from section 1. A computation shows (differentiation is in the ( variable) (XH)((, z) = C(z, () so that
K(z) = lim [ €+O
r
Jr,(z)
H((, z) >(()
1\
w(()  2
r
JS\N,(z)
C((, z) >(()dO"(()]
Since
cI>(z) = 41r 2>(z)  K(z) by Theorem 2.3, the proof will be complete if we can show that lim
(3.1)
€+O
r
Jr,(z)
H((, z) >(()
1\ w(() =
To establish (3.1), choose a unitary map U with Ue1
r
Jr,(z)
H((, z) >(()
The torus r e(e1) parametrized by
1\ w(() =
r
Jr,(et}
41r2>(z)
= z. Then for fixed E > 0,
H(TJ, ed (> 0 U)(TJ)
1\ w(TJ)
= {TJ : ITJ11 = 1  E}, oriented as the boundary of S \ Ne(ed, is
where Then on re(ed,
and
which gives
r
Jr,(z)
H((, z) >(() 1\ w(()
(3.2) where
r r 27r
27r
IIel ::; C Jo Jo
r2
11  (1 .: €)ei9112d(hd{;l2
for some C > o. An application of the Poisson integral formula shows that the first integral in (3.2) converges to 41r2(>oU)(e1) = 41r2>(z) as E + 0, while lime+o Ie = O. This completes the proof. D
30
JOHN
T.
ANDERSON AND JOHN WERMER
4. An Approximation Theorem Fix ¢ E C 1 (S). The quantity dist(¢, A(B)) = inf{ll¢  gil : g E A(B)} where I . II is the uniform norm on S measures how closely ¢ can be approximated by polynomials on S. THEOREM 4.1. There exists C > 0 so that/or all ¢ E C1(S),
dist(¢, A(B)) ~
CIIX¢II
PROOF. Let IIHl11 denote the L1  drr norm of Henkin's kernel H(·, z) (which is independent of z E S). By the representation in Theorem 2.3, there exists E A(B) so that for z E S, 14rr2¢(z)  (z) I
lis
H«(, z) 8¢«() Aw(OI
211s H«(, Z)(X¢)(Odrr(ol < 211 H lh11X¢11
o
from which the result follows.
References [1] M.L. Agranovskii and R.E. Val'skii, Maximality of Invariant Algebras of Functions, Siberian Math. J. 33 (1983),p. 227250. [2] A. Boggess, CR Manifolds and the Tangential CauchyRiemann Complex, CRC Press, 1991. [3] S.C. Chen and M.C. Shaw, Partial Differential Equations in Several Complex Variables, American Mathematical Society, 2001 [4] G. M. Henkin, The Lewy Equation and Analysis on Pseudoconvex Manifolds, Russian Math. Surveys, 32:3 (1977); Uspehi Mat. Nauk 32:3 (1977),p. 57118 [5] H. P. Lee, Orthogonal Measures for Subsets of the Boundary of the Ball in C 2 , Thesis, Brown University, 1979. [6] H. P. Lee and J. Wermer, Orthogonal Measures for Subsets of the Boundary of the Ball in C2, in Recent Developments in Several Complex Variables, Princeton University Press, 1981, pp. 277289. [7] E.L. Stout, The Boundary Values of Holomorphic Functions of Several Complex Variables, Duke Math. J. 44, 1977,p. 105108. [8] N. Th. Varopoulos, BMO functions and the aequation, Pac. J. Math. 71, no. 1 (1977). pp. 221273. DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, COLLEGE OF THE HOLY CROSS, WORCESTER, MA 016102395 Email address: andersonOradius.holycross .edu DEPARTMENT OF MATHEMATICS, BROWN UNIVERSITY, PROVIDENCE, RI 02912 Email address: wermerlDmath. brown. edu
Contemporary Mathematics Volume 328, 2003
On adual Algebras Hugo Arizmendi, Angel Carrillo, and Lourdes Palacios ABSTRACT. Let Ti(D) and TieD) be the algebras which consist of all holomorphic functions in the open unit disc D and in the closed unit disc 75, respectively. These algebras, considered as algebras of sequences, are denoted by A and B. Let A" and B" be the adual spaces of A and B, respectively; we have that A" = Band B" = A, and these sequence spaces with their normal topologies are topological algebras. A similar treatment can be applied to the algebra c of all entire functions and its adual space c" consisting of all complex functions that are analytic in some neighborhood of the origin. Here we examine a more general situation. If A(ap,n) is a matrix algebra, we establish conditions under which the adual space A" (ap,n) is a topological algebra relative to the normal topology. We also analyse some other important examples of topological algebras with such properties.
1. Introduction
A sequence space), is a vector space of complex sequences x = (Xi)~O' The vector space operations are the usual operations on the coordinates. ), can be considered as a linear subspace of the space w of all complex sequences. To each sequence space ), we assign another sequence space N", its adual. ),01. is the set of all complex sequences Y = (Yi)~O for which the scalar product <Xl
yx =
I: XiYi
converges absolutely, for each x E ),.
i=O
The normal topology on ), is the topology determined by all the seminorms defined by <Xl
Ilxlly =
2: IXkYkl
k=O where Y runs over all ),01.. An important class of sequence spaces are the echelon and coechelon spaces which have been studied by G. Kothe and O. Toeplitz. They are defined as follows: Let (ap,n), p = 1,2, ... , n = 0,1, ... be an infinite matrix of nonnegative numbers such that (1) 0:::; ap,n :::; ap+1,n (2) For every n, there exists p such that ap,n > O. 2000 Mathematics Subject Classification. Primary 46; Secondary 30. Key words and phmses. Topological Algebras, Normal Topology, Matrix Algebra.
© 31
2003 American Mathematical Society
32
HUGO ARIZMENDI, ANGEL CARRILLO, AND LOURDES PALACIOS
Let A = {(Xn) E
W :
E JXnJ ap,n < 00, p = 1,2, ... }.
A is called an echelon space
n
and N" a coechelon space. We note that A = fu!!ll(ap,n)' p
In the following we denote the echelon space A by A(ap,n)' In [1] it is proved that if A(ap,n) is an algebra under the convolution product, then this product is jointly continuous and A(ap,n) is a topological algebra. As a matter of fact, it is a metrizable locally convex complete algebra, i.e. aBoalgebra. It can be seen that a necessary and sufficient condition under which A(ap,n) is an algebra is the following: (3) For each pEN, there exists q E N such that ap,n+m S aq,n aq,m; for n, m = 0, 1,2, .... If (ap,n) satisfies (1), (2) and (3), we call A(ap,n) a matrix algebra. Let 00
A = {(an)~=o: if
JzJ < 1,
then L anz n converges in C} n=O
and 00
B = {(bn)~=o: there exists a z
E
C, JzJ > 1, and L bnzn converges in C}. n=O
These sequence spaces A and B are called analytic sequence spaces and they are algebras under the usual linear operations and the convolution product. The transformation 00
(Xk)k=O ~ L Xk zk k=O identifies those sequence algebras A and B with the function algebras H(D) and H(D) consisting of all holomorphic functions in the open unit complex disc D and in the closed unit complex disc D, respectively. Through this identification we shall 00
indistinctly write x = (Xk)k=O or x =
E
Xkzk to refer to an element x of A or B. k=O By the same fact we can consider in A the compactopen topology that is originally defined for H(D). We note that A ~ A(r;), n = 0,1, ... , p = 0,1, ... , where (rp) is an increasing sequence of positive numbers converging to 1, and the compactopen topology on A can be given by the sequence of seminorms 00
JJxJJp = L JXnJr;, n=O
00
where x
= LXnZnEA. n=O
In [5], O. Toeplitz studied the topological properties of the analytic sequences spaces A and B and proved that AO: = Band BO: = A. A similar treatment can be applied to the algebra c of all entire functions and its adual space cO: consisting of all complex functions that are analytic in some neighborhood of the origin. In [2] it is proved that A and B are topological algebras when they are endowed with the normal topology given by A and B, respectively. The same happens with the algebra of all entire functions c and its adual space cO:.
ON ",DUAL ALGEBRAS
If ap n > 0, n ,
33
= 0,1,2, ... , p = 1,2, .... then A"'(ap n) = limlOO(1) as sets. ap,n ,
t
p
The inductive limit topology is stronger than the normal topology in A"'(ap,n)' Prom now on we are going to assume: ap,n > 0, n = 0,1,2, ... ,p = 1,2, .... If A(ap,n) is nuclear, then A"'(ap,n) ~:::'n~lOO(a:.J ~ l~l1(a:.J as topological p
p
vector spaces. Here we prove that l~l1(a:.J is an algebra if A(ap,n) satisfies: p
(*) for each p there exist q > p and Mp such that ap,n ap,m :::; Mpaq,n+m (or equivalentely _ 1 _ :::; Mp_1___1_) for all n,m. aq,n+m
a p . n ap,rn
And then the convolution product is jointly continuous. We also prove that if (ap,n) does not satisfy (*), then l~l1(a:.J is not an algebra. Therefore l~l1(a:.J p
p
is an algebra if, and only if, it is a topological algebra. Thus, if A(ap,n) is nuclear, then A"'(ap,n) is an algebra if, and only if, it satisfies (*). Therefore A(ap,n) and A"'(ap,n) are topological algebras under the normal topology. This is a generalization of the properties of (A, B) and (c, c"'). We also study some other important examples of topological algebras with such properties.
2. Definitions and Notation We recall some relevant definitions. Through this section we assume that X is a commutative complex topological algebra with unit element. X is called a locally convex algebra if it is also a locally convex space. In this case its topology can be given by means of a family (11.11",)"'E;l of seminorms such that for each index Q E ~, there is an index f3 E ~ such that (2.1) for all x, y EX. If relation (2.1) can be replaced by (2.2) for all x, y EX, then we say that X is locally multiplicatively convex (shortly mconvex) algebra. X is called a Boalgebra if it is a complete metrizable locally convex algebra. In this case its topology can be given by means of a sequence (1I.lIn)~=1 of seminorms satisfying
for n = 1,2, ... and for all x,y E X. Let (ay,k), "I E r, k = 0,1, ... , be an infinite matrix of positive real numbers. Assume that for each "I E r there is a "I' E r such that
(2.3) for all k, l = 0, 1, ....
ay,k+l :::; ay',k ay',l
34
HUGO ARIZMENDI, ANGEL CARRILLO, AND LOURDES PALACIOS
The matrix algebra A(a/"k) associated with the matrix (a/"k) is the algebra of 00
all formal complex power series x =
L
Xkzk such that, for each "I E
k=O
r,
00
L a/"k IXkl <
IIxli/' =
00.
k=O By (2.3), A(a/"k) is a complete locally convex algebra under the usual linear operations and the convolution product. If r = N, then A(a/"k) is a Boalgebra. If "I = "I' in (2.3), then A(a/"k) is an mconvex algebra. We shall be mainly interested in a Bo matrix algebra A(ap,n)' We note that 00
A(>O'(ap,n)
= {(zn)~=o
1
L IZnYnl <
00,
V Y = (Yn)~=o in AO(ap,n)}'
n=O We have the following: REMARK
2.1. AOO(ap,n) = A(ap,n)'
For each p, the row (ap,n) is an element of AO(ap,n)' Therefore, if
PROOF.
00
Z E AOO(ap,n), then
L
n=O
IZnap,nl <
00
o
and hence Z E A(ap,n)'
3. A Matrix Algebra and its adual 3.1. AO(apn ) = limloo(I) , ; ap,n
1
PROPOSITION
PROOF.
M
Let Y
= {(Yn);:"=o
liml oo
E
;
(_1_);
1 13 p: sup IYn< oo} as sets. n Qp,n
then sup IYn
ap,n
n
_1_1
<
00
for some p.
If
ap,n
= sup IYnII, we have that IYn 1 :s M ap,n for each n. Therefore, for n ap,n 00
L
00
:s M L
Ixnap,nl = M Ilxlip and hence Y E AO(ap,n)' n=O n=O Conversely, let us suppose that there exists Y E AO(ap,n) such that oo (_I_); that is, yrf.loo(_I_) for each p. Since for each p, sup IYn yrf.liml = 00, + ap,n Qp,n n ap,n
x E A(ap,n)'
IXnYnl
_1_1
I'
_1_1
then there exists an increasing sequence (np) such that IYn p ap,np define the sequence x
= (xn);:"=o as follows: Xn =
{
_I_~ ap,np
P
> p2. Let us
ifn#np =n .
if n
P
We claim that x E A(ap,n): For arbitrary q, IIxllq q "
00
aq,np 1
L.J a
p=O p,n"
p2'
+ "L.J
aq,np 1
a
p=q+l ",n"
f:
p2"
f:
f:
00
" aq,np..!.. = n=O Ixnlaq,n = p=O 1~~laq,np = p=O L.J a p,np p2 v,np P
<
00
as claimed •
f:
Nevertheless n=O IXnYnl = n=O I~~ Yn,,1 = 00 since I~~pl p,n" P p,np Yn,,1 > 1 for every n. This is a contradiction to the assumption that Y E AO(ap,n)' 0 PROPOSITION 3.2. The inductive limit topology is stronger than the normal topology in AO(ap,n).
ON
oDUAL ALGEBRAS
35
PROOF. Let us recall that 11.11 is a continuous seminorm in limloo(I) + av,n p
if,
and only if,
for
each p,
there exists a
Ilyll
~ Cp lIylll"" = Cps~p IYna:.n I for every Y E loo(a:.J·
1 =
1 Let Y E 100 (_1_), then sup IYnap,n n ap,n
AI <
00
constant
Cp such that
1and therefore IYnlap,n 00
for each n.
Hence, for x E A(ap,n) we have that IIYllx
=
I:
~ AI
IXnYnl ~ <
n=O
n~o AI IXnl ap,n = AI n~o IXnl aq,n = AI IIxllp = s~p IYna:.n Illxll p = Ilxllp Ilyllloo'
D
In [4] it is proved that an echelon space A(ap,n) is nuclear if, and only if, for each p there exist q > p and a sequence u = (un);:::'=o E II such that ap,n = Un aq,n, n = 0,1, ... PROPOSITION 3.3. If A(ap,n) is nuclear, then AO(ap,n), endowed with the normal topology, is such that AO(ap,n) ~ limloo(I) ~ limll(I). + ap,n + ap,n p
p
PROOF. Due to the nuclearity of A(ap,n) it is easy to see that limll(I) ~ limloo(I). By the previous Proposition we know that the inductive + ap,n + ap,n p
p
limit topology is stronger than the normal topology in AO(ap,n). Let 11·11 be a seminorm in limll(I). Let (e n );:::'=1 be the sequence of the + ap,n p
canonical vectors and note that (e n );:::'=1 E 11(_1_) for each p. Therefore there ap,n exists Cp > 0 such that lIenll ~ Cpllenill oo = CPa :. n for each p. (Xn);:::'=1 = (1Ien ll);:::'=I' We will prove that (Xn);:::'=1 EA(ap,n)' 00
Indeed, IIxnllp
I:
00
n=1
I:
Ilenllap,n
n=1
We put
00
I: Cqllenllqunaq,n
=
1L~1 ynenll ~ n~IIYnlllenll
=
lIenllunaq,n <
n=1
00
Cq
I: Un < 00
n=1
Therefore, for any Y E loo(a:.J, IlylI
=
00
I:
IYnllxnl = IIYllx <
00.
SO II . II is continuous with respect to the normal topol
n=1
ogy.
D Let A(ap,n) be a matrix algebra. We define the following condition:
DEFINITION 3.4. (*) for each p there exist q ap,n ap,m ~ AIp aq,n+m for all n,m.
>
P and AIp such that
Let us recall that the absolutely convex hull of a subset V of a vector space is the set r(V) =
{I: .Aiai, I finite, ai EV, iEI
.Ai
EC and
I: l.Ail ~ I}.
It is a straight for
iEI
ward matter to check that if V and Ware two subsets of an algebra, then the property (x, Y E V ~ xy E W) implies the property (x, Y E r(V) ~ xy E r(W)). PROPOSITION 3.5. If the matrix algebra A(ap,n) satisfies (*), then l~ll(a:.J p
is a topological algebra under the convolution product.
HUGO ARIZMENDI, ANGEL CARRILLO, AND LOURDES PALACIOS
36
PROOF. Let x, Y E 11(_1_). From condition (*), let q > p and Mp be such that
Mp.: ••":_,, The:'il:YII, ~ "~O I(xy)" I .;," ~ I
.;," <;
f
t
Mp
C~O IXnl a:,n) C~O IYnl a:. n) =
n=O k=O
IXkYnkl a:q,n
:s n=O f (t Mp IXka:k IIYnk k=O ".
I)
l_k p.n
a
Mp IIxlip IIYllp <
Ikt IXk!lnkll·;"
<;
=
00.
So far, limll( a:) is an algebra. From the previous proof it is easy to see that p.rt.
+ p
(a:.J
(a..J,
(a:.J
1 if x E II and Y E II then xy E II and IIxYllq ::; MpllxllrllYIIs, where q and Mp are any two numbers that satisfy (*) for p = max(r, s). Now let n be a neighborhood of the origin in limll(I); then + ap,n
"
n= r
CQl V(O, II·IIp' ep)), where ep > 0 for all p ~ 1.
For each p
~
1, let qp and Mp be two numbers that satisfy (*) for p and let us
(1,
take 0 < t5p < min ~). If x E V(O, II.IIr,er) and Y E V(O, II·IIs,es), then IIxYllqp MpllxllrilYlls, where p = max(r,s); therefore IIxyllq" < eq", thus xyEV(O, II.IIqp,eq,,)' From the result stated before this proposition, it follows that,
x, Y E r
:s
CQl V(O, II·IIp' t5p)) ::::} xy r CQl V(O, II·IIp' ep)) ~ n. E
This shows that
the multiplication is continuous.
COROLLARY 3.6. If the matrix algebra A(ap,n) is nuclear and satisfies (*), then AQ(ap,n) is a topological algebra under the normal topology. PROPOSITION 3.7. If the matrix algebra A(ap,n) does not satisfy (*), then limll(I) is not an algebra. + ap,n p
PROOF. The hypothesis gives a Po with the property that for each q > Po and Mq there exist m and n such that apo,n apo,m > !llq aq,n+m, or, equivalently, _ 1 _ > M _1_ _ _ 1_. q
aq,n+m
Fix an i that
1
a pQ •n
apQ,m
.
>
Po and choose Mi = 22i. There are positive integers mi, ni such > Mi _1_ _ _1_.
ai,ni+mi
apQ,ni
a'PO·1'1~i
00
Let x
= L:
n=O
for i ~ 1 and an
00
anz n and Y
= L:
n=O
bnz n , with ani
= bn = 0 otherwise.
Then
=
IIxli po =
2ill)illpo' bmi
00
=
~ 2illz~illpo IIznili po
2illz;';1I,,0
= 1 and
.=1 00
IIyllpo = ~ .=1
2'lIz;'ql" IIzm; IIpo
= 1; but, since xy=
00
?:
J=O
00
anbmzi
= L: anibm;zni+m; .=1
n+m=j 00
we have IIxYllq ~ . L: 2i+illzni lI~o IIzmj IIp() II zn;+mj IIi = 00 .=q+l Therefore the space limll(I) can not be an algebra (see also Proposition + ap,n p
3.1).
o
ON ",,DUAL ALGEBRAS COROLLARY
37
3.8. A(a pn ) satisfies (*) if, and only if, limll(I) is a topological , + ap,n p
algebra under the convolution product. COROLLARY 3.9. If A(ap,n) is nuclear, then it satisfies (*) if, and only if, A""(ap,n) is a topological algebra under the convolution product. EXAMPLES
3.10.
(1) Let us consider the matrix algebra A(ap,n) where ap,n
In this case, A(ap n) I
= (n + l)P, n = 0, 1,2, ....
=
{(xn)
E w: lim (n n+oo
+ l)P IXnl = 0,
p
= 1,2, ... }.
A(ap,n) and A""(ap,n) are nuclear algebras and one is the adual of the other. A(ap,n) is nuclear since for each p we can take q = 3p and
(un)~=o =
((n+I)q
p) ~=o =
(( (n;l)p )
2) ~=o'
which clearly is in ll,
and satisfies ap,n = unaq,n for all n. A""(ap,n) is an algebra due to Corollary 3.6 since for each p we can take q = 2p and Mp = 1 to satisfy ap,nap,m ::; Mpaq,n+m for all n, m. (2) Let us consider the matrix algebra A(ap,n) where 1
a p,n  n 2 P
nEN.
,
In this case, A(ap,n) satisfies (*). For, if p is given, let q = p + 1 and 2 p  1 Mp=2 . Note that for arbitrary n , m 'nnt.~ < n+m = .1+..!.. < 2. Then ap,n ap,m = m nm n m aq,n+m 1
1
;:;:z=P ~ _ (n+m) ;
P
1

2p1
(nm)2
P
_ (n+m) 
2p1
(n2m2)2
P
1
2
_ (n+m) 
n2 m 2
pl
< 2TP 
1 
M p.
(n+m)
However, the algebra A(ap,n) is not nuclear. For, if p < q are arbitrary, _1_ 2 P 2 q 2q2P ~ = ~ = (.1) = (.1) ~ E II only if ~ > 1, which is aq,n 2q n n 2P q impossible. (3) If w is the linear space of all complex sequences, we note that w = A(ap,n) where ap,n
=
I { 0
ifn::;p
""
if n > p . Then A (ap,n)
= I!m(N). Both of them are
nuclear topological algebras. Here Corollary 3.9 does not apply because it is not true that ap,n > 0 for all indices p and n.
References [IJ R. Arizmendi. "Matrix Algebras and mconvexity". Demostratio Mathematica, Vol. XVII, no. 3, 1984. [2J R. Arizmendi; A. Carrillo; L. Palacios. "On continuous multiplicative Mappings on Ananlytic Sequence Spaces". Commentatione Mathematicae, XXXIX (1999). [3] G. Kothe. Topological Vector Spaces l. A series of Comprehensive Studies in Mathematics, SpringerVerlag, 1966. [4] A. Pietsch. Nuclear Locally Convex Spaces. SpringerVerlag, 1969. [5] O.Toeplitz. Die linearen volkommenen Riiume der Funktionentheorie. Comment. Math. Relv. 23 (1949), 222242.
38
HUGO ARIZMENDI, ANGEL CARRILLO, AND LOURDES PALACIOS
(Hugo Arizmendi, Angel Carrillo) INSTITUTO DE MATEMATICAS. UNIVERSIDAD NACIONAL AUTON OM A DE MEXICO, APDO POSTAL 14455, MEXICO, D.F.
Email address:hugolDservidor.unam.mx (Lourdes Palacios) DEPARTAMENTO DE MATEMATICAS, UNIVERSIDAD AlTTONOMA METROPOLITANA. Av. SAN RAFAEL ATLIXCO 186, COL. VICENTINA, 07340 MEXICO, D.F.
Email address:pafalDxanum.uam.mx
Contemporary Mathematics Volume 328, 2003
A Connected metric space that is not separably connected Richard M. Aron and Manuel Maestre ABSTRACT. We construct a subset of the unit ball of foo that is connected but not separably connected.
A topological space X is said to be separably connected if for every two points X, Y E X there exists a connected and separable subset C{x,y} such that x,y E C{x,y}' The reader may be interested, perhaps even amazed, at the fact that this concept arises naturally in economics. In fact, in their work on utility theory, J. Candeal, C. Herves, and E. Induntin [B] ask whether there is a connected metric space that is not separably connected. They observe that there are examples of connected topological spaces which are not separably connected. However, no example of a connected metric space which is not separably connected is given and, in fact, this issue is explicitly raised in [A]. The purpose of this note is to give a negative answer to this question. In fact, this problem was solved by R. Pol over 25 years ago, who produced a somewhat different construction in [C]. The construction presented here seems natural, in view of its similarity to the standard construction of a nonmeasurable subset of JR. Our example will be a subset of the Banach space (foo, 11·1100) of all sequences (Xn)~=l in lK = JR or C, satisfying II(xn)~=11l := sup{lxnl : n = 1, 2... }
< 00,
Given U E Roo and T/ > 0, we denote by B(u, T/) := {v E Roo : IIv  ull ::; T/}, i.e., B( u, T/) is the closed ball with center u and radius T/. We will single out the unit vector e := (1,0, ..... ) in Roo and the associated linear functional c.p : Roo + lK, c.p( (xn)) = x}, for all (xn) E f oo . Clearly, Ker c.p = ((xn) E Roo : c.p( (xn)) = Xl = o}. Let .N denote the set consisting of all nonempty subsets S C {2, 3, .. , n, ... }, and for each such S, let Us = (Xn)~=l E foo : Xn = 1 if n E Sand Xn = 0 if n ~ S, n = 1,2 .... The set £ == {us : S E .N} is uncountable since .N is, any element of £ has norm one and Ilu  vii = 1 for all u,v E £, u I v. Recall now 2000 Mathematics Subject Classification. Primary 54D05, 46B26. Key words and phrases. Metric spaces non separably connected, Banach spaces. The first author was partially supported by the Ministerio de Educacion y Cultura of Spain (SAB19990214). The second author was partially supported by MCYT and FEDER Project BFM200201423. © 2003 American Mathematical Society 39
RICHARD M. ARON AND MANUEL MAESTRE
40
the following equivalence relation on [0, 1], which is used to prove the existence of a nonLebesgue measurable subset of [0,1]: x '" y if and only if x  y E Q. We denote the equivalence classes of [0,1] defined by this relation by {x : x E [0, I]}. By applying the Axiom of Choice, we can choose one element from each class, in particular taking E O. Let us denote the set so obtained by A.
°
LEMMA 1.1. The following properties hold: (1) Card (A) = Card([O, 1]). (2) Given x, yEA, x =1= y and r, SEQ, x + r =1= y + s. (3) If a E [0,1]' then there exist a unique x E A and a unique r E Q with
a = x+r.
(4) Given a < b E JR and x E JR, the family {x +r : r E Q} n (a, b) is a dense subset of [a, b].
The proof of the above properties is immediate. Let us remark that (4) can be obtained by using the fact that for each x E JR, the map ¢ : JR + JR defined by ¢(t) := x + t is a homeomorphism and the fact that Q n (a  x, b  x) is always a dense subset of [a  x, b  x]. Also it is very easy to construct a bijection R : A + £, R(x) = ex. We define our set C as C = C 1 U C2 , where
U
C1 =
re+ [O,eo]
rE[O,l]nQ
and
U
(x + r)e + (0, ex].
xEA\{O}.rEQ,x+rE[O,l]
For an intuitive, geometric idea of what C is, consider a type of "comb" set in the plane given by {(s, t) : 0::; s ::; 1, s ~ Q, t ::; I} U {(s, t) : 0::; s ::; 1, s E
°: ;
Q,O>t~I}.
We endow C with the metric induced by the norm of loc. Our goal in this note is to prove the following: THEOREM 1.2. C is a connected metric space that is not separably connected. PROOF. By Lemma 1.1.(2), given x, YEA, x =1= y, r, SEQ and any u, v E Ker rp we have that rp((x + r)e + u) = x + r =1= y + s = rp((y + s)e + v). Hence for x E A \ {O} and r E Q with x + r E [0,1]' we will have that
+ r)) U rpl((x + r, +00)). Consider a connected subset DeC such that 0, e + eo E D. If there exist x E A \ {O} and r E Q with x + r E (0,1) and D n ((x + r)e + (0, ex]) = 0, then, from rp(O) = < x + r < 1 = rp(e + eo) and (1.1), we obtain that D is not connected. Hence for all x E A \ {O} and r E Q with x + r E (0,1) we have that there exists C \ {(x
(1.1)
+ r)e + (0, ex]} c
rpl(( 00, X
°
°<
Ax ::; 1 with
(x + r)e + Axex E D. Thus, {ex}xEA\{O} C span{e, D}. However, the family {ex}xEA\{O} is uncountable, and Ile x  ey II = 1 for all x, yEA \ {O}, x =1= y. As a consequence span {e, D} is not
A CONNECTED METRIC SPACE THAT IS NOT SEPARABLY CONNECTED
41
separable, which implies that D is nonseparable. Thus, it only remains to show that C is connected. If C were not connected, then there would exist a separation of C by subsets U and V; that is, there would exist open (and closed) sets U and V such that U =I
0 =I V, U U V
= C,
UnV =
0.
To simplify the notation, given Z E [0,1] we denote by z the only x E A such that = X, and we denote by Iz = ze + (0, e z] if z fj. Q and I z = ze + [0, eo] if Z E Q. Since Iz is connected we have that given Z E [0,1] if I z n U =I 0 then Iz C U (and the same with respect to V). As eo E U U V, we will assume that eo E U and hence 10 C U. Consider Z E Q n [0,1] such that I z C U. Since U is open, there exists TJ = TJ(z) > a such that the closed ball B(ze,2TJ) n C c U and moreover such that (z  TJ, z + TJ) C [0,1] if Z E (0,1) n Q. Now it is immediate that for every u E f(X), Ilull = 1, every y E (zTJ, z+TJ)n[O, 1] and every A, 0:$ A:$ TJ, lIye+Auzell :$ 2TJ. By Lemma 1.1.(3), there exist unique Xl E A and rl E Q with y = Xl +rl. If Xl = 0, then ye + [0, TJeo] = rle + [0, TJeo] C B(ze, 2TJ) n C l c U and hence rle + [0, eo] C U. Alternatively, if Xl E A\ {a}, then ye+(O, TJe X1 ] = (Xl +rl)e+(O, TJe X1 ] C B(ze, 2TJ)n C2 c U, and again it follows that (Xl + rde + (0, eX!] C U. In either case, we see that Iy C U for all y E (z  TJ,Z +TJ) n [0,1]. Let A:= {z E Qn [0,1] : Iz C U} and B:= {z E Qn [0,1] : Iz C V}. Let K := UzEA(Z  TJ, Z + TJ) n [0,1] and L := UzEB(Z  TJ, z + TJ) n [0,1] where TJ = TJ(z) is given above. K and L are open sets in [0,1] and Q n [0,1] c L U K. If t is in k, the closure of K, then given n E N we can find Zn E A such that (Zn  TJn, Zn + TJn) n (t  ~, t + ~) n [0, 1] =I 0, where 1Jn = TJ(zn). Hence there exists a < b such that (a, b) C (zn  TJn, Zn + TJn) n (t  ~, t + ~) n [0, 1]. By Lemma 1.1.(4) there exists r n E Q such that t + r n E (a, b). Thus, the preceding paragraph shows that I Hrn C U and as t + rn = f, we have (t + rn)e + ef E U. Since Irnl < ~ for all nand U is closed in C we have that te + er E U, thus It cU. The same argument applied to L implies that if S E L, then Is C V. Hence k and L are two closed disjoint sets. But as Q n [0,1] c K U L c [0,1] we have [0,1] = k U L. Since [0,1] is a connected set and a E A c K we have that k = [0,1] and hence C = U, a contradiction. Therefore C is connected. 0
z
REMARK 1.3. This result extends to any nonseparable Banach space. To see this, consider a nonseparable Banach space F. As above we can take e E F with lIell = 1 and a continuous, linear functional
a and an uncountable set c Ker
e
e
C l :=
U rEQn[O,l]
re + [0, eo] and C2 :=
U
(X + r)e + (0, ex].
x+rE[O,l] xEA\{O}, rEQ
If C is endowed with the metric induced by the norm of F, then, by a proof similar to the one given in Theorem 1.2, C is a connected metric space that is not separably connected.
42
RICHARD M. ARON AND MANUEL MAESTRE
ACKNOWLEDGEMENT. This note evolved during visits the first author made to the Departamento de Amilisis Matematico of the Universidad de Valencia, and while the second author was a Visiting Professor in the Department of Mathematical Sciences at Kent State University for the 199899 academic year. The authors express their thanks to the Departments concerned for their hospitality. In addition, the authors are very grateful to the Referee, whose careful reading of the original version of this paper led to the present considerably simplified proof of Theorem 1.2.
References [A] Balbas, A., Estevez, M., Herves, C., and Verdejo, A. Espacios separablemente conexos, Rev. R.Acas. Cien. Exact. Fis. Nat. (Spain), 92, 1, (1998), 3540. [B] J. Candeal, C. Herves and E. Indurain, Some results on representation and extension of preferences, Journal of Mathematical Economics 29 (1998), 7581. [C] R. Pol, Two examples of nonseparable metrizable spaces, ColI. Math. 33, (1975),209211. DEPARTMENT OF MATHEl\IATICS, KENT STATE UNIVERSITY, KENT. OH 44242 USA Email address: aronClmcs. kent. edu DEPARTAMENTO DE ANALISIS MATEMATlCO, FACULTAD DE MATEMATICAS, UNIVERSIDAD DE VALENCIA, 46100 BURJASOT (VALENCIA). SPAIN Email address: manuel.maestreCluv. es
Contemporary Mathematics
Volume 328, 2003
Weighted Chebyshev Centres and Intersection Properties of Balls in Banach Spaces Pradipta Bandyopadhyay and S Dutta ABSTRACT. Vesely has studied Banach spaces that admit weighted Chebyshev centres for finite sets. Subsequently, Bandyopadhyay and Rae had shown, inter alia, that Ltpreduals have this property. In this work, we investigate why and to what extent are these results true and thereby explore when a more general family of sets admit weighted Chebyshev centres. We extend and improve upon some earlier results in this general setup and relate them with a modified notion of minimal points. Special cases when we consider the family of all finite, or more interestingly, compact subsets lead to characterizations of Ltpreduals. We also consider some stability results.
1. Introduction
Let X be a Banach space. We will denote by Bx[x,r] the closed ball ofradius r > 0 around x EX. We will identify any element x E X with its canonical image in X**. Our notations are otherwise standard. Any unexplained terminology can be found in either [6] or [10). In this paper we continue the study of Banach spaces that admit weighted Chebyshev centres that began with [3). DEFINITION 1.1. Let Y be a subspace of a Banach space X. For A ? ~+, define
~
Y and
p:A
¢A,p(X) = sup{p(a)llx 
all : a E A}
A point Xo E X is called a weighted Chebyshev centre of A in X for the weight p if ¢A,p attains its minimum at Xo. When A is finite, Vesely [18) has shown that if X is a dual space, A admits weighted Chebyshev centres in X for any weight p, that the infimum of ¢ A,p over X and X** are the same, and THEOREM 1.2. [18, Theorem 2.7) For a Banach space X and aI, a2, ... ,an E X, the following are equivalent : 2000 Mathematics Subject Classification. Primary 4IA65, 46B20; Secondary 41A28, 46B25, 46E15, 46E30. Key words and phrases. Weighted Chebyshev centres, minimal points, central subspaces, Icomplemented subspace, I PI,oo, Ll_preduals. © 43
2003 American Mathematical Society
44
BANDYOPADHYAY AND DUTTA (a) Ifrl, r2, ... , rn > 0 and ni=IBx " [ai, riJ =10, then ni=IBx[ai, riJ =10. (b) {al,a2, ... ,an } admits weighted Chebyshev centres for all weights rl,r2, ... ,rn > O. (c) {aI, a2,' .. , an} admits fcentres for every continuous monotone coercive f : IR+. 4 IR (see [18J for the definitions).
In this work, we investigate why and to what extent are these results true and thereby explore when a more general family of sets admit weighted Chebyshev centres. Extending the notion of central subspaces introduced in [3], we define an ACsubspace Y of a Banach space X with the centres of the balls coming from a given family A of subsets of Y, the typical examples being those of finite, compact, bounded or arbitrary sets. The first gives us the central subspace a la [3J and the last one is related to the Finite Infinite Intersection Property (I Pj,oo) [8J. We extend and improve upon some results of [3, 18J in this general setup and relate them with a modified notion of minimal points. We also improve upon one of the main results of [4J on the structure of the set of minimal points of a compact set. As in [3], special cases when we consider the family of all finite, or more interestingly, compact subsets lead to characterizations of Llpreduals. We also consider some stability results.
2. General Results We first extend VeselY's result in [18J on dual spaces from finite sets to all the way upto bounded sets and also strengthens its conclusions. We need the following notions. DEFINITION 2.1. Let X be a Banach space and A ~ X. (a) We define a partial ordering on X as follows : for Xl, X2 EX, we say that Xl :SA X2 if IlxI  all IIx2  all for all a E A. We will denote by mx (A) the set of points of X that are minimal with respect to the ordering :SA and often refer to them as :SAminimal points of X. Note that :SA defines a partial order on any Banach space containing A and we will use the same notation in all such cases. f(x2) (b) A function f : X 4 IR+ is said to be Amonotone if f(xd whenever Xl :SA X2· (c) Let Y be a subspace of X and A ~ Y. Following [9], we say X E X is a minimal point of A with respect to Y if for any y E Y, Y :SA X implies y=x. We denote the set of all minimal points of A with respect to Y in X by Ay,x, Note Ay,x ;2 A. For A ~ X, the set Ax,x will be called minimal points of A in X, and will be denoted simply by min A. (d) For A ~ X bounded, the Chebyshev radius of A in X is defined by
:s
:s
r(A) = inf sup Ilx xEX aEA
all.
THEOREM 2.2. (a) If A ~ X is bounded and X ~ A + r(A)B(X), then there exists y E X such that y :SA x. (b) If X = Z· is a dual space and A is bounded, then every Amonotone and w*lower semicontinuous (henceforth, lsc) f : X 4 IR+ attains its minimum. In particular, for every p, ¢ A,p attains its minimum.
CHEBYSHEV CENTRES AND INTERSECTION PROPERTIES OF BALLS
45
(c) If X = Z* is a dual space, for every Xo E X, there is a Xl E mx(A) such that Xl ::;A Xo. In particular, the minimum in (b) is attained at a point ofmx(A). PROOF. (a). Let X ~ A + r(A)B(X). Then, there exists c > 0 such that IIx  all > r(A) + c for all a E A. By definition of r(A), there exists y E X such that sUPaEA Ily  all < r(A) + c. Clearly, y ::;A x. (b). By (a), if X ~ A + r(A)B(X), there exists y E X such that y ::;A X, and hence, f(y) ::; f(x). Thus, the infimum of f over X equals the infimum over A + r(A)B(X). Moreover, since X is a dual space and f is w*lsc, it attains its minimum over any w*compact set. Thus f actually attains its minimum over X as well. Since the norm on X is w*lsc, so is cPA,p for every p. (c). Consider {x EX: x ::;A xo}. Let {Xi} be a totally ordered subset. Let z be a w*limit point of Xi. Since the norm is w*lsc, we have liz 
all ::; lim inf Ilxi  all =
inf Ilxi
 all
for all a E A.
Thus the family {Xi} is ::;Abounded below by z. By Zorn's lemma, there is a Xl E mx(A) such that Xl ::;A Xo. Now let Xo be a minimum for f. There is a Xl E mx(A) such that Clearly, f attains its minimum also at Xl.
Xl
::;A Xo. 0
REMARK 2.3. (a) It follows that for any bounded set A, minA C A + r(A)B(X). This improves the estimates in [9] or [18]. (b) Apart from cPA,p, there are many examples of Amonotone and w*lsc f : X = Z*  lR+. One particular example that has been treated extensively in [4] is the function cPf.,l defined by cPf.,l(x) = fA Ilx  aI1 2 dJ.L(a), where J.L is a probability measure on a compact set A ~ X. (c) Observe that though minimal points of A are ::;Aminimal, there is some distinction between the two notions. The two notions coincide if X is strictly convex. See Proposition 3.1 below. Now, if A is a bounded subset of a Banach space X, then by Theorem 2.2, A has a weighted Chebyshev centre in X**. But what about a weighted Chebyshev centre in X? When A is finite, Vesely [18] has shown that the infimum of cPA,p over X and X** are the same, and A admits weighted Chebyshev centres in X for any weight p if and only if X satisfies Theorem 1.2(a). We now show that both of these are special cases of more general results. We need the following definition. DEFINITION 2.4. Let Y be a subspace of a Banach space X. Let A be a family of subsets of Y. (a) We say that Y is an almost ACsubspace of X if for every A E A, X E X and c > 0, there exists y E Y such that
(1)
Ily  all::; IIx  all + c forall a E A. (b) We say that Y is an ACsubspace of X if we can takec = 0 in (a). (c) If A is a family of subsets of X, we say that X has the (almost) AIP if X is an (almost) ACsubspace of X**.
BANDYOPADHYAY AND DUTTA
46
Some of the special families that we would like to give names to are : (i) F = the family of all finite sets, (ii) K = the family of all compact sets, (iii) B = the family of all bounded sets, (iv) 'P = the power set. Since these families depend on the space in which they are considered, we will use the notation F(X) etc. whenever there is a scope of confusion. REMARK 2.5. (a) Note that FCsubspaces were called central (C) subspaces in [3], 'PCsubspaces were called almost constrained (AC) subspaces in [1, 2]. Also if X has the FIP, it was said to belong to the class (GC) in [18, 3], and the 'PIP was called the Finite Infinite Intersection Property (IPj,oo) in [7, 2]. (b) The definition of almost ACsubspace is adapted from the definition of almost central subspace defined in [17]. The exact analogue of the definition in [17] would have, in place of condition (1), sup Ily aEA
all::; sup Ilx  all + c. aEA
Clearly, our condition is stronger. We observe below (see Proposition 2.7) that this definition is more natural in our context. (c) By the Principle of Local Reflexivity (henceforth, PLR), any Banach space has the almost FIP. More generally, if Y is a!l ideal in X (see definition below), then Y is an almost FCsubspace of X. DEFINITION 2.6. A subspace Y of a Banach space X is said to be an ideal in X if there is a norm 1 projection P on X* with ker(P) = y.l. PROPOSITION 2.7. Let Y be a subspace of a Banach space X. Let A be a family of bounded subsets of Y. Then the following are equivalent: (a) Y is an almost ACsubspace of X (b) for all A E A and p : A t lR.+, if naEABx [a, p(a)] ¥ 0, then for every e > 0, naEABy[a, p(a) + c] ¥ 0. (c) for every bounded p, the infimum of ¢A,p over X and Yare equal. PROOF. Equivalence of (a) and (b) is immediate and does not need A to be bounded. (a) ::::} (c). Let Y be an almost ACsubspace of X, A E A and p : A t lR.+ be bounded. Let M = supp(A). Let e > O. By definition, for x E X, there exists y E Y such that lIy  all::; Ilx  all + e for all a E A. It follows that
p(a)lly  all ::; p(a)llx 
all + p(a)e ::; p(a)llx  all + Me for all a E A.
and hence,
¢A,p(Y) ::; ¢A,p(X)
+ Me.
Therefore, inf ¢A,p(Y) ::; inf ¢A,p(X)
+ Me.
As e is arbitrary, the infimum of ¢A,p over X and Yare equal.
CHEBYSHEV CENTRES AND INTERSECTION PROPERTIES OF BALLS
47
(c) =} (a). Let A E A, x E X and e > O. We need to show that there exists y E Y such that Ily  all IIx  all + e for all a E A. If x E Y, nothing to prove. Let x E X \ Y. Let N = sUPaEA IIx  all. Let p(a) = l/llx  all. Since x r/: Y and A ~ Y, p is bounded. Then ¢A.p(X) = 1, and therefore, inf ¢A,p(X) S 1. By assumption, inf ¢A,p(Y) = inf ¢A,p(X) S 1, and so, there exists y E Y, such that ¢A,p(y) 1 + e/N. This implies Ily  all S Ilx  all + ell x  all/N S Ilx  all + e for all a E A. 0
s
s
As noted before, by PLR, any Banach space has the almost FIP. And therefore, the result of [18J follows. PROPOSITION 2.8. Let A and Al be two families of subsets of Y such that for every A E A and e > 0, there exists Al E Al such that A ~ Al + cB(Y). If Y is an almost AICsubspace of X, then Y is an almost ACsubspace of X as well. Consequently, any ideal is an almost KC subspace and any Banach space has the almost KIP. In particular, if A is a compact subset of X and p : A + lR+ is bounded, then the infimum of ¢A,p over X and X** are the same. PROOF. Let A E A and e > O. By hypothesis, there exist Al E Al such that Al + eB(Y). Let x E X. Since Y is an almost AICsubspace of X, there exists y E Y such that A
~
lIy  alii S Ilx  alII
+ e/3 for all al
E AI.
Now fix a E A. Then there exists al E Al such that lIa  alii < e/3. Then Ily  alii + lIa  alii S Ilx  alii + 2e/3 Ilx  all + Iia  alii + 2e/3 S IIx  all + e. Therefore, Y is an almost ACsubspace of X as well. Since any Banach space has the almost FIP, by the above, it has the almost KIP too. The rest of the result follows from Proposition 2.7. 0 Ily  all
S S
EXAMPLE 2.9. Vesely [18J has shown that if A is infinite, the infimum of ¢A,p over X and X** may not be the same. His example is X = co, A = {en: n ;::: I} is the canonical unit vector basis of Co and p == 1. Then inf ¢A,p(X) = 1 and inf ¢A,p(X**) = 1/2. The example clearly also excludes countable, bounded, or, taking Au {O}, even weakly compact sets. Thus Co fails the almost BIP, almost PIP and if A is the family of countable or weakly compact sets, then Co fails the almost AIP too. Stronger conclusions are possible for AIP. LEMMA 2.10. Let Y be a subspace of a Banach space X. For A ~ Y, the following are equivalent : (a) For every Amonotone f : A + lR+ and x E X, there exists y E Y such that f(y) S f(x). ( b) For every p : A + lR+ and x EX, there exists y E Y such that ¢A,p(Y) S ¢A,p(X). (c) For every continuous p : A + lR+ and x EX, there exists y E Y such that ¢A,p(Y) S ¢A,p(X). (d) For every bounded p : A + lR+ and x EX, there exists y E Y such that ¢A,p(Y) S ¢A,p(X).
BANDYOPADHYAY AND DUTTA
48
(e) Any family of closed balls centred at points of A that intersects in X also intersects in Y. (f) for any x EX, ther·e exists y E Y such that y :SA x. It follows that whenever any of the above conditions is satisfied, for every Amonotone f : A + lR+, the infimum of f over X and Yare equal and if A has a weighted Chebyshev centre in X, it has a weighted Chebyshev centre in Y.
PROOF. (a) => (b) => (c), (b) => (d) and (e) ¢} (f) => (a) are obvious. (c) or (d) => (f). As in the proof of Proposition 2.7, let p(a) = l/llxall. Then p is continuous and bounded and
:s
:s
We now conclude the discussion so far by obtaining the extension of Theorem 1.2. THEOREM 2.11. For a Banach space X and a family A of bounded subsets of X, the following are equivalent: (a) X has the AIP. (b) For every A E A and every f : X** + lR+ that is Amonotone and w*lsc, the infimum of f over X** and X are equal and is attained at a point of (c) For every A E A and every p, the infimum of
x.
We now study different aspects of ACsubspaces. DEFINITION 2.12. Let Y be a subspace of a Banach space X. Let A x E X and X* E B(X*), define U(x,A,x*)
= inf{x*(y) + IIx  yll : yEA}
L(x, A, x*)
=
~
Y. For
sup{x*(y) lix  yll : yEA}
The following lemma is in [1]. We include the proof for completeness. LEMMA 2.13. Let Y be a subspace of a Banach space X and A ~ Y. For Xl, X2 EX, X2 :SA Xl if and only if for all x* E B(X*), U(X2, A, x*) U(XI, A, x*).
:s
:s
PROOF. If X2 :SA Xl, then for all x* E B(X*), x*(y) + IIx2  yll x*(y) + IIxI YII. And therefore, U(x2,A,x*):S U(xI,A,x*). Conversely, suppose IIx2  Yo II > IIxI  Yo II for some Yo E A. Then there exists c > 0 such that IIx2  Yo II  c ~ IIxI  Yoli. Choose x* E B(X*) such that IIxI  Yo II IIx2  yolI c < X*(X2  Yo)  c/2. Thus U(XI, A, x*) x*(yo) + IIxI Yoll < X*(X2)  c/2 < U(X2, A, x*). 0
:s
:s
REMARK 2.14. Instead of B(X*), it suffices to consider the unit ball of any norming subspace of X*. We compile in the following propositions several interesting facts about ACsubspaces and the AIP. PROPOSITION 2.15. Let Y be a subspace of a Banach space X. For a family A of subsets of Y, the following are equivalent: (a) Y is an ACsubspace of X
CHEBYSHEV CENTRES AND INTERSECTION PROPERTIES OF BALLS
49
(b) for every x E X and A E A, there exists y E Y such that U(y, A, x*) ::; U(x,A,x*) for every x* E B(X*). (c) for any A E A, Ay,x ~ Y.
PROOF. This follows from Lemma 2.13 and the definition of Ay,x.
COROLLARY 2.16. X has the PIP if and only if for every x** E X**, there exists x E X such that x is dominated on B(X*) by the 1~pper envelop of x** considered as a function on B(X*) equipped with the w*topology. PROOF. Observe that for any x E X, U(x,X,·) == x on B(X*) and for x** E X**, U(x**, X, x*) is the upper envelop of x** considered as a function on B(X*) equipped with the w*topology (see [8]). 0 PROPOSITION 2.17. (a) Let X be a Banach space and let Y be a subspace of X. Let A be a family of subsets of Y and let Al be a subfamily of A. If Y is a AC subspace of X, then Y is a Al C subspace of X as well. In particular, PIP implies BIP implies /CIP implies FIP. (b) 1complemented subspaces are ACsubspaces for any A. (c) Let Z ~ Y ~ X and let A be a family of subsets of Z. If Z is an ACsubspace of X, then Z is an ACsubspace of Y. And, if Y is an ACsubspace of X, then the converse also holds. PROOF. The proof follows the same line of argument as in [3, Proposition 2.2]. We omit the details. 0 PROPOSITION 2.18. For a family A of subsets of a Banach space X, the following are equivalent : (a) X has the AIP (b) X is a ACsubspace of some dual space. (c) for all A E A and p : A + lR+, ni=l Bx [ai, p(ai) + c] =f. 0 for all finite subset {aI, a2, ... ,an} ~ A and for all c > 0 implies naEABx [a, p(a)] =f. 0. In particular, any dual space has the AIP for any A. Let S be any of the families F, /C, B or P. The SIP is inherited by SCsubspaces, in particular, by 1complemented subspaces. PROOF. Clearly, (a) =} (b), while (c) =} (a) follows from the PLR. (b) =} (c). Let X be an ACsubspace of Z*. Consider the family {Bz. [a, p(a)+ c] : a E A, c > O} in Z*. Then, by the hypothesis, any finite subfamily intersects. Hence, by w*compactness, naEABz • [a, p(a)] =f. 0. Since X is an ACsubspace of Z*, we have naEABx[a,p(a)] =f. 0. 0 The following result significantly improves [3, Proposition 2.8] and provides yet another characterization of the AIP. PROPOSITION 2.19. Let Y be an almost FC subspace of a Banach space X. Let A be a family of subsets of Y. If Y has the AIP, then Y is an AC subspace of X. In particular, the conclusion holds when Y is an ideal in X. PROOF. Let x EX, A E A. Since Y be an almost FC subspace of X, for all finite subset {aba2, ... ,an} ~ A and for all c > 0, nb:IBy[ai, Ilx.,... aill + c] =f. 0. Since Y has the AIP, by Proposition 2.18(c), naEABy[a, Ilx  alll =f. 0. 0 Since X is always an ideal in X**, the following corollary is immediate.
BANDYOPADHYAY AND DUTTA
50
COROLLARY 2.20. For a Banach space X and a family A of subsets of X, the following are equivalent : . (a) X has AIP. (b) X is an ACsubspace of every superspace Z in which X embeds as an almost F C subspace. (c) X is an ACsubspace of every superspace Z in which X embeds as an ideal.
3. Strict convexity and minimal points PROPOSITION 3.1. If a Banach space X is strictly convex, then for every A X, min A = mx(A).
~
PROOF. As we have already observed, min A ~ mx(A). Let Xo E mx(A) and Xo ~ minA. Then there is an x E X such that x f= Xo and x :S::A Xo. Since Xo E mx(A), we must have Ilx  all = Ilxo  all for all a E A. Since X is strictly convex, II (x + xo)/2  all < IIxo  all for all a. This contradicts that Xo E mx(A). Hence Xo EminA. D REMARK 3.2. If X is strictly convex, by a similar argument, for every Xo EX, there is at most one Xl E mx(A) such that Xl :S::A Xo. Thus for a strictly convex dual space, for every Xo E X*, there is a unique xi E mx· (A) such that xi :S::A Xo' PROPOSITION 3.3. Let X be strictly convex. Let A be a compact subset of X. For each continuous p, A admits at most one weighted Chebyshev centre. PROOF. Suppose A admits two distinct weighted Chebyshev centres Xo, Xl E X. Then
3.4. Let X be a Banach space such that (i) X has the FIP; and (ii) for every compact set A ~ X, mx(A) is weakly compact. Then X has the KIP. Moreover, if X** is strictly convex, then the converse also holds. THEOREM
Let X have the FIP and for every compact set A ~ X, let mx(A) be weakly compact. Observe that for any B ~ A, we have mx(B) ~ mx(A). Let A ~ X be compact and let X** E X**. By Lemma 2.10, it suffices to show that there is a Zo E X such that Ilzo  all :s:: Ilx**  all for all a E A. Let {an} be a norm dense sequence in A. Take a sequence Ck + O. By compactness of A, for each k, there is a nk such that A ~ U~k BX[an,ck]. Since X has the FIP, there exists Zk E nlk Bx [an' Ilx**  anll] and Zk E mx( {al, a2 ... ank }) ~ mx(A). Then IIzkall :s:: Ilx**all+2ck for all a EA. Now, by weak compactness of m x (A), we have, by passing to a subsequence if necessary, Zk + Zo weakly for some Zo EX. Since the norm is weakly Isc, we have Ilzo all :s:: lim inf IIZk all :s:: Ilx** all for all a E A. Conversely, let X have the KIP and X** be strictly convex. Let A ~ X be compact. It is enough to show that any sequence {xn} ~ mx(A) has a weakly convergent subsequence. Without loss of generality, we may assume that {xn} are PROOF.
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all distinct. By Remark 2.3 (a), mx(A) ~ A + r(A)B(X) is bounded. Let x** be a w*cluster point of {x n } in Xu. It suffices to show that x** EX. Suppose x** E X** \ X. Since X has the KIP, there exists Xo E mx(A) such that Ilxo  all ::; Ilx**  all for all a E A. Since X** is strictly convex, lI(x** +xo)/2  all < IIx**  all for all a E A. Since (x** +xo)/2 E X** \X, by KIP again, there exists Zo E mx(A) such that Ilzo all::; II(x** +xo)/2all < IIx** all for all a E A. Since A is compact, there exists c > 0 such that Ilzo  all < IIx**  all  c for all a E A. Observe that
Ilzo  all < Ilx**  all c ::; liminf IIx n n
all c for all a E A.
Therefore, for every a E A, there exists N(a) EN such that for all n
~
N(a),
IIzo  all < Ilxn  all c. By compactness, there exists N E N such that IIzo  all < IIx n  all  c/4 for all n ~ N and a E A. Thus, Zo ::;A Xn for all n ~ N. Since Xn E mx(A) and X is strictly convex, Zo = Xn for all n completes the proof.
~
N. This contradiction
REMARK 3.5. In proving sufficiency, one only needs that {Zk} has a subsequence convergent in a topology in which the norm is lsc. The weakest such topology is the ball topology, bx . So it follows that if X has the FIP and for every compact set A ~ X, mx(A) is bxcompact, then X has the KIP. Is the converse true? COROLLARY 3.6. [4, Corollary 1] Let X be a reflexive and strictly convex Banach space. Let A ~ X be a compact set. Then min(A) is weakly compact. REMARK 3.7. Clearly, our proof is simpler than the original proof of [4]. If Z is a nonreflexive Banach space with Z*** strictly convex, then X = Z· is a nonreflexive Banach space with KIP such that X** is strictly convex. Thus, our result is also stronger than [4, Corollary 1].
4. L l preduals and PIspaces Our next theorem extends [3, Theorem 7], exhibits a large class of Banach spaces with the KIP and produces a family of examples where the notions of FCsubspaces and KCsubspaces are equivalent. DEFINITION 4.1. (a) [12] A Banach space X is called an Llpredual if X* is isometrically isomorphic to £1(Jt) for some positive measure Jt. (b) [11] A family {Bx[xi,ri]} of closed balls is said to have the weak intersection property iffor all x* E B(X*) the family {Ba[x*(xd,ri]} has nonempty intersection in JR. THEOREM 4.2. For a Banach space X, the following are equivalent: ( a) X is a K C subspace of every superspace (b) X is a KC subspace of every dual superspace ( c) X is a F C subspace of every superspace (d) X is an almost F C subspace of every superspace (e) X is a FCsubspace of every dual superspace (f) X is an almost FCsubspace of every dual superspace (g) X is an Llpredual.
BANDYOPADHYAY AND DUTTA
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PROOF. Observe that if X ~ Y ~ y** and X is a ACsubspace of Y**, then X is a ACsubspace of Y. Thus (a) {::} (b) and (c) {::} (e). And clearly, (a) ::::} (c) ::::} (d) ::::} (f). (f) ::::} (g). Since the definition of almost central subspaces in [17J is weaker than our definition of almost FCsubspaces, this follows from [17, Theorem 1, 2::::} 3J (g) ::::} (a). Suppose X is an LIpredual, and let X ~ Y. Let A ~ X be compact with at least three points. Let Yo E Y. Then the family of balls {Bx[a, Ilyo  allJ : a E A} have the weak intersection property. Since X is an £lpredual and since the centres of the balls are in a compact set, by [14, Proposition 4.4J, naEABx [a, IlyoallJ ¥ 0. If A has two points, observe that two balls intersect if and only if the distance between the centres is less than or equal to the sum of the radii, it is independent of the ambient space. 0 COROLLARY 4.3. Every LIpredual has the /CIP and hence also the FIP.
Y
~
PROPOSITION 4.4. Suppose X is an LI predual space. Then for a subspace X, the following are equivalent : (a) Y is an ideal in X (b) Y is a /CCsubspace of X (c) Y is a FCsubspace of X (d) Y is an almost FCsubspace of X (e) Y itself is an LI predual
PROOF. (e) ::::} (b) follows from Theorem 4.2 and (e) ::::} (a) follows from [16, Proposition IJ. And clearly, (b) ::::} (c) ::::} (d) and (a) ::::} (d). (d) ::::} (e). This again is an easy adaptation of the proof of [17, Theorem 1, 2 ::::} 3J. We omit the details. 0 The analog of Theorem 4.2 for 'PCsubspaces involves 'PIspaces. DEFINITION 4.5. Recall that a Banach space is a 'PIspace if it is 1complemented in every superspace. THEOREM 4.6. For a Banach space X, the following are equivalent: (a) X is a 'PI space ( b) X is icomplemented in every dual space that contains it ( c) X is a 'P C subspace of every superspace (d) X is a 'P C subspace of every dual space that contains it ( e) X is isometric to C (K) for some extremally disconnected compact Hausdorff space K. PROOF. (a) {::} (b) and (c) {::} (d) follow as in the first paragraph of Theorem 4.2. And clearly, (a) ::::} (c). (d) ::::} (a). By Proposition 2.18 and Theorem 4.2, (d) implies X is an LI_ predual with 'PIP. Recall that [12, Theorem 3.8J a Banach space X is a 'PIspace if and only if every pairwise intersecting family of closed balls in X intersects. And that X is a LIpredual if and only if X** is a 'PIspace. Now given a pairwise intersecting family of closed balls in X, since X** is a 'PIspace, they intersect in X**. And since X has 'PIP, they intersect in X too. (a) {::} (e) is also observed in [12, Section 11J. 0
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PROPOSITION 4.7. Let A be a family of subsets of X such that F ~ A. Then, the following are equivalent : (a) X is an Ll predual with AIP (b) X is an ACsubspace of every superspace (c) for every A E A, every pairwise intersecting family of closed balls in X with centres in A intersects. PROOF. (a) =? (b). Since X has the AIP, it is an ACsubspace of every superspace in which it is an ideal (Proposition 2.20) and since X is an L1predual, it is an ideal in every superspace [16, Proposition 1]. Thus (b) follows. (b) =? (a). Since F ~ A, this is immediate. (a) =? (c). This is similar too the proof of Theorem 4.6 (d) =? (a). (c) =? (a). If every finite family of pairwise intersecting closed balls in X intersects, then X is an L1predual. And that X has the AIP follows from Propo0 sition 2.18 (c). Let C(T, X) be the space of all Xvalued bounded continuous functions on a topological space T equipped with the sup norm. We now characterize when C(T, X) is a real L1predual. First we need the following lemma. LEMMA 4.8. Suppose Y is a subspace of a Banach space X and Y is a real L1predual. Let A ~ Y be a compact set and r : A 7 lR,+ be such that naEABx[a,r(a)] =I 0. Let y E naEABy[a,r(a) + c] for some c > O. Then there exists z E naEABy [a, r(a)] such that Ily  zll S c. PROOF. Since naEABx [a, r(a)] =I 0, and intersection of intervals is an interval, for any y* E B(Y*), naEABJR[y*(a), r(a)] =I 0 and is a closed interval. As y*(y) E naEABJR[y*(a),r(a) + c] for any y* E B(Y*), the family {By[y,c].By[a,r(a)] : a E A} is a weakly intersecting family of balls in Y. Since Y is a L1predual,
By[y,C]nnaEABy[a,r(a)] =10.
PROPOSITION 4.9. A Banach space X is a real Ll predual if and only if for each paracompact space T, C(T, X) is a real L1predual. PROOF. Since X is Icomplemented in C(T, X), hence a KCsubspace, by Proposition 4.4, if C(T, X) is an L1predual, then so is X. Conversely, suppose X is a real L1predual. Let Z = C(T, X), {!I, 12,···, fn} ~ Z and rl, r2,···, rn > 0 be such that the family {Bz[fi, ri] : i = 1, ... , n} intersects weakly. Then for each t E T, the family {B x [fi (t), r i] : i = 1, ... , n} intersects weakly, and since X is a real L Ipredual, they intersect in X. Consider the multivalued map F: T 7 X given by F(t) = nf=lBx[Ji(t),ri]. Note for each t, F(t) is a nonempty closed convex subset of X. CLAIM : F is lower semicontinuous, that is, for each U open in X, the set V = {t E T: F(t) n U =I 0} is open in T. Let to E V. Let Xo E F(to) n U. Let c > 0 be such that lI:r  xoll < c implies x E U. Let W be an open subset of to such that t E W implies 111;(t) fi(to) II < c/2 for all i = 1, ... , n. We will show that W ~ V. Let t E W. Then for any i = 1, ... , n, Ilxu  fi(t)11 S Ilxo  f.i(tO) II + 111;(to) fi(t)11 Sri + c/2. Therefore, Xo E nf=lBx[fi(t),rj + c/2]. By Lemma 4.8, there exists z E F(t) = nf=lBx[fi(t), ri] such that Ilxozll S c/2 < c. Then z E F(t)nU, and hence, t E V. This completes the proof of the claim.
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Now since T is paracompact, by l\Iichael's selection theorem, there exists 9 E Z such that g(t) E F(t) for all t E T. It follows that 9 E ni'=l Bz[f;, ril, 0 REMARK 4.10. For T compact Hausdorff, this result follows from [13, Corollary 2, p 43]. But our proof is simpler.
5. Stability Results In this section we consider some stability results. With a proof similar to [3, Proposition 14], we first observl' that PROPOSITION 5.1. KIP is a separably determined property. i.e .. if every sepamble subspace of a Banach space X have KIP. then X also has KIP. DEFINITION 5.2. [10] A subspace Y of a Banach space X is called a semiLsummand if there exists a (nonlinear) projection P : X > Y such that P(>..x
+ Py) Ilxll
+ Py, and IIPxl1 + Ilx  VI'II
>"P.r
for all x, y EX. >.. scalar. In [3]. it was shown that semiL snmmands are .1'Csubspaces. Basically the same proof actually shows that PROPOSITION 5.3. A semiLsummand is an ACsubspace for any A. Our next result concerns proximinal subspaces. DEFINITION 5.4. A subspace Z of a Banach space X is called proximinal if for every :r E X, there exists Zo E Z such that Ilx  zoll = d(J;, Z) = infzEz Ilx  zll· The map PZ(J:) = {zo E Z : Ilx  zoll = infzEz lI:r  zll} is called the metric projection. PROPOSITION 5.5. Let Z ~ Y ~ X, Z proximinal in X. (a) Let A be a family of subsets of Y / Z. Let A' be a family of subsets of Y s'ltch that for any :1; E X and A E A. there exists A' E A' such that for any a + Z E A. {a + Pz(x  a)} n A' i= 0. Suppose Y is a A'C subspace of X. Then Y/Z 'is a ACsubspace of X/Z. Let S be any of the families .1', l3 or P. (b) IfY is a S(Y)Csubspace of X. then Y/Z is a S(Y/Z)C8'ubspace of X/Z. (c) Suppose the metric projection has a continuous selection. Then. if Y is a K(Y)Csubspace of X, Y/Z is a K(Y/Z)Csubspace of X/Z. (d) Let Z ~ Y ~ X*, Z w*closed in X*. 11' Y is a S(Y)Csubspace of X*, then Y/Z is a S(Y/Z)Csubspace of X* /Z. and hence. has the S(Y/Z)IP. (e) Let X have the S(X)IP. Let l'd ~ X be a reflexive subspace. Then X/M has the S(X/l'd)IP. PROOF. (a). Let A E A and x + Z E X/Z. Choose A' as above. Then, for a + Z E A, there exists z E Pz(x  a) (depending on .1: and a) such that a + zEA'. Since Y is a A' Csubspace of X, there exists Yo E Y snch that Ilyo  a  zll ~ Ilx  a  zll for all a + Z E A. Clearly then Ilyo  a + ZII ~ Ilyo  a  zll ~ 11:1;  a  zll = IIx  a + ZII·
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If S is the family under consideration in (b) and (c) above and A = S(Y/Z), then for any choice of A' as above, S(Y) ~ A'. Hence, (b) and (c) follows from (a). For (d), we simply observe that any w*closed subspace of a dual space is proximinal. And (e) follows from (d). 0 As in [3, Corollary 4.6], we observe PROPOSITION 5.6. Let Z ~ Y ~ X, Z proximinal in Y and Y is a semiLsummand in X. Then Y/Z is a PCsubspace of X/Z. Let us now consider the Co or fp sums. THEOREM 5.7. Let r be an index set. For all 0: E r, let Ya: be a subspace of Xa:. Let X and Y denote resp. the Co or fp (1 :::; p :::; 00) sum of Xa: 's and Ya: 'so (a) For each 0: E r, let Aa: be a family of subsets ofYa: such that {O} E Aa: and for any A E Aa:. there exists B E Aa: such that Au {O} ~ B. Let A be a family of subsets of Y such that for any 0: E r, the 0:section of any A E A belongs to Aa:. Then Y is an ACsubspace of X if and only if for each 0: E r, Ya: is an Aa:Csubspace of Xa:. Let S be any of the families F, 1C, B or P. (b) Y is a S(Y)Csubspace of X if and only if for any 0: E r, Ya: is a S(Ya:)Csubspace of Xa:. (c) The SIP is stable under fpsums (1 :::; p:::; 00). PROOF. (a). The proof is very similar that of to [3, Theorem 4.7]. We omit the details. (c). Xa: has SIP if and only if Xa: is a SCsubspace of some dual space Y';. Now the fpsum (1 :::; P :::; 00) of Y';'s is a dual space. 0 REMARK 5.8. The result for FIP has already been noted by [18] with a much different proof. The stability of the PIP under f1sums is noted in [15] again with a different proof. [18] also notes that FIP is stable under cosum. And Corollary 4.3 shows that Co has the ICIP. However, we do not know if the ICIP is stable under Cosums. As for the BIP or PIP, we now show that cosum of any infinite family of Banach spaces lacks the BIP, and therefore, also the PIP. This is quite similar to Example 2.9. PROPOSITION 5.9. Let r be an infinite index set. For any family of Banach spaces Xa:, 0: E r, X = fficoXa: lacks the BIP. PROOF. For each
0: E
r, let Xa:
be an unit vector in Xa: and define ea: E X by if f3 = 0: otherwise
Then the set A = {ea: : 0: E r} is bounded and the balls Bx[ea:, 1/2] intersect at the point (1/2xa:) E X**, but the balls Bx[ea:, 1/2] cannot intersect in X. 0 REMARK 5.10. As before, taking AU{O}, it follows that X lacks the AIP even for A = weakly compact sets. Coming to function spaces, we note the following general result.
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BANDYOPADHYAY AND DUTTA
PRDPOSITION 5.11. Let Y be a subspace of a Banach space X and A be a family of subsets of Y. (a) For any topological space T, if C(T, Y) is a ACsubspace of C(T, X), then Y is a AC subspace of X. Moreover, if C(T, X) has AIP, X has AIP. (b) Let (n, L, fl) be a probability space. If for some 1 ~ P < 00, LP(fl, Y) is aACsubspace of LP(fl, X), then Y is aACsubspace ofX. Moreover, if LP(fl, X), has AIP, then X has AIP. PROOF. For (a) and (b), let F(X) denote the corresponding space of functions and identify X with the constant functions. In (a), point evaluation and in (b), integral over n gives us a norm 1 projection from F(X) onto X. Thus X inherits AIP from F(X). Now suppose F(Y) is an ACsubspace of F(X). Let P : F(Y) + Y be the above norm 1 projection. Let x E X and A E A. Then, there exists g E F(Y) such that IIgali ~ IIxall for all a E A. Let y = Pg. Then, lIyall ~ IIgall ~ IIxall for all a E A. 0 The following Proposition was proved in [3]. PROPOSITION 5.12. (a) Let X has Radon Nikodym Property and is 1complemented in Z* for some Banach space Z. Then for 1 < p < 00, LP(IL,X) is 1complemented in U(fl, Y)* (l/p+ l/q = 1), and hence has the 'PIP. (b) Suppose X is separable and 1complemented in X** by a projection P that is w*w universally measurable. Then for 1 ~ P < 00 LP(fl, X) is 1complemented in Lq(fl,X*)* (l/p+ l/q = 1), and hence has the 'PIP. Since the BIP or 'PIP is inherited by Icomplemented subspaces and the BIP, the next result follows essentially from the arguments of [17].
Co
lacks
PROPOSITION 5.13. (a) Let X be a Banach space containing Co and let Y be any infinite dimensional Banach space. Then X 181" Y fails the BIP and 'PIP. (b) If C(K, X) has the BIP, then either K is finite or X is finite dimensional. C(K,X) has the 'PIP if and only if either (i) K is finite and X has the 'PIP or (ii) X is finite dimensional and K is extremally disconnected. (c) For· any nonatomic measure space (n,L,/L) and a Banach space X containing co, £1 (fl, X) fails the BIP. In the next Proposition, we prove a partial converse of Proposition 5.11(a) when Y is finite dimensional and K is compact and extremally disconnected. PROPOSITION 5.14. LetS be any ofthefamiliesF, IC, B or'P. LetY be a finite dimensional a S(Y)Csubspace of a Banach space X. Then for any extremally disconnected compact space K, C(K, Y) is a S(C(K, Y))Csubspace of C(K, X). PROOF. We argue similar to the proof of [3, Proposition 4.11]. Let K be homeomorphically embedded in the StoneCech compactification .B(r) of a discrete set r and let ¢ : .B(r) + K be a continuous retract. Let A E S(C(K, Y)) and g E C(K, X). Note that since Y is finite dimensional, by the defining property of .B(r), any Yvalued bounded function on r has a norm preserving extension
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in C(,B(r), Y). Thus C(,B(r), Y) can be identified with EBeoc(r) Y. Lift A to this space. In view of Theorem 5.7, this space is S(Y)Csubspace of EBooX. This latter space contains C(,B(r) , X). Thus by composing the functions with 4>, we get a f E C(K, Y) such that Ilf  hll ~ Ilg  hll for all h E A. Hence the result. 0 And now for a partial converse of Proposition 5.11(b). THEOREM 5.15. Let Y be a separable subspace of x. IfY is a PCsubspace of X, then for any standard Borel space 0 and any afinite measure jl, Lp(jl, Y) is a PCsubspace of Lp(jl,X). PROOF. Let f E Lp(jl, X). Since Y is a PCsubspace of X, for each x E X, nyEyBy[y, IIx  yll] =1= 0. Define a multivalued map F : 0 + Y, by
F(t) = {
nyEy
By[y, Ilf(t)
{f(t)}
 ylll
if
f(t) EX \ Y
if
f(t) E Y
Let G = {(t, z) : z E F(t)} be the graph of F. Claim: G is a measurable subset of 0 x Y. To establish the claim, we show that GC is measurable. Since Y is separable, let {Yn} be a countable dense set in Y. Observe that z ~ F(t) if and only if either f(t) E Y and z =1= f(t) or f(t) E X \ Y and there exists Yn such that liz  Ynll > IIf(t)  Ynll· And hence,
G C = {f(t) E Y and z
=1=
f(t)} u
U {f(t) EX \ Y and liz  Ynll > Ilf(t) 
Ynll}
n;:::l
is a measurable set. By von Neumann selection theorem, there is a measurable function 9 : 0 + Y such that (t,g(t)) E G for almost all tEO. Observe that Ilg(t)11 ~ IIf(t)1I for almost all t. Hence 9 E Lp(jl, Y). Also for any h E Lp(jl, Y) we have Ilg(t)  h(t)11 ~ Ilf(t)  h(t)1I for almost all t. Thus, Ilg  hll p ~ IIf  hll p for all h E Lp(jl, Y). 0 QUESTION 5.16. Suppose Y is a separable lCCsubspace of X. Let (0, ~,jl) be a probability space. Is LP(jl, Y) a lCC subspace of LP(jl, X)? REMARK 5.17. This question was answered in positive in [3] for FCsubspaces and we did it for PCsubspaces. Both the proofs are applications of von Neumann selection Theorem. The problem here is for a compact set A in LP(jl, Y) and wE 0 the set {f(w) : f E A} need not be compact in Y. ACKNOWLEDGEMENTS. Partially supported by a DSTNSF grant no. RP041/2000. The firstnamed author availed this grant to visit Southern Illinois University at Edwardsville, USA in MayJune 2002 and attended the Fourth Conference on Function Spaces, where he presented a talk based on this work. He would like to thank Professor K. Jarosz for the warm hospitality and a wonderful conference. We also thank the referee for suggestions that improved the paper.
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References [1] P. Bandyopadhayay, S. Basu, S. Dutta and B. L. Lin Very nonconstmined subspaces of Banach spaces, Preprint 2002. [2] P. Bandyopadhayay and S. Dutta, Almost constmined subspaces of Banach spaces, Preprint 2002. [3] Pradipta Bandyopadhyay and T. S. S. R. K. Rao, Centml subspaces of Banach spaces, J. Approx. Theory, 103 (2000),206222. [4] B. Beuzamy and B. Maurey, Points minimaux et ensembles optimaux dans les espaces de Banach, J. Functional Analysis, 24 (1977), 107139. [5] J. Diestel, Geometry of Banach Spaces, selected topics, Lecture notes in Mathematics, Vo1.485, SpringerVerlag (1975). [6] J. Diestel and J. J. Uhl, Jr., Vector' measures, Mathematical Surveys, No. 15, Amer. Math. Soc., Providence, R. 1. (1977). [7] G. Godefroy, Existence and uniqueness of isometric preduals: a survey, Banach space theory (Iowa City, lA, 1987), 131193, Contemp. Math., 85, Amer. Math. Soc., Providence, RI, 1989. [8] G. Godefroy and N. J. Kalton, The ball topology and its applications, Banach space theory (Iowa City, lA, 1987), 195237, Contemp. Math., 85, Amer. Math. Soc., Providence, RI, 1989. [9] G. Godini On minimal points, Comment. Math. Univ. Carotin., 21 (1980), 407419. [10] P. Harmand D. Werner and W. Werner, Mideals in Banach spaces and Banach algebras, Lecture Notes in Mathematics, 1547, SpringerVerlag, Berlin, 1993. [11] O. Hustad, Intersection properties of balls in complex Banach spaces whose duals are L1 spaces, Acta Math., 132 (1974), 283313. [12] H. E. Lacey, Isometric theory of classical Banach spaces, Die Grundlehren der mathematischen Wissenschaften, Band 208, SpringerVerlag, New YorkHeidelberg, 1974. [13] J. Lindenstrauss, Extension of compact opemtors, Mem. Amer. Math. Soc., No. 48, 1964. [14] A. Lima, Complex Banach spaces whose duals are L1spaces, Israel J. Math., 24 (1976), 5972. [15] T. S. S. R. K. Rao, Intersection properties of balls in tensor products of some Banach spaces II, Indian J. Pure Appl. Math., 21 (1990),275284. [16] T. S. S. R. K. Rao, On ideals in Banach spaces, Rocky Mountain J. Math., 31 (2001), 595609. [17] T. S. S. R. K. Rao Chebyshev centers and centmble sets, Proc. Amer. Math. Soc., 130 (2002), 25932598. [18] L. Vesely, Genemlized centers of finite sets in Banach spaces, Acta Math. Univ. Comen., 66 (1997),83115. (Pradipta Bandyopadhyay) STATMATH DIVISION, INDIAN STATISTICAL INSTITUTE, 203, B. T. ROAD, KOLKATA 700 108, INDIA, Email: pradiptaClisical.ac.in (S Dutta) STATMATH DIVISION, INDIAN STATISTICAL INSTITUTE, 203, B. T. ROAD, KOLKATA 700 108, INDIA, Email: sudipta.rGlisical.ac . in
Contemporary Mathematics Volume 328, 2003
The boundary of the unit ball in HItype spaces Paul Beneker and Jan Wiegerinck ABSTRACT. This is a survey on the extreme, exposed and strongly exposed boundary points of the unit baH in various Hltype spaces.
1. Introduction
Theorems like the KreinMilman theorem and Phelps' theorem assert the existence of many extreme points in certain convex sets. However, to decide whether a particular point in the boundary of a convex set is extreme, exposed or strongly exposed is quite a different question. In this paper we survey what is known in the very special situation of the unit ball of certain Hardy spaces. Of course we refer to the literature for the majority of the proofs. But we have included some, and sketched others, hoping that this will clarify the exposition. After introducing extreme, exposed and strongly exposed points, we end the introduction with definitions of the Hardyspaces in which we study our problem. These are: the standard Hl(]I))) of the disc ]I)) in e, HI(0.) of a domain of finite connectivity 0. c e, the Bergman space Al(]I))) of the disc and Hl(lB n ) of the unit ball in en. In Section 2 we will study the boundary points of the unit ball of HI (]I))). Here good function theoretic descriptions of extreme and strongly exposed points are possible. Exposed points seem to escape such a description. The ball of HI (0.) is the topic of Section 3. We will see that results for the disc in general do not extend to this case. In Section 4 we will turn to the Bergman space. Now all boundary points of the unit ball are exposed and many wellbehaved functions, like polynomials, are strongly exposed. However, there are boundary points that are not strongly exposed. The final section is devoted to the little we know about the higher dimensional situation. The paper borrows heavily from the expository parts of the PhDthesis of the first author, [3]. 1.1. Generalities. Let X be a Banach space with (only in this section) real dual space X* and let A be a bounded closed subset of X. We recall some notions that are connected with convexity properties at a boundary point of A. Research of the first author was supported by the Dutch research organization NWO.
© 59
2003 American Mathematical Society
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DEFINITION 1.1. A point x E A is called extreme if x = tp + (1  t)q for some p, q E A and 0 < t < 1 implies that p = q = x. A point x E A is called exposed if there exists L E X* such that Lx = 1 while Ly < 1 for every yEA \ {x}. The functional L is called an exposing functional for x.
For L E X* we set
M(L, A) = sup{Lx : x E A}
(1.1) and (1.2) If IILII
(1.3)
1\1(A) = sup{llxll : x
= 1 and a > 0 we
E
A}.
define a slice of A as
S(L,a,A) = {x E A: Lx> 1\1(L, A)  a}.
Let B be the closed unit ball of X. DEFINITION 1.2. A subset A of X is called dentable if for every c > 0 there is a point x E A such that x is not in the normclosed convex hull of A \ (x + cB). Equivalently, A is dent able if for every c > 0 there is a slice S(L, a, A) of diameter less than c. DEFINITION 1.3. A point x E A c X is called a denting point if for every c > 0 there exists a slice S(L, a, A) of diameter less than c that contains x. A point :1: E A c X is called a strongly exposed point if there exists L E X* such that for every c > 0 there exists a > 0 such that the slice S(L, a, A) contains x and has diameter less than c. In other words, there exists L E X* such that Lx = 1\1(L, A) and if LX n + 1\1(L, A) for {x n } C A, then Xn + x in the norm of X.
With these definitions at hand we make some observations. A strongly exposed point in A is exposed. Normalizing the functional L in the previous definition will yield an exposing functional for x. An exposed point of A is extreme and an extreme point of A belongs to the boundary of A. A theorem of Phelps, [38], states that if every bounded subset of X is dentable, then every bounded closed convex set is the convex hull of its strongly exposed points. It is known that separable dual spaces have the property that every bounded set is dentable, cf. [10], Chapter 6. However, neither Phelps' theorem nor its proof gives us any information about, say, which of the exposed points of a convex set C are strongly exposed points. In the rest of the paper it will be more convenient to use an equivalent, more analytic definition of strongly exposed point: An exposed point x of the set A C X with exposing functional L is strongly exposed if and only if every sequence (xn)in A with the property that LX n + 1 converges strongly to x. Moreover, we will work over C, so that the definition of exposed point becomes slightly adapted. A point x E A is an exposed point of A, if there exists a (complex) functional L on X with the following property: Lx = 1 and for all yEA, Y =1= x ~ Re L(y) < 1. (For A E C, Re A denotes the real part of A.) Again we say that the functional L is an exposing fUIlctional for x, or simply that L exposes the point x.
THE BOUNDARY OF THE UNIT BALL IN HI_TYPE SPACES
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1.2. Hardy spaces. Next we briefly recall the definition and some properties of Hardy spaces. Excellent introductions are [12, 16, 24], and for domains of finite connectivity [13]. We will denote the space of holomorphic functions on a domain D c en by H(D). Let !Dl be the unit disc in e and 'JI' its boundary. DEFINITION 1.4. Let 0 < p < 00. The Hardy space HP = HP(!Dl) consists of all holomorphic functions I on !Dl for which 11111itv:= sup
OCrCl
1
2~
dO II(re i9 W  <
2w
00.
The Hardy space HOC(!Dl) consists of the bounded functions in H(!Dl). It is a closed subspace of LOC(!Dl) under the supnorm 11.lIoc. If IE HP(!Dl), then
(1.4) exists for almost allO. The function 1* belongs to U('JI'). For alII::; p::; 00, II.IIHP is a norm, IIIIIHP = 111*IILp and HP(!Dl) becomes a Banach space that is a subspace of U('JI'). We will simply write 1IIIIp instead of 11111Hv. Alternatively, the Hardy space HP(!Dl) (1::; P < 00) may be described by
HP(!Dl) = {f E H(!Dl) : IIIP admits a harmonic majorant} = closure of holomorphic polynomials in £P ('JI'). The value at 0 of the least harmonic majorant of IIIP,
uf(z) = inf{h(z) : h harmonic and h> IIIP} equals the norm, 1IIIIp = u(O?/p. Equivalent norms are given by (uf(z)?/P, (z E
!Dl) . These descriptions are useful for defining HPtype spaces on a domain n in
e:
= {f E H(n) : IIIP admits a harmonic majorant}.
HP(n)
Norms, all equivalent, can be defined via the least harmonic majorant uf by (Uf(Z))l/P, (z En). If n has smooth boundary an then HP(n) is again a closed subspace of U(an). Similarly on the unit ball ~n of (n ~ 2):
en
HP(~n)
= closure of holomorphic polynomials in
LP(a~n).
We end this section by defining the Bergman space AP(!Dl). Let 1 ::; p < IE H(!Dl) is such that
00.
If
1111I~p:= i'D[11(zW dxdy < 00, W then we say that I belongs to the Bergman space AP(!Dl). In other words, AP(!Dl) = H(!Dl) n U(!Dl). It is a closed subspace of U(!Dl). One motivation for looking at the Bergman space Al(!Dl) is that
Al(!Dl) ~ {f E Hl(~2): I(Zl,Z2) = I(Zl'O)}. Therefore B(Al(!Dl)) may serve as a testing ground for B(H1(!Dl)).
PAUL BENEKER AND JAN WIEGERINCK
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2. The boundary of unit ball in H1(]]))) 2.1. Extreme points. The subject started with a famous theorem of deLeeuw and Rudin, [26], that characterizes the extreme points of B(H 1 (]])))). It will be presented below. Recall the innerouter decomposition for f in H1(]]))), cf. [16]:
f(z) = I(z)U(z). Here I E H(]]))) is an inner function, that is, I E HOC(]]))) with 1I*(ei9 )1 = 1 a.e. and U E H(]]))) is an outer function, i.e. log IUI(z) = P[log IUI](z), where P is the Poisson operator:
Pu z = [ ]()
1
211"
1  Izl2 1  2Re (ze i9 )
+ Izl2 u (e
.
t9
d() . ) 27r
The inner factor thus contains all the zeros of f. In fact, every inner factor is of the form
I(z) = B(z)S(z),
(2.1)
with B a Blaschke product and S is the singular factor (2.2) where J.L is a positive, singular measure on 'JI' and  denotes harmonic conjugation. EXAMPLES 2.1. Outer functions are zero free, and e.g. a holomorphic function on]])) that is continuous and away from 0 on iij is outer. Taking a root and applying a limit argument shows then that f E HI (]]))) that assumes values only in C \ (00,0) is outer too. EXAMPLE 2.2. If I is an inner function then (1
+ 1)2
is outer.
DeLeeuw and Rudin proved THEOREM 2.3 ([26]). A function f is an extreme point of the unit ball of HI (]]))) if and only if f is an outer function and IIfl11 = 1. PROOF. Suppose f is an outer function of unit norm and suppose g in H1 is such that IIf + gill = Ilf  gill = 1. Let dJ.L be the probability measure If I on the unit circle. Then, with k = gl f on 'JI', the relations Ilf + gill = Ilf  gill = 1 imply that J;11" 11 + kl + 11  kldJ.L = 2. Because for all z E C: 11 + zl + 11  zl 2 2, with equality if and only if z E [1,1], we conclude that for almost dJ.Levery ~ E 'JI' : g(~)1 f(O E [1,1]. Observe that this inclusion also holds for almost d()every ~ E 'JI', because ~ = If I i: 0 almost d()everywhere on 'JI'. In particular, Igl :::; If I d() a.e. on 'JI'. By the fact that f is outer, then also for all zED: Ig(z)1 :::; If(z)l. Hence gI f E HOC has real boundary values on 'JI'. The Poisson integral representation of glf yields that glf is constant. Finally, because Ilf + gill = 1, this constant is zero, i.e., g == 0, so f is extreme. In the other direction, suppose f = I· F is of unit norm, but I is a nontrivial inner function. Because for every ei9 E 'JI' \ {±1}:
g!
±(1 ± ei9 )2 e
.
':.,,::'  = 2 ± 2Re (e t9 ) = 2 ± 2 cos( ()) > 0, t17
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the functions I and (1 ± I)2 have the same argument a.e. on'lI'. Let 9 = We have III ± gill =
dB
l±t .F.
dB
h[2'" 1F1(1 ± Re (I)) 2w = 1± Re (h[2'" IFI' I ~ ).
Now if we replace I by AI throughout the preceding, where A E 'lI' is arbitrary, then we obtain: 2", dB II/±glll=I±Re(A IFI·I). o 2w
1
Therefore, we can choose A in such a way that Re (A J:'" IFI . I quently, III ± gill = 1, but 9 "¥= 0, so I is not extreme.
g!)
=
o.
ConseD
EXAMPLE 2.4. Normalized polynomials without zeros in j[)) are extreme points. 2.2. Exposed points in B(Hl(j[)))), function theoretic methods. As far as the authors know, there is no clearcut function theoretic characterization of exposed points of B«Hl(j[)))). The necessary and sufficient condition of Helson, [21] given below in Theorem 2.8 probably comes closest. The following lemma identifies the exposing functionals. LEMMA 2.5.
II I
is exposed in B(Hl(j[)))), then Lf:
9
~
J7 9
dB UT2w
is the unique exposing functional. PROOF. Suppose I is an exposed point of B(HI(j[)))) with exposing functional L. By the HahnBanach theorem there exists a function cp E V"', Ilcplloo=l, which represents the action of L: [21< dB L(g) = gcp 2w'
io
Because L(f) = 11/111 = 1, and because the function I has mass (almost) everywhere on 'lI', there is only one cp that has the desired properties, namely cp = 711/1 = exp( i arg(f)). In particular we see that the exposing functional for I is unique and of the above form. D DEFINITION 2.6. A function I E HI (j[))) is called rigid if 9 E HI (j[))) and arg 9 = arg I implies that 9 = cl for some c > O. An HIfunction is rigid if its argument on 'lI' determines the function up to a multiplicative constant. By the previous lemma exposed points are precisely the rigid functions of norm 1. Not all outer functions are rigid as the following example demonstrates. EXAMPLE 2.7. By the DeLeeuwRudin theorem (Theorem 2.3), no HIfunction with nontrivial inner factor is rigid. Inspection of the proof of the theorem explicitly gives another HIfunction with the same argument: if I = I· F with F outer, then 9 = (1 + 1)2. F is an outer function, cf. Example 2.2 that has the same argument as f. This argument can be reversed: suppose I "¥= 0 is divisible in HI by the outer function (1 + I)2, where I is any nontrivial inner function, then I
PAUL BENEKER AND JAN WIEGERINCK
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and I I / (1 + I) 2 have the same argument, so nontrivial inner functions I we have (2.3)
I
If I is rigid, then 1/(1
is not rigid. In other words, for all
+ 1)2 rt HI
Nakazi [28, 29] conjectured that an outer function with fI(l + cz)2 rt HI for every c E 'JI' must be rigid. E. Hayashi [17] gave a counter example. Later Sarason [44, 45] conjectured that (2.3) characterizes rigid functions. Inoue [22] subsequently gave an example of a nonrigid function I E HI with the property that 1/(1 + 1)2 rt HI for all nontrivial inner functions I. Helson showed that a variant of (2.3) indeed characterizes rigid functions; THEOREM 2.8 ([21]). Let I be an outer lunction in HI. Then I is rigid il and only il lor all inner lunctions p and q, not both constants, I/(p + q)2 rt HI. The drawback of Helson's theorem is that his condition is difficult to check in practice. More user friendly are the following results of Yabuta. THEOREM 2.9. (1) II I E HI is invertible in HI, then I is rigid. [51] (2) Suppose ther'e exists h E Hoo \ {O} such that Re(h!) 2: 0 a.e. on 'JI'. Then I is rigid. [52] The proof uses a result of Neuwirth and Newman [36], that is interesting in its own right. LEMMA 2.10 ([20],[36]). Every lunction I E H 1 / 2 that is positive a.e. on'JI' is constant. Nakazi [29] gives another proof of Yabuta's results. EXAMPLE 2.11 ([23]). Normalized polynomials P without zeros on II)) and at most simple zeros on 'JI' are exposed. Indeed, the argument of such a function makes jumps of height 7r at the zeros in 'JI', hence by adding a smooth function
+iv> yields a function with positive real part and the second of Yabuta's criteria applies. 2.3. Strongly exposed points. Whereas it seems impossible to give a characterization of exposed points that allows to check if a given function in aB(HI (II)))) is an exposed point, it is possible to characterize the strongly exposed points. This is the content of work of Nakazi and, independently, the first author. Let us describe strongly exposed points in terms of properties of the exposing functional. THEOREM 2.12 ([2,4]). II alunction IE 8B(H 1 (1I)))) is strongly exposed, then d( Tfr, HOO) < 1. (d = Loodistance). PROOF. Suppose that the LOOdistance of
g!
The space HOO + G('JI') that appears in the next theorem, has been studied extensively, see [42] and cf. [16]. It is a closed subalgebra of LOO('JI').
THE BOUNDARY OF THE UNIT BALL IN HI_TYPE SPACES THEOREM
2.13 ([47]). Let IE B(Hl(]])))) be exposed. Ild(
65
&!. Hoo+C(1I')) < 1,
then I is strongly exposed. We will explain how the proof goes if 'P := 111/1 E Hoo + C(1I'). The exposing 7r functional L for I is defined by L(g) = g'Pg!. Suppose that (fn)f is a sequence in the unit ball of Hl such that L(fn) ) 1 as n ) 00. Clearly this implies that II In I 1 > 1. We claim that there is a subsequence (fnk) that converges to I in HI. By weak* compactness of the unit ball of (C(1I'))* the sequence of measures Ing! has a weak* convergent subsequence Ink g! with limit dJ.L({}) , and IIJ.LII :::; 1. For all r 27r em . (J n = 1,2,3, ... , we have Jo dJ.L({}) = O. Now the F. & M. Riesz theorem [16] precisely states that J.L is of the form dJ.L( (}) = F g! for some F in the unit ball of HI . Write 'P = 'Pl + 'P2 with 'Pl E H[J = zHoo and 'P2 E C. Because H[J annihilates HI, the weak* convergence of Ink g! implies that
fg
We conclude that F = I because I is exposed. Again, by weak* convergence Ink converges pointwise to I. The following result of D.J. Newman ensures that the convergence is in Hl with limit I. The argument shows that every subsequence of (fn) has a subsubsequence that converges in HI to I, and therefore In ) I in HI. 2.14 ([35]). Let (fn)i be a sequence 01 functions in the unit ball Suppose In converyes pointwise to I E Hl on ]])) and I In I 1 > 11/111 as Then In converges to I in HInorm.
PROPOSITION
01 Hl. n
> 00.
The method for the general case in [47] is to work in the maximal ideal space of LOO(1I') and use the HelsonLowdenschlager generalization of the F. & M. Riesz theorem. Cf. [30] for a related approach. 2.4. Strong exposedness and HelsonSzego weights. In this section we will explicitly describe the strongly exposed points in the unit ball of HI: they are the outer functions induced by socalled HelsonSzego weights on the unit circle. We will discuss three different (albeit related) roads to this result. The first one quickly establishes that strongly exposed points are induced by HelsonSzego weights and uses little background on exposed points and Hardy space theory. The second one uses function theory to show that HelsonSzego weights give rise to strongly exposed points and is essentially due to T. Nakazi [31]. The third approach (to be discussed in Section 2.5) uses operator theory and the relation between exposed points and Toeplitz operators [2] to prove that strongly exposed points and HelsonSzego weights are "the same". These last two proofs use the characterization of strong exposedness obtained in the previous section (Theorem 2.13). To put the results in perspective, we will mention some surprising properties of strongly exposed points. Let J.L be a finite (positive) measure on 1I'. Let P be the collection of polynomials in z and let Q be the space of polynomials in z vanishing at the origin. On the unit
PAUL BENEKER AND JAN WIEGERINCK
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circle P and Q consist of the trigonometric polynomials of the form Ln>o ane inlJ and Ln
p:= sup I iPqd/Ll, where the supremum is taken over all pEP and q E Q restricted by Ilpll£2(/J) ~ 1 and Ilqll£2(/J) ~ 1. By CauchySchwarz, 0 ~ P ~ 1. We see that the spaces P and Q are orthogonal if and only if P = O. On the other hand, when the L 2 (/L)closures P of P and Q of Q have a nontrivial intersection, then clearly p = 1. The size of the number p is related to the question when the sum P + Q is closed in L 2 (/L). Namely, when p < 1, then for all p E P, q E Q in L 2 (/L)norm, lip + ql12 ~ IIpl12
+ IIqll2 
2p· Ilpll· Ilqll ~ (1 p)(llpl12
+ IlqIl2),
which implies that P + Q is closed in When p < 1 we say that P and Q are at the positive angle cosI(p) > O. When the spaces P and Q are at positive angle, the projection
L 2 (/L).
N
N
P+(Lane inlJ ) = LaneinlJ , N
L 2 (/L),
which is densely defined on extends to a bounded operator on L 2 (/L). Conversely, if the intersection of P and Q in L 2 (/L) is trivial, then the definition of P+ acting on trigonometric polynomials is welldefined and extends to a bounded operator on L 2 (/L) if and only if P and Q are at positive angle. DEFINITION 2.15. We say that a function w ~ 0 on 'll' is a HelsonSzego weight (on 'll') ifthere exist real valued u,v E Loo('ll') with Ilvll oo < I such that w = eU+V. (Here v is the boundary function of the harmonic conjugate of the Poisson integral of v to JIll.) The following theorem of H. Helson & G. Szego elegantly describes all measures /L for which P and Q are at positive angle. THEOREM 2.16 ([19]). The subspaces P and Q are at positive angle in L 2 (/L) if and only if the measure /L is of the form d/L = wdO, for some HelsonSzego weight won'll'. COROLLARY 2.17. If the function f is strongly exposed in the unit ball of HI, then If I is a HelsonSzego weight on'll'. PROOF. Assume f is an exposed point, such that If I is not a HelsonSzego weight. We will show that f is not strongly exposed. Let /L be the probability By the theorem of Helson and Szego the spaces P and Q are at measure If I zero angle (p = 1). Thus we can find sequences (Pn) and (qn) in the L 2(/L)unit balls of P and Q respectively, such that
g!.
1
211'
(2.4)
Pnqn d/L
+
1,
as n + 00. For n = 1,2, ... , let fn be the HIfunction Pnqnf. These functions are contained in the unit ball of HI:
THE BOUNDARY OF THE UNIT BALL IN Hl_TYPE SPACES
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If we set
1
211"
L(g) =
dO g
Now (2.4) expresses that L(fn) ~ 1 as n ~ 00, yet the functions fn do not converge to f in HI because fn(O) = qn(O) = O. We conclude that f is not strongly exposed. D Already we can mention a surprising property of strongly exposed points. COROLLARY 2.18 ([2]). If the function f is strongly exposed in the unit ball of HI, then for all small c > 0, the functions f and 1/ f are contained in H1+ e . The corollary together with Theorem 2.13 show that if f E B(HI(][})) satisfies the distance condition in Theorem 2.13, then f is strongly exposed if and only if 1/ f E HI. (It is worth noting that there exist exposed points f E HI such that 1/ f is in no HP, p > 0, [37, 44].) Since f and l/f are outer functions, the corollary is immediate from the following lemma (which is actually an exercise in [16]). LEMMA 2.19. If the function w is a HelsonSzego weight on 'JI', then for all small c > 0, we have w E £1+ e ('JI') and l/w E L1+e('JI'). LEMMA 2.20. (Cf. [16], Lemma IV. 3. 3, p. 148.) If'lf; is a measurable real function, then the LOOdistance of ei.p to Hoo is less than 1 if and only if there exist c > 0 and h E H oo such that 7r (2.5) Ihl 2:: c and I'lf; + arghl $ "2  c (mod 27r)
almost everywhere on 'JI'. THEOREM 2.21 ([2, 31]). Let f be an extreme point of the unit ball of HI. Then f is strongly exposed if and only if If I is a HelsonSzego weight. PROOF. ([31]) By Corollary 2.17 we need only to prove the backward implication. Let us assume that the outer function f is such that If I is a HelsonSzego weight on 'JI', say, If I = exp(u + v), where Ilvll oo < l We remark that the function is exposed by Theorem 1 and Lemma 2.19. Next, again using the fact that f is outer, we notice that
f(z) = eu(z)+iu(z) . ev(z)iv(z),
because the right hand side is an outer function with the appropriate absolute values on 'JI'. We set
O. Recall Corollary 2.18: if f is strongly exposed then for all small c > 0, the functions p+e and 1/ f1+ e are contained in HI. This result can be sharpened somewhat using Theorem 2.21:
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PAUL BENEKER AND JAN WIEGERINCK
PROPOSITION 2.23 ([2]). Let f be a strongly exposed point in the unit ball of HI. Then for all sufficiently small c > 0, the (normalized) functions ce f1+E: and del f1+E: of unit norm are again strongly exposed in the unit ball of HI. The Proposition explains the following examples from [47]. EXAMPLES 2.24. A polynomial is rigid in HI if and only if its zeros on the unit circle are single zeros (and it is zerofree on llJ), obviously). Any normalized polynomial P with at least one single zero on '][' is not strongly exposed however, because 1/ P ¢ HI Let f#(z) be the extreme point c(1 + z) log2(1 + z) in the unit ball of HI. Because f# is outer and because 1/lf#1 E LI, we conclude that f# is exposed (Theorem 2.9. 1). However, 1/lf#1 ¢ L1+E:, so f# is not strongly exposed.
2.5. Toeplitz operators and De BrangesRovnyak spaces. Let P+ be the orthogonal projection of L2('][') onto H2: 00 00 P+(Lane inll ) = Lane inll , 00 0 and let P_ be the orthogonal projection of L2 onto (H2).l. 00 1 P_(Lane inll ) = Laneinll . 00 00
=
H~:
DEFINITION 2.25. Given a bounded function '¢ E LOC the Toeplitz operator T", is the bounded map T", : H2 _ H2 given by T",(f) = P+('¢f).
We say that '¢ is the defining function of the Toeplitz operator T",. We see that the norm of the Toeplitz operator T", is at most 1I,¢1I00. It is not difficult to show that the norm of T", is in fact equal to 1I'¢1I00, but we will not need this result. Also, it is a routine exercise to verify that the adjoint of T", is the Toeplitz operator Ttji. Clearly, if'¢ E Hoo, then T",(f) = '¢f. Combining these two observations we have the following result: LEMMA 2.26. If cp or'¢ is contained in H oo , then TqiT", = TVi",. Given a function '¢ E Loo, the Hankel operator H", (with defining function '¢) is the bounded operator H",(f)
= '¢f  T",(f) = (I  P+)(,¢f) = P_('¢f)
from H2 into (H2).l.. By the same reasoning the norm of H", is at most 1I,¢1I00. If two functions cp and '¢ in Loo differ by an element of H OO , then the associated Hankel operators coincide. Hence the operator norm of H", is at most Loodist('¢,HOO). The basic fact about Hankel operators, due to Z. Nehari, is that equality holds: THEOREM 2.27 ([34]). The operator norm of H", equals the Loo distance of '¢ to HOO.
THE BOUNDARY OF THE UNIT BALL IN
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The following result (and its corollaries) will be of great importance. We only have as reference Sarason's notes [43]. THEOREM 2.28. (DevinatzRabindranathan) If'IjJ is unimodular, then T1fJ is leftinvertible if and only if Loodist('IjJ,Hoo) < 1.
A bounded linear operator L : X + Y is said to be leftFredholm if the range of X under L is closed and of finite codimension in Y. For leftFredholm operators one has the following result. COROLLARY 2.29 ([11]). (DouglasSarason) If'IjJ is unimodular, then T1fJ is leftFredholm if and only if the Loo distance of'IjJ to HOO + C is less than 1. See [43], p. 119 ff. also for a proof of the next result COROLLARY 2.30 ([9, 49]). (DevinatzWidom) If'IjJ is unimodular, then T1fJ is invertible if and only if'IjJ can be written as ei(uH), where u and v are real functions in Loo such that IIvll oc < 'IT". We will now explain the relation between Toeplitz operators and rigidity (exposedness) of functions in HI. LEMMA 2.31. Let f be an outer function in HI. Then f is rigid if and only if the Toeplitz operator with defining function
{af, ag)M := (f, g)2.
This makes M(a) a Hilbert space, and the Toeplitz operator Ta : H2 + M(a) unitary. Clearly, when lIall oo ::; 1, the inclusion i : M(a) + H2 is a contraction.
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PAUL BENEKER AND JAN WIEGERINCK
Let b be a nonconstant function in the unit ball of Hoo. We will construct the space 1t(b), the socalled complementary space to M(b), in a similar fashion; the narile reflects the fact that M(b) + 1t(b) = H2 (although the intersection M(a) n 1t(b) is trivial only when b is an inner function). Observe that I  TbTj) is a positive contraction on H2, hence the operator (I  nTj»)l/2 is welldefined on H2 and contractive. As a linear space 1t(b) consists of all H 2functions in the range of the operator (I  TbTj»)l/2 on H2. We also use this map to give 1t(b) a Hilbert space structure. Namely, if lor 9 is orthogonal in H2 to ker(I TbTj»)l/2 = ker(I TbTj»), we set
((I  TbTj»)l/2 I, (I  TbTj»)l/2g)b:= (/,g)2. As a consequence (I TbTj)?/2 is a coisometry from H2 onto 1t(b) and the inclusion map i : 1t(b) + H2 is another contraction. Moreover, we see that if Ilbll oo < 1, then 1t(b) is all of H2 with an equivalent norm. Given an outer function I E Hl (not a constant) Sarason constructs three auxiliary holomorphic functions: 27T
(2.6)
(2.7) (2.8)
G(z) = {
io
ifJ e, + z I/(eifJ)1 dO, e,fJ  z 211'
( ) _ 2v'f(Z)
a z  G(z)
+ l'
G(z)  1 b( z) = G ( z) + 1 '
where we may take any branch of IT Observe that we can recover the function I from a and b: 1= F2, with F = a/(1  b) E H2. The function G is reasonably wellbehaved on JI): because ReG(z) ~ I/(z)1 ~ 0, G is contained in HP for all o < p < 1. The real part of G is positive and majorizes hence a and bare contained in the unit ball of Hoo. The fact that Re G = III a.e. on 'II' gives us that lal 2 + IW = 1 a.e. on 'II'. By a theorem in [26] the functions a and b are not extreme in the unit ball of Hoo. It is known in such cases that the polynomials are contained and dense in 1t(b) (cf. [46], IV3).
Jill.
We shall also encounter Toeplitz operators on H2 with unbounded defining functions. For a function 't/J E L2, the Toeplitz operator T,p (with defining function 't/J) acts as follows. Take a function 9 E H2, then 't/Jg is an L l ('II')function with Fourier series E~oo cn['t/Jg]einfJ and we set 00
T,p(g) =
L en ['t/Jg]zn
E
H(D).
n=O
For bounded 't/J this coincides with the Toeplitz operators as defined earlier. For unbounded 't/J, the map T,p is a densely defined operator from H2 to H2 and is continuous from H2 into the space of holomorphic functions on JI) with the topology of uniform convergence on compact subsets. In this light it is perhaps surprising that the product of the operators Tlb and Tp (the former is ordinary multiplication by 1  b) is bounded on H2. This is a consequence of the following theorem of Sarason. The proof uses that G = ~ is the Herglotz integral of an absolutely continuous measure (cf. (2.6)); it may be found in [46], IV13, p. 30.
THE BOUNDARY OF THE UNIT BALL IN HI_TYPE SPACES
71
THEOREM 2.32. The operator T1bTF is unitary from H2 onto 1i(b). Next one can show that TIbTFTF/F = TIbTFF/F = TIbTF = Ta
(compare with Lemma 2.26). Therefore T1bTF maps the range ofTF/F in H2 onto M(a) C 1i(b) (Theorem 2.32). Consequently, M(a) is dense in 1i(b) if and only if the range ofTF/F on H2 is dense in H2, or equivalently ifTF/F = T;/F is injective on H2. Combining this with Lemma 2.31 we find Sarason's result:
I
THEOREM 2.33 ([44]). Let I be an extreme point is exposed il and only il M(a) is dense in 1i(b).
01 the
unit ball in HI. Then
This fundamental result has several striking consequences, one of them being that if I is exposed, then so are the squares of the H 2functions FA = a/(1  )"b) ().. E 1l'), cf. [44, 46] As a next step one may even replace).. by any inner function u. Sarason proves that the square of Fu = a/(I ub) is exposed in HI. We now formulate a variant on Theorem 2.33 that deals with strongly exposed points.
I
THEOREM 2.34 ([2]). Let I be an extreme point is strongly exposed il and only il M(a) = 1i(b).
01 the
unit ball
01 HI.
Then
PROOF. First observe that for any FE H2 \ {O}, the Toeplitz operator TF/F is injective. Sarason's reasoning preceding Theorem 2.33 shows that (with 1= F2 as before) (2.9)
M(a) = 1i(b) if and only if TF/F maps H2 onto H2.
Hence, (2.10)
M(a) = 1i(b)
¢:}
TF/F is invertible
¢:}
TF/F is invertible.
Suppose I is strongly exposed. By Theorem 2.13, the LOOdistance of F / F to H OO + C is less than 1. The DouglasSarason theorem (Corollary 2.29) implies that the operator TF/F has closed range in H2. As we have seen its adjoint TF/F is injective, hence the range of TF/F is dense in H2 too. Thus TF/F : H2 + H2 is surjective. The operator is also injective by the rigidity of I = F2 (Lemma 2.31). By (2.10), M(a) = 1i(b). To establish the other half of the theorem suppose M(a) = 1i(b). We observe that I is exposed by Theorem 2.33. By (2.10), the operator TF/F is invertible, in particular, it is leftinvertible. Theorem 2.28 implies that the LOOdistance of 711/1 to HOC is less than 1. We conclude that I is strongly exposed by Theorem 2.13. 0 There is an alternative proof of the implication
"I strongly exposed =* 1/1 E
HI" that lies on the surface: if I is strongly exposed, then 1 E 1i(b) = M(a), because 1i(b) contains all polynomials. Thus l/a E H2 and 1/1= (lb)2/ a2 E HI. lt should come as no surprise that l/a is actually in HP for slightly larger p > 2. Indeed, equality of the spaces M (a) and 1i(b) occurs if and only if the operator Ta/a
is invertible and the nonextreme pair (a, b) satisfies the socalled Corona condition: inf
zED
la(zW + Ib(zW > 0,
72
PAUL BENEKER AND JAN WIEGERINCK
see [44] and [46], IX5, p. 66. We see in particular that for strongly exposed f, a2 Illall~ is also strongly exposed (cf. (2.10)), and that lla E H2+£, 1/(Ib) E Hi+£ for sufficiently small E > O. Let us go back to the distance condition used in Theorem 2.13 and assume f is an extreme point such that the LOOdistance of 7/1fl to HOO + C is less than 1. Starting with the DouglasSarason theorem (Corollary 2.29) and exploiting the full strength of the statement that TF/F is (left)Fredholm it follows that the operator TF/F has closed range in H2 of finite codimension, say, N. Hence M(a) is closed and of finite co dimension N in 1i(b). Using [20], Theorem 6 or [46], X18, p. 77, we conclude that f is of the form f = p2g, where 9 is strongly exposed in the unit ball of Hi and p is a polynomial of degree N that has all its zeros on 11'. Conversely, if f is of the above form, then
distallfl, H oo
+ C) = dist(z2N . gllgl, H oo + C) = dist(gllgl, H oo + C) < 1, by the algebra structure of Hoo + C, but dist(7/lfl,H OO ) = 1 when N > O. This calculation serves to illustrate that if for a given extreme point f the distance of 7Ilfl to H oo + C is less than 1, the ("only") thing that can prevent f from being exposed (and hence strongly exposed) is the divisibility of f in Hi by functions of the form (1  U)2 with u(z) = >.z (>. E 11') a particularly simple inner function. Functions in Hi that lack this divisibility property are called strong outer functions. We see that for any strong outer function that is not exposed, like Inoue's example, [22], it is true that the distance of 7/1fl to H OO +C is 1. Alternatively, the strongly exposed points are the (normalized) strong outer functions f that satisfy distallfl, H oo + C) < 1.
3. The boundary of B(Hi(O)) Having studied the sets of exposed and strongly exposed points in the unit ball of the classical Hardy space Hi of the unit disc, we will investigate how these results hold up for the Hardy space Hi of a domain of finite connectivity ("finite domain") 0 c C. Such domains are e.g. conformally equivalent to domains with a smooth boundary consisting of finitely many components or to a disc from which a finite number of disjoint slits are deleted, cf. [33]. We defined Hi (0) in Section 1.2. There are two important differences with the classical Hardy space that make the analysis very different. On domains of finite connectivity, Hifunctions may not allow a classical factorization using Blaschke products, (singular) inner functions and outer functions. Indeed, the argument principle prevents the existence of a holomorphic function f on the annulus A = {I < Izl < 2}, such that If I = 1 a.e. on 8A and f has exactly one zero on A. Secondly, and in a way related, there now exist extreme points in the unit ball with (finitely many) zeros (Section 3.1). While such zero sets are somewhat generic for extreme points, their location plays a surprisingly crucial role in its being a (strongly) exposed point (Section 3.2). As in Section 2.1 we call a function I E HOO(O) an inner function if 11*1 = 1 almost everywhere with respect to arc length (dO') on 80. Consequently, II(z)1 ::; 1 for all z E O. Let w(·, z) denote harmonic measure on 80 at z. We call a function F E Hi(O) an outer function if for every z in 0 (equivalently, for at least one
THE BOUNDARY OF THE UNIT BALL IN Hl_TYPE SPACES
73
zEn): log IF(z)1 = (
lao.
log IF*(~)I dw(~, z).
An outer function is zero free on n. Green's function G(z; zo) and the Blaschke factor Bzo(z) = t:z:oz for are related by G(z; zo) = log IBzo(z)l. This suggests the following definition.
1:°1
]I))
DEFINITION 3.1. Let G(z; zo) be Green's function on a finite domain n with pole at Zo and let G(z; zo) denote its (multiple valued) harmonic conjugate. Suppose that Zl, Z2, . .. satisfy the Blaschke condition En G(z; Zn) < 00. Then the Blaschke product with zeros at Zl, Z2, ... is the multiple valued function
(3.1)
B(z) = exp(  L G(z; zn)" iL G(z; Zn». n
n
DEFINITION 3.2. A multiple valued function F on n is modulus automorphic if • IF(z)1 is welldefined; • locally on n, IFI coincides with the absolute value of a holomorphic function. If in addition, IF(z)IP admits a harmonic majorant on n (0 < p < oo), then we say that F E MHP(n}. If IFI is bounded, we say that F E MHOO(n}. I E MHOO(n} is called innerif 11*1 = 1 a.e on r. I is singular inner if I is in addition zerofree. Blaschke products are modulus holomorphic inner functions. One can show that if FE MHP(n}, then IFI has nontangential limits (denoted IF*!) a.e. on an, IF*I E p(an,da}, and unless F == 0, 10glF*1 E LI(an,da) and (3.2)
log IF*(z)l:::; (
lao.
log IF*(~}ldw(~, z),
for all zEn.
If equality holds in (3.2) at all points zEn (equivalently, for at least one zEn), we call F an outer function in MHI(n). While a single valued innerouter decomposition is impossible in HP(n), a decomposition in modulus automorphic inner and outer factors is possible.
THEOREM 3.3 ([48]). Let f E HP(n) be not identically zero. Then there exist a Blaschke product BE MHOO(n), a singular inner function 8 E MHOO(n}and an outer function F E MHP (n) such that for all zEn
If(z}1 = IB(z)I·18(z}I·IF(z}l· This factorization is unique in the following sense: if B 1 , 8 1 and FI are a Blaschke product, a singular inner function and an outer function on n for which If(z)1 = IB1 (z}I·181 (z}I·IFI(z)l, then IBI = IBII, 181 = 181 1 and IFII = IFI· 3.1. Extreme points in HI(n}. The question arises which functions are extreme in the unit ball of Hl(n). After the deLeeuwRudin theorem 2.3, the following result is elementary: LEMMA 3.4. Iff E HI(n} is an outer function of unit norm, then f is extreme in the unit ball of HI(n}. Any attempt to copy the proof of the deLeeuwRudin theorem in the other direction will break down. For suppose f = I· F where I is a nontrivial inner function (in MHOO(n)}. Then deLeeuw and Rudin look at the function 9 = (1+12)F
74
PAUL BENEKER AND JAN WIEGERINCK
and show that IIf ± gill = 1. However, unless I is a single valued inner function, 1 + 12 is not a welldefined modulus automorphic function. There is no remedy for this problem, because as we have already mentioned, when m ~ 2 there exist extreme points with a nontrivial inner part. The following theorem of F. Forelli is crucial in understanding which inner functions can appear in the innerouter factorizations of an extreme points. THEOREM 3.5 ([14]). Let f be an extreme point of the unit ball of H1(n). Then the codimension of the H 1closure of f . Hoo in H1 is at most T' when n is bounded by m + 1 closed smooth curves. All functions in the H 1closure of f . HOO(n) inherit the zeros of f, hence Forelli's theorem implies that an extreme point can only have a limited number of zeros. In fact one has: COROLLARY 3.6 ([14]). If f is an extreme point of the unit ball of H1 (n), then zeros. the inner part of f is a finite Blaschke product with at most
T
Inspection of the proof of the deLeeuwRudin theorem gives us the following criterion for extremity (where we identify H 1functions with their boundary values): LEMMA 3.7. Let f E H1(n) be of unit norm. Then f is not extreme if and only if there exists a nonconstant real function k E LOO(an) for which kf E H1(n). Suppose f = I . F is of unit norm where I is a finite Blaschke product, and F outer. Let us suppose that f is not an extreme point of the unit ball of H1(n). Then let k be as in the lemma, and let 9 E H1(n) have boundary values kf. Because F is an outer function, for all zEn: Ig(z)1 $ IIkll oo ·1F(z)l. Hence, because also III = 1 everywhere on an, the meromorphic function h = g/ f is bounded near an, realvalued (a.e.) on an, and has its poles in the zeros of f (with corresponding mUltiplicities). Conversely, if h is a meromorphic function on n with these three properties, then with k := h on an and 9 := hf E H 1(n), we have 9 = kf on an, so by the previous lemma, f is not an extreme point. We come to the following definition. DEFINITION 3.8. Let I be a finite Blaschke product. We say that I is an extremal Blaschke product if there exists no meromorphic function h on n that is bounded near an, realvalued on an and has its poles in the zeros of I, with no greater multiplicity than the zeros of I. The conclusions of the previous paragraphs may thus be summarized as follows: PROPOSITION 3.9. Let the norm of f E H1(n) be 1. Then f is an extreme point of the unit ball of H1(n) if and only if the inner part of f is an extremal Blaschke product. We wish to stress that Forelli's theorem also gives us an upper bound for the number of zeros of an extremal Blaschke product on n. Also, by the previous proposition we see that it is only the location of the zeros of a function in H1 (n) and not so much the outer factor that decides whether or not the function is extreme in the unit ball (after normalization). The problem of determining the extreme points of H1 (n) has thus been reduced to a problem on meromorphic functions on n with predescribed poles, that is: a problem concerning meromorphic divisors on n.
THE BOUNDARY OF THE UNIT BALL IN Hl_TYPE SPACES
75
Let I be a finite Blaschke product with zeros Zl, Z2, ... , Zn repeated according to multiplicity. Thus I has n zeros on O. Let 8 := 1'ZI +1'Z2+' +1,zn be the divisor on 0 associated with I. If 8' = L:zEf! d'(z) . Z is another divisor on 0 we say that 8' ~ 8 if at every Z E 0: 8'(z) ~ 8(z). The space of all meromorphic differentials w on 0 that are realvalued on 00 and for which the associated divisor (w) satisfies (w) ~ 8 is a real linear space of dimension MD(8). Using a theorem of H.L. Royden [39], based on the RiemannRoch theorem, T.W. Gamelin & M. Voichick proved the following result: THEOREM 3.10 ([15]). The Blaschke product I with zeros Zl, Z2, ... , Zn and associated divisor 8 is extremal if and only if MD(8) + 2n = m. In particular, using only the fact that the inner factor of an extreme point is a finite Blaschke product (as shown by Forelli), Gamelin & Voichick also arrived at Forelli's upper bound (W) for the number of zeros of an extremal Blaschke product. They proved that this upper bound is also sharp. THEOREM 3.11 ([15]). The HIclosure of the set of extreme points in the unit ball of HI(O) is the collection of all functions in HI(O) that have unit norm and no more than W zeros. There is a special type of finite domains where Gamelin and Voichick described the zero sets of extreme points explicitly, namely the so called real slit domains. These are usually defined as the extended complex plane with a finite number of intervals deleted. In our situation we prefer a (conformly equivalent) definition. DEFINITION 3.12. We will call any domain n of the form JDl \ (It u ... U 1m), where It, 12', • .• ,Im are disjoint, bounded and closed intervals in (1, 1) a real slit domain.
n
THEOREM 3.13 ([15]). Let be a real slit domain and let Zl, Z2, ... , Zn be points of (not necessarily distinct). Then the Blaschke product with zero set Zl, Z2, ... ,Zn is extremal if and only if:
n
• n::; W and • for all i, j: Zi =I Zj. (In particular, none of the Zi is real.)
We omit most of the proof, and only observe that because the meromorphic function Z~Zi + z~z; + l~iz + lkz is bounded and realvalued on an c JR., no zeros of an extremal Blaschke product are conjugated.
3.2. Strong exposedness and the location of zeros. In this section we investigate exposed and strongly exposed points in the unit ball of HI(O). We give several examples and criteria for (strongly) exposed points. Also we show that nontrivial properties of strongly exposed points in HI (JDl) (for example: Llinvertibility on the boundary) have no analogue for finite domains. Finally, we again look at the zero sets of extreme points and the question of divisibility of extreme functions by functions of the form (1 + U)2, where u is a nonconstant inner function. The HahnBanach theorem again gives that for f an exposed point of the unit ball of HI(O) the exposing functional L for f is unique and given by;
L : 9 E HI
1+
fan 9 I~I da.
PAUL BENEKER AND JAN WIEGERINCK
76
Hence, like H1(1I))), a function I in the boundary of the unit ball of H1(n) is exposed if and only if it is rigid: apart from (positive) constant multiples of I there is no H 1function with the same argument a.e. (dO") on an. The following criteria for rigidity of H 1 (1I)))functions carryover to finite domains word for word: • If IE H1 and 1/1 E Hl, then I is rigid (Theorem 2.9. 1). • If there is agE Hoo such that Re(fg) > 0 a.e. on an, then I is rigid (Theorem 2.9. 2). • If u is a nonconstant inner function such that 1/(1 + u)2 is in Hl, then I is not rigid (or I == 0). A priori the first two conditions can only be used to demonstrate rigidity of outer functions. In both cases III cannot be too small near the boundary of 0.: if I satisfies the second condition, then 1/1 E H1 "'(an) for all E: > 0, so 1/1 is "nearly" in H1. The first condition can be modified to allow for exposed points with zeros on n. PROPOSITION
3.14 ([4]). II I is extreme in H1(n) and 1/1/1 E L1(an) then I
is exposed.
Similar to Theorem 2.13 is: THEOREM 3.15. Let I be a lunction in H1(n). Then I is strongly exposed in the unit ball 01 H1 il and only il I is exposed and L oodist(7 /1/1, Hoo+C(an)) < 1. Throughout the remainder of this section the domain R will be a real slit domain with m slits that contains the origin. Note that on 11'\ {i} the function (z+i)2 has the same argument as iz so (z+i)2 is not rigid in H1 (II))). LEMMA 3.16 ([4]). For all m ~ 2, the normalized lunction I(z) = c(z + i)2 is strongly exposed in the unit ball 01 H 1(R). The proof is an amusing exercise in elementary function theory. One supposes that 9 E H1 has the same argument as I a.e. on oR and sets h = g/ f. Schwarz's reflection principle eventually shows that h extends to a rational function which turns out to have no poles at all. Hence I is exposed and by Theorem 3.15 strongly exposed. REMARK 3.17. Similarly, if m > k + 1, then the normalized function hk(Z) = c(z + i)2k is strongly exposed in the unit ball of H 1 (R). In particular we see that there exist strongly exposed points in the unit ball of H1 (R) that are "small" on the boundary: 1/lhkl rt L 1 / 2k (OR). We recall that for I = I . F to be an extreme point the only requirement is that the inner part I of I is an extremal Blaschke product  a generic zero set; "most" of the properties of the function I then follow from its outer factor, i.e., the size of Ilion the boundary an. It is reasonable to ask whether exposedness is also essentially a property of the outer factor. We make this question precise in the following sense: if I E H1 is a rigid outer function, I is an extremal Blaschke product on 0., and 9 is invertible in MHOO(n) and such that Ig E HOO(n) (a single valued function), is the extreme point Ig· 1/IIIg . 1111 also exposed? (Compare
THE BOUNDARY OF THE UNIT BALL IN Hl_TYPE SPACES
77
with the first example in Section 2.2.) Proposition 3.14 tells us the answer is yes if 11f E Ll(80). Theorem 3.18 below shows that in general the answer is no. We mentioned in Section 2.2 Helson's criterium for exposedness of outer functions, 2.8. It fails for finite domains: THEOREM
3.18 ([4)). For m = 3, there exists ~
f(z)
=
E
R such that the function
c(z  ~)(z + i)2
is extreme in the unit ball of Hl(R), but not exposed.
We will only describe a nontrivial HI (R)function 9 with the same argument
as
f a.e. on an, for suitable ~ E R \ lR and refer to [4J for details.
Suppose the three slits are the intervals [Xl,YlJ, [X2,Y2J, [X3,Y3J. Let k(z) be a rational function with poles of order 2 in ~,~, 1/~, 1/~, and poles of order 4 in ±i l , and double zeros at ±1, while k(oo) = 1. and zeros at the 12 points Next, let q be the (welldefined) square root of k on R with q(oo) = i. Now with a suitable choice of ~ and an appropriate polynomial p of degree 8 that is positive on lR U '][', one can show that the function
xtl, yt
g(z)
p(z) + q(z) (z  ~)(z  i)2
1 (1 ~z)(l ~z) belongs to HOO(R) and has the same argument as f on 8R \ {i}, which implies that f is not rigid. Because the inner part of f is the extremal Blaschke product with zero at ~, f is extreme, however. By Lemma 3.16, (z + i)2 is rigid. We conclude that if Helson's criterion were valid, for any two inner functions u, v not both constant on R, (z + i)2/(u(z) + v(z))2 (j. Hl(R), hence also (z~)(z+i)2/(u(z)+v(z))2 (j. Hl(R), a contradiction! =
4. The Bergman space AI(JD)) Recall from Section 1.2 the definition of the Bergman space
AP(JD)) = H(JD)) n LP(JD)) In the Bergman space extreme and exposed points are extremely simple. Indeed, every f E 8B(Al(JD))) is exposed, and (hence) extreme. The exposing functional is
Lf :
gl+
kgl~ldA(Z).
If Lfg = 1 for some 9 E 8B(Al(JD))), the argument of 9 would be equal to that of f, (a.e) and this clearly implies f = g. However, to study strongly exposed points we need some machinery.
4.1. Bergman projection and Bloch space. An important tool for the study of strongly exposed points will be the Bergman projection; this is the orthogonal projection It is given by an integral operator (4.1)
(4.2)
Pcp(z)
= =
(f, kz)
[
=
f(w)
iD (1 _ zw)2 dA(w)
fUn+ 1)· 1m f(w)'W'dA(w))zn. n=O
D
PAUL BENEKER AND JAN WIEGERINCK
78
Elementary properties are, see e.g. [18], • If cp has compact support in]l)), then Pcp is holomorphic on a neighborhood of Jij. • If cp is Coo on Jij, then Pcp is Coo on Jij. The effect of the Bergman projection on Loo(]I))) and on C(Jij) will be very important for us. We need the Bloch spaces. DEFINITION 4.1. The Bloch space B consists of all holomorphic functions f on ]I)) with the property that (1lzI2)lf'(z)1 is bounded on]l)). Equipped with the norm
(4.3)
IlfllB := If(OI
+ sup (1 lzI 2)1f'(z)l, zED
B becomes a Banach space. The set of all functions f in B for which the expression (1lzI2)1f'(z)1 Izl > 1 is a closed subspace of B, called the little Bloch space Bo.
>
0 as
Let Co denote the continuous functions on Jij that are zero on T. We have the following theorem of R. Coifman, R. Rochberg and G. Weiss: THEOREM 4.2 ([8]). The Bergman projection P maps Loo(.]I))) boundedlyonto B. Furthermore, P maps both C(D) and Co boundedlyonto Bo. The result proved in [8] is much more general than Theorem 4.2. As we state it, the theorem may be found in [18], Theorem 1.12, with an elementary proof. There we also find the following results THEOREM 4.3. (1) The dual space of Al is the Bloch space B under the following pairing:
(4.4)
g E B : f E Al !+lim ( fr(z)g(z)dA(z). rTl
JD
(2) The dual space of the little Bloch space Bo is the Bergman space A l under the pairing: f
E
Al : g
E
Bo
1+
lim ( f(z)gr(z)dA(z). rTl
JD
REMARK 4.4. When one identifies (A1)* with the Bloch space in Theorem 1 B, the dual norm on B yields a norm that is equivalent with, but not equal to the norm 11.1113 that we have previously defined on B. Hence, there exists a norm 11.11. on the Bergman space A I that is equivalent to 11.111 and is such that the dual norm of 9 E B = (AI)* equals IIgIiB. The strongly exposed points in the unit ball of Al with the norm 11.11. have been described by C. Nara ([32]), who also showed that up to isometrical isomorphisms, Al with the 11.11. norm is the unique predual of B. 4.2. Strongly exposed points of AI(.]I))). The following theorem is a consequence of Theorems 5.3 and 5.2, the fact that Al(]I))) may be identified with the subspace of HI (lI£2) consisting of functions that depend only on one variable, and the fact that these functions are exposed in B(HI(lI£2). We set
(4.5)
THE BOUNDARY OF THE UNIT BALL IN HlTYPE SPACES
79
THEOREM 4.5. Let I E Al be 01 unit norm. Then I is strongly exposed in Ball(AI) il and only il the LOOdistance 01 filII to the space (AI)1. + C(D) is less than one. Henceforth we will simply write (AI)1.
+C
instead of (AI)1.
+ C{D).
The question now is: how can we estimate the distance in Loo of cp = 1/111 to (AI)1. +C, where I is a given function in AI? (Clearly the distance cannot exceed one. ) Let us first look at polynomials of a particularly simple form: I (z) = c( z  0:) n , where c is normalizing. We will assume n 2: 1 because the constant functions are strongly exposed by Theorem 4.5. We distinguish three cases in order of increasing difficulty: 10:1 > 1, 10:1 < 1 and 10:1 = 1. The case where 10:1 > 1 is very easy: 1/111 is continuous on D, so I is strongly exposed. In fact, we may even take noninteger powers n and products of such functions and we always obtain strongly exposed points after normalization. When 10:1 < 1, let us write cp = 1/111 = 'l/Jo + 'l/JI, where 'l/JI is compactly supported in 1D> and cp == 'l/JI on a neighborhood of 0:, while 'l/Jo is smooth on D. From (4.1) we see that P'l/JI is holomorphic across the unit circle because 'l/JI is compactly supported in 1D>. Next, because 'l/Jo is smooth on D, also P'l/Jo is smooth on D. Hence Pcp is continuous on D. Now cp  Pcp is bounded, so cp = (cp  Pcp) + Pcp is contained in (AI)1. + C. By Theorem 4.5 I is strongly exposed. Again, our reasoning readily shows that the normalized product I of functions (z  O:i)n;, for all ni and all O:i ft 'll', is strongly exposed. Now suppose 10:1 = 1; we may take 0: = 1. Let us write In{z) = cn (1 z)n and CPn = In/l/nl· Introducing polar coordinates and applying Cauchy's theorem one finds that the exposing functional L for h is given by
L(g) = [ g(z) ~1 
J"D
1
z~: dA{z) = Z
[ Ig{z) (1 
JD 
z
z) dA{z) =
Cog{O)
+ CIg'{O).
But then there exists a polynomial P2 such that L(g) = fD gP2 dA. Therefore CP2  P2 is contained in (A I) 1., hence CP2 E (A I) 1. + C so h is strongly exposed. Similarly, for all even n, CPn is contained in (A I) 1. + C and In is strongly exposed in AI. Again, we may introduce noninteger exponents. Let 1f3 = cf3(1  z)f3, where (3 > 2 to ensure that I f3 E A I; the constant Cf3 > 0 is normalizing. Set
cpf3 = 1f3/I/f3I· PROPOSITION 4.6 ([5]). For all (3 > 1, the Loodistance 01 CPf3 to (AI)1. + C is at most Isin( f327i") I· In particular, lor all (3 > 1, (3 =I 1,3,5, ... , the lunction I f3
is strongly exposed in the unit ball 01 A I. The proof consists of observing that if 1(3  2nl < 1, then IIcpf3  CP2nli00
=
IsinCf)1 < 1. We will come back to odd exponents in Section 4.4 and end this section with and example of a boundary points of B (A I (1D>)) that is not strongly exposed. 2
EXAMPLE 4.7 ([5]). The normalized function I(z) = (Iz)2~~g2(Icz) is not strongly exposed in the unit ball of A I. The functions I f3 tend pointwise to O. However, limf3!2 fD /{3"(j5 dA = 1.
PAUL BENEKER AND JAN WIEGERINCK
80
·4.3. The space (AI).1 + C. We have observed that (AI).1 + C plays the same role in Theorem 4.5 with respect to the Bergman space as (HI).1 + C('1I') = Hoo +C('1I') with respect to the Hardy space HI(JI))) (Theorem 2.13). We mentioned already in Section 2.3 that Hoo + C is closed in L oo . From this then it followed relatively easily that Hoo + C('1I') is in fact an algebra, cf. [41], Theorem 6.5.5. How far do these results extend to the space (A I ).1 + C? From [5] we quote (1) Al (JI))).1 + c(il~) is a closed subspace of LOO(JI))). (It equals PI(80 )!) (2) AI(JI))).1 + C(ii)) is a C(ii)) module. (3) A I (JI))).1 + C(ii)) is not an algebra. (4) The space (A 1).1 + C is invariant under composition with holomorphic automorphisms of JI)). EXAMPLE 4.8. [5] As for (3), let J{3 = (1 z){3 and let 'P{3 = J{3/IJ{31 for {3 E R Then 'P2 and 'P4 E (AI).1, but 'P2 is not contained in (AI).1 + C. The space (A I ).1 + C is not an algebra. The properties of (A 1).1 proposition.
+C
lead in a straightforward way to the following
PROPOSITION 4.9. Let J be a strongly exposed point in AI. Then (a) iJu is an automorphism oJJI)), then the normalized Junction FI = CI(fou) is strongly exposed; (b) iJ v E A(JI))) is zeroJree on the circle, then the normalized Junction F2 = C2!v is strongly exposed. Furthermore, the functions J IIJI, FdlFII and F2/1F21 have the same Loodistance to (AI).1 + C. 4.4. Strong exposedness of (1  z){3. We saw in Section 4.2 that the functions J{3 = c{3(1  z){3 are strongly exposed in the unit ball of Al for all {3 > 1 except possibly when {3 = 1,3,5, .... This was deduced from rather straightforward estimates of the L 00 distances of the functions 'P = ff3 I IJ{31 to the space (A 1).1 + C (Proposition 4.6). In [5] a much sharper result is proved. THEOREM 4.10. For all {3 ::::: 0, the Bloch distance oj the Junction P'P{3 to 8 0 4I sin({h)1
equals 1r
{3+T .
SKETCH OF PROOF. It is convenient to rewrite 'P{3 as 'P{3(w) = (1 w){3/2/(1w){3/2. Using the series expansions for the Bergman kernel 1/(1 zw)2 (see (4.2)), as well as for (1  W){3/2, and 1/(1  w){3/2, we evaluate the Bergman projection P'P{3' One obtains P'P{3 = E~=o c{3,nzn, where
c{3,n =
(n + 1) sin(~) ~ r(m + ~)r(m + n  ~) 271" ~ m!(m + n + I)! .
It is proved in [5] that for fixed (3 > 0:
(4.6)
~r(m+~)r(m+n~) =:0 m!(m + n + I)!
where the o(I)term tends to zero as n
4
= n2{3({3 + 2) (1 + 0(1)),
+ 00.
2sin(~)
This implies that
c{3,n = 7I"({3 + 2)n (1
+ 0(1)),
THE BOUNDARY OF THE UNIT BALL IN where the o{I)term vanishes as n
> 00.
!::tv
(3,n
SPACES
81
But then,
so the Bloch distance of PCP(3 to Bo is at least large N, I{ ~ c
HITYPE
41:~~~1I.
On the other hand, for
zn)'1 < ~ nlc 1.l z ln1 < 21 sin{~)ll + 0(1)  n~ (3,n  7r(,B + 2) 1  Izl '
where the o{I)term tends to zero as N increases. Using the fact that the polynomials are contained in Bo it follows that the Bloch distance of PCP(3 to Bo is at most 4I sin (¥)1 o ... ((3+2) . COROLLARY 4.11. Let d(cp(3, (Al)1. (Al)1. + C. Then for all,B 2: 0, (4. 7)
+ C)
denote the LOOdistance of CP(3 to
~ 1sin( !?f) 1 < d( (A 1) 1. + C) < ~ 1sin( ~) 1 < ~ 2 ,B + 2 CP(3,  7r ,B + 2  7r'
In particular, all f(3 are strongly exposed for ,B 2: O. SKETCH OF PROOF. Let q: B > B/Bo be the quotient map. By Theorem 4.2, the map q 0 P : L OO > B/Bo is continuous and surjective. The kernel of the map q 0 P is the space (Al)1. + C, cpo Property (I) from Section 4.3. Hence the derived map
P* : L oo / ((Al)1.
+ C)
>
B /Bo
is bijective and bounded (by ~ as follows from the proof in [18] of Theorem 4.2). This gives the lower bound for d(cp(3, (Al)1.
+ C), because
IIP*cp(311 =
~ ISi~~y)l.
By the closed graph theorem, the inverse P* 1 of P* is also bounded. Actually, one can show directly that II P* 111 ::; 1, which in turn yields the upper bound for d{cp(3, (Al)1.+C). Supposing that FE B/Bo has norm 1, we will show that P*I(F) has norm at most 1 in L oo /((Al )1.+C). For any c > 0, we can find a representative fEB of the coset F such that IIfl18 < 1 + c. In the proof of Theorem 4.2 in [18] one finds that
'¢(w) = (I IWJ2) . f'(w) ~ f'(O)
E
L oo
w satisfies f(z)  P'¢{z) = f(O) + j'(O)z E Bo. Thus '¢ is a representative of P*I(F) in LOO. As a consequence IIp*l{F)IILOO/((AI)J.+C) ::; d(,¢, (Al)1.
=
+ C)
::; lim esssuPr
o The module structure of (Al)1.
+C
and the previous corollary lead to
82
PAUL BENEKER AND JAN WIEGERINCK
COROLLARY 4.12. Suppose that g E A(JI1» vanishes nowhere on'll'. Let Zl, Z2, E 'll' be distinct and let {31, {32, ... ,(3n be real numbers greater than  2. Then the normalized Junction J(z) = cg(z) n~=l (1 ZZi),8i is strongly exposed in the unit ball oj A I iJ and only iJ all Junctions J,8i = C,8i (1  z ),8i are strongly exposed. In particular, all choices oj {3i > 1 yield strongly exposed points and all normalized
•.. , Zn.
polynomials are strongly exposed in the unit ball oj A I. We end this section with a conjecture on the functions J,8 for 2 < {3 < 0, for which strong exposedness is already implied by Proposition 4.6 when 1 < {3 < O. In [5] it is shown that
~lsin(!?f)I
2< (.J
with equality in the limit as {3 ! 2. This leads us to surmise the following: CONJECTURE 4.13. For all2 < (3 < 0, d(cp,8,(AI)1. +C) = particular, the Junctions J,8 are strongly exposed Jor all said (3.
~lsi~~y)J. In
5. Scattered results for B(HI (lRn )) The geometry of B(HI(lR n )) is still quite out of reach. At one time it was conjectured that all boundary points would be extreme, [41], but this conjecture was refuted as soon as the existence of inner functions on lR n was proved, [1, 27]. Just as in Theorem 2.3 one finds that inner functions, or more generally products with an inner factor cannot be extreme. However, it is well known that there is no innerouter decomposition possible for B(HI(lR n )), because the zero sets of functions in various HP classes are essentially different in size. Rudin already observed in [41] that a function J E 8B(H 1 (lR n )) which is continuous on an open set in 8lRn must be extreme. The proof goes by considering the intersection of lR n with appropriate complex lines and applying Idimensional theory. The same idea can be used to show that such functions are in fact exposed, [50]. EXAMPLE 5.1. Not all extreme points are exposed: c(1 has the same argument as I.
+ 1)2
is extreme but
Example 4.7 also is an example of an exposed point in B(HI(lR2)) that is not strongly exposed. Let us write S = 8lRn , HI = HI(lR n ) and HJ for the functions in HI that vanish at o. We define
(HI)1. =
N
E Loo(S) :
Is
{JjdA = 0 for allJ E HI}
Analogously we define (HJ)1.. Let d denote the Loodistance on S. Theorems 2.13 and 2.12 have a full analogue in B(H 1 ). THEOREM 5.2 ([50]). Let J be an exposed point in B(H 1 ). Then exposed iJ and only il d(J 11/1, C(S) + (HI)1.) < 1.
I
is strongly
The proof again uses methods from uniform algebra, but is more involved. Noticing that (HJ)1.) is a fairly small subspace of (HI)1. +C(S), this theorem can be strengthened in one direction. THEOREM 5.3. Suppose that d(J IIJI, (HJ)1.) < 1.
I
is a strongly exposed point in B(H 1 ). Then
THE BOUNDARY OF THE UNIT BALL IN Hl_TYPE SPACES
83
EXAMPLES 5.4. Homogeneous polynomials in 8B(H 1 ) are strongly exposed, see [50J. Corollary 4.12 shows that polynomials of one variable in 8B(H 1 (B 2 )) are strongly exposed.
It seems natural to try to generalize ideas of Section 4 to HI. The Bergman projection should be replaced by the Szego projection P and the Bloch spaces 8 and 8 0 by BMOA, the space of analytic functions of bounded mean oscillation and VMOA, the space of functions of vanishing mean oscillation (with respect to the nonisotropic balls in S). It is known that P maps Loo(S) surjectively to BMOA and C(S) surjectively to VMOA, see [25J for an overview of these kind of results.
So we again have a map P*: Loo(S)/«H1)1. +C)
>
BMOA/VMOA. However,
we have no idea how to estimate (p*)l, nor how to estimate the norm of P*cp for cp = p/ipi with p for example a polynomial. Answers to these questions would be very interesting. References [1] A.B. Aleksandrov, A. B. The existence of inner functions in a ball, (Russian) Mat. Sb. (N.S.) 118(160) (1982), no. 2, 147163,287. (32A40 46J15) [2] P. Beneker, Strongly exposed points, HelsonSzego weights and Toeplitz operotors, J. Int. Eq. Oper. Th. 31 (1998), 299306. [3] P. Beneker, Strongly exposed points in unit balls of Banach spaces of holomorphic functions, Academisch Proefschrift, Universiteit van Amsterdam, 2002. Electronically available at http://remote.science.uva.nl/ ~beneker /thesis.ps [4] P. Beneker, J. Wiegerinck, Exposedne'ss in Hardy spaces of domains of finite connectivity, Indag. Math., N.S., 11 (4),2000,487497. [5] P. Beneker, J. Wiegerinck, Strongly exposed points in the ball of the Bergman space, http://arXiv.org/abs/math.CV /0208234 to appear. [6] L. de Branges, J. Rovnyak, Square Summable Power Series, Holt, Rinehart and Winston, New York, 1966. [7] L. de Branges, J. Rovnyak, Appendix on square summable power series, in "Perturbation Theory and its Applications in Quantum Mechanics", John Wiley and Sons, New York (1966), 347392. [8] R.R. Coifman, R. Rochberg, G. Weiss, Factorization theorems for Hardy spaces in severol variables, Ann. of Math. (2), 103 (1976), 611635. [9] A. Devinatz, Toeplitz operotors on H2 spaces, Trans. Amer. Math. Soc. 112 (1964), 304317. [10] J. Diestel, Geometry of Banach paces  Selected topics LNM 485, Springer, Berlin etc. 1975. [11] R.G. Douglas, D.E. Sarason, Fredholm Toeplitz operotors, Proc. Amer. Math. Soc. 26 (1970), 117120. [12] P.L. Duren, Theory of HPspaces, Academic Press, 1970. [13] S.D. Fisher, Function Theory On Planar Domains, John Wiley & Sons, 1983. [14] F. Forelli, Extreme points in Hl(R), Can. J. Math. 19 (1967), 312320. [15] T.W. Gamelin, M. Voichick, Extreme points in spaces of analytic functions, Can. J. Math. 20 (1968), 919928. [16] J. Garnett, Bounded holomorphic functions, Pure & Applied Mathematics, 96, Academic Press, Inc., New YorkLondon, 1981. [17] E. Hayashi, The solution of extremal problems in Hl, Proc. Amer. Math. Soc. 93 (1985), 690696. [18] H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman Spaces, Springer Graduate Texts in Mathematics 199, 2000. [19] H. Helson, G. Szego, A problem in prediction theory, Ann. Mat. Pura Appl. 51 (1960), 107138. [20] H. Helson, D. Sarason, Past and future, Math. Scand. 21 (1967), 516. [21] H. Helson, Large analytic functions II, Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., 122, Dekker, New York (1990), 217220.
84
PAUL BENEKER AND JAN WIEGERINCK
[22] J. Inoue, An example of a nonexposed extreme function in the unit ball of Hl, Proc. Edinburgh Math. Soc. 37 (1993), 4751. [23] J. Inoue; T. Nakazi, Polynomials of an inner function which are exposed points in Hl. Proc. Amer. Math. Soc. 100 (1987), no. 3, 454456. [24] P. Koosis, Introduction to HP spaces, Cambridge University Press, 1980. [25] S.G. Krantz, Geometric Analysis and FUnction Spaces CBMS Regional Conference Series in Mathematics, 81. Amer. Math. Soc., Providence, RI, 1993. [26] K. de Leeuw, W. Rudin, Extreme points and extremum problems in Hl, Pacific J. Math. 8 (1958), 467485. [27] E. Ll1lw,A construction of inner functions on the unit ball in CP. Invent. Math. 67 (1982), no. 2, 223229. [28] T. Nakazi, Exposed points and extremal problems in Hl, J. Funet. Anal. 53 (1983), 224230. [29] T. Nakazi, Exposed points and extremal problems in Hl, II, T6hoku Math. J. 37 (1985), 265269. [30] T. Nakazi, Existence of solutions of extremal problems in Hl , Proc. Edinburgh Math. Soc. (2) 34 (1991), no. 2, 99112. [31] T. Nakazi, personal communication. [32] C. Nara, Uniqueness of the predual of the Bloch space and its strongly exposed points, Illinois J. Math, 34 (1990), no. 1,98107. [33] Z. Nehari, Conformal Mapping, McGrawHill, 1952. [34] Z. Nehari, On bounded bilinear forms, Ann. of Math. 65 (1957), 153162. [35] D.J. Newman, Pseudouniform convexity in HI, Proc. Amer. Math. Soc. 14 (1963), 676679. [36] J. Neuwirth, D.J. Newman, Positive Hl/2 functions are constant, Proc. Amer. Math. Soc. 18 (1987),958. [37] A. Nicolau, The coefficients of Nevanlinna's pammetrization are not in HP, Proc. Amer. Math. Soc 106 (1989), 115117. [38] R.R. Phelps, Dentability and extreme points in Banach spaces, J. Funet. Anal. 17 (1974), 7890. [39] H.L. Royden, The boundary values of analytic and harmonic functions, Math.Z. 78, (1962), 124. [40] W. Rudin, Analytic functions of class H p , Trans. Amer. Math. Soc. 78 (1955), 4666. [41] W. Rudin, FUnction Theory in the Unit Ball of en, Grundlehren der Mathematischen Wissenschaften, 241, SpringerVerlag, New YorkBerlin, 1980. [42] D.E. Sarason, Algebras of functions on the unit circle, Bull. Amer. Math. Soc. 79 (1973), 286299. [43] D.E. Sarason, Function Theory on the Unit Circle, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 1979. [44] D.E. Sarason, Exposed points in Hl, I, Operator Theory: Advances and Applications 41 (1989), Birkhiiuser Verlag Basel, 485496. [45] D.E. Sarason, Exposed points in Hl, II, Operator Theory: Advances and Applications 48 (1990), Birkhiiuser Verlag Basel, 333347. [46] D.E. Sarason, SubHardy Hilbert Spaces in the unit disc, Univ. of Arkansas Leeture Notes in the Math. Sc. 10, WileyInterscience, 1995. [47] D. Temme, J. Wiegerinck, Extremal properties of the unit ball of Hl, Indag. Mathern., N.S., 3 (1) (1992), 119127. [48] M. Voichick, L. Zalcman, Inner and outer functions on Riemann surfaces, Proc. Amer. Math. Soc. 16 (1965), 12001204. [49] H. Widom, Inversion of Toeplitz matrices III, Notices Amer. Math. Soc. 7 (1960), 63. [50] J. Wiegerinck, A chamcterization of strongly exposed points of the unit ball of Hl, Indag. Mathern., N.S., 4 (4) (1993), 509519. [51] K. Yabuta, Unicity of the extremum problems in Hl(un), Proc. Amer. Math. Soc. 28 (1971), 181184. [52] K. Yabuta, Some uniqueness theorems for HP(un) functions, T6hoku Math. J. 24 (1972), 353357. FACULTY OF MATHEMATICS, UNIVERSITY OF AMSTERDAM, PLA:'ITAGE I\[UIDERGRACHT
1018
TV, AMSTERDAM, THE NETHERLANDS
addre.~s:benekerClscience.uva.n1;janwiegClscience.uva.n1
24,
Contemporary Mathematics
Volume 328, 2003
Complete isometries  an illustration of noncommutative functional analysis David P. Blecher and Damon M. Hay ABSTRACT. This article, addressed to a general audience of functional analysts, is intended to be an illustration of a few basic principles from 'noncommutative functional analysis', more specifically the new field of operator spaces. In our illustration we show how the classical characterization of (possibly nonsurjective) isometries between function algebras generalizes to operator algebras. We give some variants of this characterization, and a new proof which has some advantages.
1. Introduction
The field of operator spaces provides a new bridge from the world of Banach spaces and function spaces, to the world of spaces of operators on a Hilbert space. For researchers in the new field, the philosophical starting point is the combination of the following two obvious facts. Firstly, by the HahnBanach theorem any Banach space X is canonically linearly isometric to a closed linear subspace of C(K), where K is the compact space Ball(X*). Secondly, C(K) is a commutative C*algebra. Thus one defines a noncommutative Banach space, or operator space, to be a closed linear subspace X of a possibly noncommutative C* algebra A. This simplistic idea becomes much more substantive with the addition of some additional metric structure. The point is that if A is any C*algebra, then the *algebra Mn(A) of nx 11, matrices with entries in A has a unique norm 11·lln making it a C*algebra (this follows from the well known nnicity of C*norms on a *algebra). If X c A then Mn(X) inherits this norm II· lin, and more precisely we think of an operator space as the pair (X, {11·lln}n). We usually insist that maps between operator spaces are completely bounded, where the adjective 'completely' means that we are applying our maps to matrices too. Thus if T : X 4 Y, then T is completely contractive if Tn is contractive for all n E N, where Tn is the map [Xij] 1+ [T(x;j )]. Similarly T is completely isometric if II[T(xij)]1I = I [Xij] I for all n E Nand [Xij] E 1I1n(X). It is an easy exercise (using one of the common expressions for the operator norm of a matrix in Mn = .Mn(C)) to prove that a linear map T : X 4 Y between 1991 Mat/;tematics Subject Classification. Primary 46L07, 46L05, 47L30; Secondary 46JlO. This research was supported in part by a grant from the National Science Foundation.. © 85
2003 American Mathematical
So(~iety
86
DAVID P. BLECHER AND DAMON M. HAY
subspaces of C(K) spaces is completely contractive if and only if it is contractive. Consequently such a T is isometric if and only if it is completely isometric. The identification of the term 'noncommutative Banach space' with 'operator space' may be thought of as a relatively recent entry in the well known 'dictionary' translating terms between the 'commutative' and 'noncommutative' worlds. We spend a paragraph describing some other entries in this dictionary. Although t.hese items are for the most part well known to the point of being tedious, it will be helpful to collect them here for the dual purpose of establishing notation, and for ease of reference later in the paper. The most. well known item is of course the fact that the noncommutative version of a C(K) space is a unital C*algebra B. The noncommutative version of a unimodular function in C(K) is a unitary u E B (Le. u*u = uu* = 1). The noncommutative version of a function algebra A C C(K) 'containing constant functions' is a closed subalgebra A of a C* algebra B, with 18 E A. We call such A a unital operator algebra. For a unital subset S of a C*algebra B, we will take as a simple noncommutative version of the assertion'S C C(K) separates points of K', the assertion 'the C*subalgebra of B generated by S (namely, the smallest C*subalgebra of B containing S) equals B'. The analogue of a closed subset E of a compact set K is a quotient B / I, where I is a closed twosided ideal in a unital C* algebra B. More generally, unital *homomorphisms 7r between unital C* algebra.'i are the noncommutative version of continuous functions T between compact spaces. Indeed clearly any such T : Kl > K2 gives rise to the unital *homomorphism C(K2 ) > C(Kd of 'composition with T', and conversely it is not much harder to see that any unital *homomorphism C(K2 ) > C(Kd comes from a continuous T in this way. Moreover such 7r is 11 (resp. onto) if and only if the corresponding T is onto (resp. 11). Thus the noncommutative version of a homeomorphism between compact spaces is a (surjective 11) *isomorphism between unital C*algebras. Coming back to 'noncommutative functional analysis', it is convenient for some purposes (but admittedly not for others) to view 'complete isometries' as the noncommutative version of isometries. It is very important in what follows that a 11 *homomorphism 7r : A  B between C*algebras, is by a simple and well known spectral theory argument, automatically an isometry, and consequently (by the same principle applied to 7r n ), a complete isometry. Similarly, a *homomorphism 7r : A  B (which is not a priori assumed continuous) is aut.omatically completely contractive, and has a closed range which is a C* algebra *isomorphic to the C* algebra quotient of A by the obvious twosided ideal, namely the kernel of the *homomorphism. The entries we have just described in this 'dictionary' are all easily justified by well known theorems (for example Gelfand's characterization of commutative C*algebras). That is, if one applies the noncommutative definition in the commutative world, one recovers exactly the classical object. Similarly one sometimes finds oneself in the very nice 'ideal situation' where one can prove a theorem or establish a theory in the noncommutative world (i.e. about operator spaces or operator algebras), which when one applies the theorem/theory to objects which are Banach spaces or function algebras, one recovers exactly the classical theorem/theory. An illustration of this point is the BanachStone theorem. The following is a much simpler form of Kadison's characterization of isometries between C*algebras [17]:
COMPLETE ISOMETRIES  AN ILLUSTRATION
87
THEOREM 1.1. (Folklore) A surjective linear map T : A + B between unital C* algebras is a complete isometry if and only if T = U7r('), for a unitary u E B and a *isomorphism 7r : A + B. PROOF. (Sketch.) The easy direction is essentially just the fact mentioned earlier that 11 *homomorphisms are completely isometric. The other direction can be proved by first showing (as with Kadison's theorem) that T(I) is unitary, so that without loss of generality T(I) = 1. The well known Stinespring theorem has as a simple consequence the KadisonSchwarz inequality T(a)*T(a) ~ T(a*a). Applying this to T 1 too yields T(a)*T(a) = T(a*a), and now the result follows immediately from the 'polarization identity' a*b = ~ E~=o(a + ikb)*(a + ikb). 0 Note that if one takes A = C(Kd and B = C(K2 ) in Theorem 1.1, and consults the 'dictionary' above, then one recovers exactly the classical BanachStone theorem. Indeed as we remarked earlier, in this case complete isometries are the same thing as isometries, unit aries are unimodular functions, and a *isomorphism is induced by a homeomorphism between the underlying compact spaces. Indeed consider the following generalization of the BanachStone theorem: THEOREM 1.2. [15,22, 1, 20] Let fl be compact and Hausdorff, and A a unital function algebra. A linear contraction T : A + C(fl) is an isometry if and only if there exists a closed subset E of fl, and two continuous functions "f : E + '][' and r.p: E + 8A, with r.p surjective, such that for all y E E T(f)(y) = "f(y)f(r.p(y))·
Here 8A is the Shilov boundary of A (see Section 2). We have supposed that. T maps into a 'selfadjoint function algebra' C(fl); however since any function algebra is a unital subalgebra of a 'selfadjoint.' one, the theorem also applies to isometries between unital function algebras. If A is a C(K) space too, then 8A = K and then t.he theorem above is called Holsztynski's theorem. We refer the reader to [16] for a survey of such variants on the classical BanachStone theorem. Often the transition from the 'classical' to the 'noncommutative' involves the introduction of much more algebra. Next we appeal to our dictionary above to give an equivalent restatement of Theorem 1.2 in more algebraic language. THEOREM 1.3. (Restatement of Theorem 1.2) Let A, B be unital function algebras, with B selfadjoint. A linear contraction T : A + B is an isometry if and only if (A) there exists a closed ideal I of B, a unitary u in the quotient C*algebra B / I, and a unital 11 *homomorphism 7r : A + B / I, such that qI (T( a)) = u7r(a) for all a E A. Here qI is the canonical quotient *homomorphism B
+
B/I.
In light of Theorems 1.1 and 1.3 one would imagine that for any complete isometry T : A + B between unital operator algebras, the condition (A) above should hold verbatim. This would give a pretty noncommutative generalization of Theorem 1.3. Indeed if Ran T is also a unital operator algebra, then this is true (see ego B.l in [3]). However, it is quite easily seen that such a result cannot hold generally. For example, let Mn = Mn(C); for any x E Mn of norm 1, the map A 14 Ax is a complete isometry from C into Mn. Now Mn is simple (Le. has no
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nontrivial twosided ideals), and so if the result above was valid then it follows immediately that x = u. This is obviously not satisfactory. To resolve the dilemma presented in the last paragraph, we have offered in [5] several alternatives. For example, one may replace the quotient B / I by a quotient of a certain *subalgebra of B. The desired relation qJ(T(a)) = U7r(a) then requires u to be a unitary in a certain C* triple system (by which we mean a subspace X of a C* algebra A with X X* X c X). Or, one may replace the quotient B / I by a quotient B / (J + J*), where J is a onesided ideal of B. Such a quotient is not an algebra, but is an 'operator system' (such spaces have been important in the deep work of Kirchberg (see [18, 19] and references therein). Alternatively, one may replace such quotients altogether, with certain subspaces of the second dual B** defined in terms of certain orthogonal projections of 'topological significance' (Le. correspond to characteristic functions of closed sets in K if B = C(K)) in the second dual B** (which is a von Neumann algebra [25]). The key point of all these arguments, and indeed a key approach to BanachStone theorems for linear maps between function algebras, C* algebras or operator algebra." is the basic theory of C* triple systems and triple morphisms, and the basic properties of the noncommutative Shilov boundary or triple envelope of an operator space. These important and beautiful ideas originate in the work of Arveson, Choi and Effros, Hamana, Harris, Kadison, Kirchberg, Paulsen, Ruan, and others. Indeed our talk at the conference spelled out these ideas and their connection with the BanachStone theorem; and the background ideas are developed at length in a book the first author is currently writing with Christian Le Merdy [7] (although we do not characterize nonsurjective complete isometries there). Moreover, a description of our work from this perspective, together with many related results, may be found in [12]. Thus we will content ourselves here with a survey of some related and interesting topics, and with a new and selfcontained proof of some characterizations of complete isometries between unital operator algebras which do not appear elsewhere. This proof has several advantages, for example the projections arising naturally with this approach seem to be more useful for some purposes. Also it will allow us to avoid any explicit mention of the theory of triple systems (although this is playing a silent role nonetheless). We also show how such noncommutative results are generalizations of the older characterizations of into isometries between function algebras or C(K) spaces. We thank A. Matheson for telling us about these results. In the final section we present some evidence towards the claim that (general) isometries between operator algebras are not the correct noncommutative generalization of isometries between function algebras. For the reader who wants to learn more operator space theory we have listed some general texts in our bibliography.
2. The noncommutative Shilov boundary At the present time the appropriate 'extreme point' theory is not sufficiently developed to be extensively used in noncommutative functional analysis. Although several major and beautiful pieces are now in place, this is perhaps one of the most urgent needs in the subject. However there are good substitutes for 'extreme point' arguments. One such is the noncommutative Shilov boundary of an operator space. Recall that if X is a closed subspace of C(K) containing the identity fUIlction lK on
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K and separating points of K, then the classical Shilov boundary may be defined to be the smallest closed subset E of K such that all functions J E X attain their norm, or equivalently such that the restriction map J f+ JIE on X is an isometry. This boundary is often defined independently of K, for example if A is a unital function algebra then we may define the Shilov boundary as we just did, but with K replaced by the maximal ideal space of A. In fact we prefer to think of the classical Shilov boundary of X as a pair (aX, i) consisting of an abstract compact Hausdorff space ax, together with an isometry j : X + C(aX) such that j(IK) = lax and such that j(X) separates points of ax, with the following universal property: For any other pair (0, i) consisting of a compact Hausdorff space 0 and a complete isometry i : X + C(O) which is unital (i.e. i(IK) = IA), and such that i(X) separates points of 0, there exists a (necessarily unique) continuous injection r : ax + 0 such that i(x)(r(w)) = j(x)(w) for all x E X, w E ax. Such a pair (aX, i) is easily seen to be unique up to an appropriate homeomorphism. The fact that such ax exists is the difficult part, and proofs may be found in books on function algebras (using extreme point arguments). Consulting our 'noncommutative dictionary' in Section 1, and thinking a little about the various correspondences there, it will be seen that the noncommutative version of this universal property above should read as follows. Or at any rate, the following noncommutative statements, when applied to a unital subspace X c C(K), will imply the universal property of the classical Shilov boundary discussed above. Firstly, a unital operator space is a pair (X, e) consisting of an operator space X with fixed element e EX, such that there exists a linear complete isometry Il, from X into a unital C*algebra C with Il,(e) = Ie. A 'noncommutative Shilov boundary' would correspond to a pair (B,j) consisting of a unital C*algebra B and a complete isometry j : X + B with j(e) = IB, and whose range generates B as a C* algebra, with the following universal property: For any other pair (A, i) consisting of a unital C* algebra and a complete isometry i : X + A which is unital (Le. i(e) = IA), and whose range generates A as a C* algebra, there exists a (necessarily unique, unital, and surjective) *homomorphism IT : A + B such that IT 0 i = j. Happily, this turns out to be true. The existence for any unital operator space (X, e) of a pair (B,j) with the universal property above is of course a theorem, which we call the ArvesonHamana theorem [2, 13] (see [3] for complete details). As is customary we write C;(X) for B or (B,j), this is the 'C*envelope of X'. It is essentially unique, by the universal property. If X = A is a unital operator algebra (see Section 1 for the definition of this), then j above is forced to be a homomorphism (to see this, choose an i which is a homomorphism, and use the universal property). Thus A may be considered a unital subalgebra of C;(A). If A is already a unital C* algebra, then of course we can take C; (A) = A. To help the reader get a little more comfortable with these concepts, we compute the 'noncommutative Shilov boundary' in a few simple examples. Example 1. Let Tn be the upper triangular n x n matrices. This is a unital subspace of M n , and no proper *subalgebra of Mn contains Tn. Let (B,j) be the C* envelope of Tn. By the universal property of the C* envelope, there is a surjective *homomorphism IT : Mn + B such that IT(a) = j(a) for a E Tn. The kernel of IT is a twosided ideal of Mn. However Mn has no nontrivial twosided ideals. Hence IT is 11, and is consequently a *isomorphism, and we can thus identify Mn with B. Thus Mn is a C*envelope of Tn.
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Example 2. Consider the linear subspace X of M3 with zeroes in the 13, 23, 21, 31 and 32 entries, and with arbitrary entries elsewhere except for the 33 entry, which is the average of the 11 and 22 entries. It is easy to see that the C* algebra generated by X inside M3 is M2 EB C. However this is not the C*envelope. Indeed the 33 entry here is redundant, since the norm of x E X is the norm of the upper left 2 x 2 block of x. The canonical *homomorphism from M2 EB C onto Nh when restricted to X is a unital complete isometry from X onto T2 (see Example 1). Thus if one takes the quotient of M2 EB C by the kernel of this homomorphism, namely the ideal 02 EB C, then one obtains M 2 , which by Example 1 is the C* envelope. Indeed this is typical when calculating the C* envelope of a unital subspace X of Mn. The C*algebra generated by X is a finite dimensional unital C*algebra. However such a C* algebras is *isomorphic to a finite direct sum B of full 'matrix blocks' M nk • Some of these blocks are redundant. That is, if p is the central projection in B corresponding to the identity matrix of this block, then x t+ x(IBp) is completely isometric. If one eliminates such blocks then the remaining direct sum of blocks is the C* envelope. Example 3. Let B be a unital C*algebra. Consider the unital subspace S(B) of the C*algebra M 2 (B) consisting of matrices
[~;
:1]
for all x,y E B and >.,Ji, complex scalars. We claim that M2(B) is the C*envelope C of S(B), and we will prove this using a similar idea to Example 1 above. Namely, first note that M 2(B) has no proper C*subalgebra containing S(B), Thus by the ArvesonHamana theorem there exists a *homomorphism 11' : M 2 (B) + C which possesses a property which we will not repeat, except to say that it certainly ensures that 11' applied to a matrix with zero entries except for a nonzero entry in the 12 position, is nonzero. It suffices as in Example 1 to show that Ker 11' = {a}. Suppose that 11'(x) = 0 for a 2 x 2 matrix x E M 2(B). Let Eij be the four canonical basis matrices for M 2, thought of as inside M2(B). Then 11'(E1i XEj2 ) = 11'(Eli)11'{X)11'(Ej2) = 0 for i,j = 1,2. Thus by the fact mentioned above about the 12 position, we must have EliXEj2 = O. Thus x = O. In fact a variant of the C*envelope or 'noncommutative Shilov boundary' can be defined for any operator space X. This is the triple envelope of Hamana (see [14]). This is explained in much greater detail in [3], together with many applications. For example it is intimately connected to the 'noncommutative .Mideals' recently introduced in [4]. This 'noncommutative Shilov boundary' is, as we mentioned in Section 1, a key tool for proving various BanachStone type theorems. However in the present article we shall only need the variant described earlier in this section. 3. Complete isometries between operator algebras We begin this section with a collection of very well known and simple facts about closed twosided ideals] in a C* algebra A, and about the quotient C* algebra AI]. We have that ]1.1. is a weak* closed twosided ideal in the von Neumann algebra A**, and there exists a unique orthogonal projection e in the center of A** with ]1.1. = A**(1  e). The projection 1  e is called the support projection for I, and
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1  e may be taken to be the weak* limit in A ** of any contractive approximate identity for I. Thus it follows that A ** / I J..J.. ~ A ** e as C* algebras. Therefore also A/Ie (A/ 1)** ~ A** / IJ..J.. ~ A**e
as C*algebras. Explicitly, the composition of all these identifications is a 11 *homomorphism taking an a + I in A/I, to ae = eae in A **. Here' is the canonical embedding A + A** (which we will sometimes suppress mention of). Thus A/I may be regarded as a C*subalgebra of A**, or of the C*algebra eA**e. We next illustrate the main idea of our theorem with a simple special case. (The following appeared as part of Corollary 3.2 in the original version of [5], with the proof left as an exercise). Suppose that T : A + B is a complete isometry between unital C*algebras, and suppose that T is unital too, that is T(I) = 1. Let C be the C*subalgebra of B generated by T(A). Applying the ArvesonHamana theorem 1 we obtain a surjective *homomorphism () : C + A such that (}(T(a)) = a for all a E A. If I is the kernel of the mapping (), then C / I is a unital C* algebra *isomorphic to A. Indeed there is the canonical *isomorphism 'Y : A + C / I induced by (), taking a to T(a) + I. The next point is that C/I may be viewed as we mentioned a few paragraphs back, as a C* subalgebra of C** , and therefore also of B**. Indeed if e is the central projection in C** mentioned there, then C / I may be viewed as a C* subalgebra of eC** e C eB** e C B**. In view of the last fact, the map 'Y induces an 11 *homomorphism 7r : A + B** taking an element a E A to the element of B** which equals (1)
  
T(a)e = eT(a)
eT(a)e

(these are equal because e is central in C**). Conversely, if T : A + B is a complete contraction for which there exists a projection e E B** such that eT(a)e is a 11 *homomorphism 7r, then for all a E A,
IIT(a)1I
~ Ile~ell
=
117r(a)11 = Iiall
using the fact mentioned earlier that 11 *homomorphisms are necessarily isometric. Thus T is an isometry, and a similar argument shows that it is a complete isometry. Thus we have characterized unital complete isometries T : A + B. If H is a Hilbert space on which we have represented the von Neumann algebra B** as a weak* closed unital *subalgebra, then B may be viewed also as a unital C*subalgebra of B(H), whose weak* closure in B(H) is (the copy of) B**. In this case we shall say that B is represented on H universally. (The explanation for this term is that the wellknown 'universal representation' tru of a C*algebra is 'universal' in our sense, and conversely if 7r is a representation which is 'universal' in our sense then 7r(B)" is isomorphic to 7ru(B)" ~ B**. See [27] Section 1.) If, further, e E B** is a projection for which (1) holds, then with respect to the splitting H = eH EEl (1 e)H we may write T(a) =
[7ro(·)
0]
SO '
for all a E A. We will see that this is essentially true even if T(IA) lIB: 1We remark in passing that one does not need the full strength of the ArvesonHamana theorem here, one may use the much simpler [8] Theorem 4.1.
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DAVID P. BLECHER AND DAMON M. HAY
THEOREM 3.1. Let T : A > B be a completely contractive linear map from a unital operator algebra into a unital C* algebra. Then the following are equivalent:
(i) T is a complete isometry, (ii) There is a partial isometry u E B** with initial projection e E B**, and a (completely isometric) 11 *homomorphism 1r : C;(A) 1r(1) = e, such that for all a E A T(a)e
= tL1r(a)
and 1r(a)
>
eB**e with

= u*T(a).
Moreover e may be taken to be a 'closed projection' (see [25] 3.11, and the discussion towards the end of our proof). (iii) If H is a Hilbert space on which B is represented universally, then there exist two closed subspaces E, P of the Hilbert space H, a 11 *homomorphism 1r : C;(A) > B(E) with 1r(1) = IE, and a unitary u : E > P, such that T(a)IE = u1r(a), and T(a)IE.L C p.l., for all a E A. Here E.l. for example is the orthocomplement of E in H. (iv) If H is as in (iii), then there exists two closed subspaces E, P of H, a unital 11 *homomorphism 1r : C;(A) > B(E), a complete contraction S : C; (A) > B( E.l. , p.l.), and unitary operators U : E EEl p.l. > Hand V : H > E EEl E.l., such that T( )  U [ 1r(a)
a 
S(a)
] V
for all a E A.
(v) There is a left ideal J of B, a 11 *homomorphism 1r from C;(A) into
a unital subspace of B I (J + J*) which is a C* algebra, and a 'partial isometry' u in B I J such that qJ(T(a)) = u1r(a)
&
for all a E A, where qJ is the canonical quotient map B
>
BIJ.
Before we prove the theorem, we make several remarks. First, we have taken B to be a C* algebra; however since any unital operator algebra is a unital subalgebra of a unital C* algebra this is not a severe restriction. We also remark that there are several other items that one might add to such a list of equivalent conditions. See [5, 6]. Items (ii)(iv), and the proof given below of their equivalence with (i), are new. We acknowledge that we have benefitted from a suggestion that we use the Paulsen system to prove the result. This approach is an obvious one to those working in this area (Ruan and Hamana used a variant of it in their work in the '80's on complete isometries and triple morphisms [28, 14]). However we had not pushed through this approach in the original version of [5] because this method does not give several of the results there as immediately. Statement (v) above has been simply copied from [5, 6] without proof or explanation. We have listed it here simply because Theorem 1.3 may be particularly easily derived from it as the special case when A and B are commutative (see comments below). Note that (iii) above resembles Theorem 1.2 superficially.
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PROOF. The fact that the other conditions all imply (i) is easy, following the idea in the paragraph above the theorem, namely by using the fact that a 11 *homomorphism is completely isometric. In the remainder of the proof we suppose that T is a complete isometry. We view A as a unital subalgebra of C;(A) as outlined in Section 3. We define a subset S(B) of M 2(B) as in Example 3 in Section 2. Similarly define a subset S(T(A)) of S(B) using a similar formula (note that S(T(A)) has 12 entries taken from T(A) and 21 entries taken from T(A)*). Similarly we define the subset S(A) of the COalgebra M2(C;(A)) (Le. S(A) has scalar diagonal entries and off diagonal entries from A and A*). We write 1 EEl 0 for the matrix in S(A) with 1 as the 12 entry and zeroes elsewhere. Similarly for 0 EEl 1. We also use these expressions for the analogous matrices in S(B). The map q, : S(A) + S(T(A)) c M 2(B) taking
[~!
:1] ~
[T~)* ~~~)]
is well known to be a unital complete isometry (this is the well known Paulsen lemma, see the proof of 7.1 in [23]). Let C be the C' subalgebra of M2 (B) generated by S(T(A)). The COenvelope of S(A) is well known to be M2(C;(A)) (see Example 3 in Section 2 where we proved this in the case that A is already a C* algebra, or for example [3] Proposition 4.3 or [30]). Thus by the ArvesonHamana theorem we obtain a surjective *homomorphism () : C + 1V/z(C;(A)) such that () 0 q, is simply the canonical embedding of S(A) into M2(C;(A)). As in the special case considered above the theorem, we let 10 be the kernel of the mapping (), then C /10 is a unital COalgebra *isomorphic to M2(C;(A)). Indeed there is the canonical *isomorphism "(: M2(C;(A)) + Clio induced by (), taking
[~!
:1] ~
[T~:)* T~~)]
+
10 ,
As in the simple case above the theorem, C /10 may be viewed as a C* subalgebra of PoC"Po, for a central projection Po E C** (namely, the complementary projection to the support projection of 10)' Now PoC**Po C C** C M2(B)**, and it is well known that M2(B)** ~ M2(B**) as COalgebras. Thus we may think of C** as a C* subalgebra of .1112 (B**). Also, C'* contains C as a C* subalgebra, and the projections 1 EEl 0 and 0 EEl 1 in C correspond to the matching diagonal projections 1 EEl 0 and 0 EEl 1 in M2(B**). These last projections therefore commute with Po, since Po is central in C**, which immediately implies that Po is a diagonal sum fEEl e of two orthogonal projections e, f E B**. Thus we may write the C* algebra POM2(B**)po as the C*subalgebra [ fB** f eB** f
f B**e ] eB*'e
of M2(B**). We said above that Clio may be regarded as a C*subalgebra of the subalgebrapoM2(B**)po of M2(B*'). Thus the map "( induces a 11 *homomorphism III : M2(C;(A)) + .!I1z(B**). It is easy to check that 1lI(1 EEl 0) = fEEl 0 and III (0 EEl 1) = 0 EEl e. Since III is a *homomorphism it follows that III maps each of the four corners of M 2(C;(A)) to the corresponding corner of POM2(B**)po C M2(B**). We let R: C;(A) + fB**e be the restriction of III to the '12corner'. Since III is 11, it follows that R is 11. If 7r is the restriction of III to the '22corner', then 7r is a *homomorphism C;(A) + eB**e taking lA to e. Applying the *homomorphism
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111 to the identity
[~ ~][~ ~]=[~ ~] we obtain that u = R(I) is a partial isometry, with u*u = 71"(1) = e. Similarly uu* = f. A similar argument shows that R(a) = R(I)7I"(a) for all a E C;(A). Thus u* R(a) = u*u7l"(a) = 7I"(a) for all a E C;(A). Next, we observe that 111 takes the matrix z which is zero except for an a from A in the 12corner, to the matrix w = Pocp(z)Po. Since cp(z) E C** and Po is in the center of that algebra, we also have w = cp(z)Po = Pocp(z). Also w viewed as a matrix in M2(B**) has zero entries except in the .12corner, which (by the last sentence) equals
 
ff(;;) e = f(;;) e = ff(;;). Also using these facts and a fact from the end of the last paragraph we have ...
* 
*
u*T(·) = R(I)*T(·) = (fT(I)e)*T(·) = eT(I) fT(·) = eT(I) T(·)e = u* R(·) = 71". Thus



T(·)e = fT(·) = uu*T(·) = U7l"(')' We have now also established most of (ii). One may deduce (iii) from (ii) by viewing B c B** c B(H), and setting E = eH, and F = (uu*)H. We also need to use facts from the proof above such as u*u = e. Clearly (iv) follows from (iii). As we said above, we will not prove (v) here. Claim: if e is the projection in (ii) above, then 1  e is the support projection for a closed ideal I of a unital *subalgebra D of B. Equivalently (as stated at the start of this section), there is a (positive increasing) contractive approximate identity (bd for I, with bt ~ 1  e in the weak* topology. This claim shows that 1  e is an 'open projection' in B**, so that e is a closed projection, as will be obvious to operator algebraists from [25] section 3.11 say. For our other readers we note that for what comes later in our paper, one can replace the assertion about closed projections in the statement of Theorem 3.1 (ii) with the statement in the Claim above. To prove the Claim, recall from our proof that Po = fEB e = Ie  Pi, where Pi is the support projection for a closed ideal 10 of C. Thus Pi = (1  f) EB (1  e). As stated at the start of Section 3, Pi is the weak* limit in C**, and hence also in M 2(B**), of a contractive approximate identity (et) of 10, By the separate weak* continuity of the product in a von Neumann algebra, it follows that the net bt = (OEB l)et(OEB 1) has weak* limit (OEB I)Pl (OEB 1) = OEB (1 e). Viewing these as expressions in B, the above says that bt ~ 1  e weak* in B* *. View (0 EB 1) C (0 EB 1) as a *subalgebra D of B, and view (0 EB 1)10(0 EB 1) as a two sided ideal I in D. It is easy to see that (bd is a contractive approximate identity of I. Thus it follows that 1  e is the support projection of the ideal I. 0 Some applications of results such as Theorem 3.1 may be found in [6]. Next we discuss briefly the relation between our noncommutative characterization of complete isometries (for example Theorem 3.1 above), and Theorem 1.3. Our point is not to provide another proof for Theorem 1.3  the best existing proof is certainly short and elegant. Rather we simply wish to show that the noncommutative result contains 1.3. Indeed Theorem 1.3 quite easily follows from Theorem
COMPLETE ISOMETRIES  AN ILLUSTRATION
95
3.1 (v). Since however we did not prove Theorem 3.1 (v), we give an alternate proof. COROLLARY 3.2. Let A, B be a unital function algebras, with B selfadjoint. Then condition (ii) in Theorem 3.1 implies condition (A) in Theorem 1.3. PROOF. By hypothesis, T(·)e = U1l"('), and u*u = e = 11"(1) so that u = u1l"(1) = T(I)e. Thus eT(I)*T(·)e = u*U1l"(') = 11"(1)11"(') = 11", so that Ran 11" C eBe = Be (note B** is commutative in this case). From [25] 3.11.10 for example, the 'closed projection' e in B** corresponds to a closed ideal J in B whose support projection is 1  e. Alternatively, to avoid quoting facts from [25], we will also deduce this from the 'Claim' towards the end of the proof of Theorem 3.1. If I is the ideal in that Claim, let J be the closed ideal in B generated by I. Since J = Bl, the contractive approximate identity of I is a right contractive approximate identity of J. Thus J has support projection 1  e too, by the first paragraph of Section 3 above. By facts in the just quoted paragraph, we have a canonical unital 11 map '11 : B / J ~ B** taking the equivalence class b + J of b E B to ebe. Indeed in this commutative case we see by inspection that '11 is a *homomorphism from the C*algebra B/J onto the C*subalgebra M = eBe of B**. Define O(a) = '111(1I"(a)), this is a 11 *homomorphism A ~ B/J. Since 11"(1) = e, 0 is a unital map too. Since uu* = u*u = e, u is unitary in M, and so 'Y = '111(U) is unitary in B/J. Note also that T(a)e = '11(T(a) + J). Applying '11 1 to the equation T(·)e = U1l"('), we obtain qJ(T(a)) = 'Y O(a), that is, condition (A) in Theorem 1.3. D If one attempts to use the ideas above to find a characterization analogous to condition (A) from Theorem 1.3 but in the noncommutative case, it seems to us that one is inevitably led to a condition such as (v) in Theorem 3.1. We address a paragraph to experts, on generalizations of the proof of Theorem 3.1. Consider a complete isometry between possibly nonunital C* algebras. Or much more generally, suppose that T is a complete isometry from an operator space X into a C* triple system W. One may form the so called 'linking C* algebra' of W, with the identities of the 'left and right algebras of W' adjoined. Call this C'(W). As in the proof of Theorem 3.1 we think of S(W) c .c'(W). Similarly, if Z is the 'triple envelope' of X (or if X = Z is already a C*algebra or C*triple system), then we may consider S(X) C S(Z) c .c'(Z). As in the proof of Theorem 3.1 we obtain firstly a unital complete isometry : S(X) ~ S(T(X)) c .c'(Z), and then a unital 11 *homomorphism 11" : .c'(Z) ~ .c'(W)**. By looking at the 'corners' of 11" we obtain projections e, f in certain second dual von Neumann algebras, so that fT( ·)e is (the restriction to X of a completely isometric) a 11 triple morphism into W**. In fact we have precisely such a result in [5] (see Section 2 there), but the key point is that the new proof gives different projections e, f, which are more useful for some purposes.
4. Complete isometries versus isometries Finally, as promised we discuss why we believe that in this setting of nonsurjective maps between C* algebras say, general isometries are not the 'noncommutative analogue' of isometries between function algebras. The point is simply this. In the function algebra case we can say thanks to Holsztynski's theorem that the isometries are essentially the maps composed of two disjoint pieces Rand S, where R is
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DAVID P. BLECHER AND DAMON M. HAY
isometric and 'nice', and S is contractive and irrelevant. However at the present time it looks to us unlikely that there ever will be such a result valid for general nonsurjective isometries between general C* algebras. The chief evidence we present for this assertion is the very nice complementary work of Chu and Wong [9] on isometries (as opposed to complete isometries) T : A + B between COalgebras. They show that for such T there is a largest projection p E B** such that T(·)p is some kind of Jordan triple morphism. This appears to be the correct 'structure theorem', or version of Kadison's theorem [11], for nonsurjective isometries. However as they show, the 'nice piece' R = T(·)p is very often trivial (Le. zero), and is thus certainly not isometric. Thus this approach is unlikely to ever yield a characterization of isometries. A good example is A = M 2 , the smallest noncommutative C* algebra. Simply because A is a Banach space there exists, as in the discussion in the first paragraph of our paper, a linear isometry of A into a C (K) space. However it is easy to see that there is no nontrivial *homomorphism or Jordan homomorphism from A into a commutative C* algebra. Such an isometry is uninteresting, and this is perhaps because the interesting 'nice part' is zero. Thus we imagine that the 'good noncommutative notions of isometry' are either complete isometries or the closely related class of maps for which the piece T(·)p from [9] is an isometry. This leads to three questions. Firstly, can one independently characterize the last mentioned class? Secondly, if T is a complete isometry, then is the projection p in the last paragraph equal (or closely related) to our projection e above? Finally, H. Pfitzner has remarked to us, there is already a gap between the isometry and the 2isometry cases (not only isometries and complete isometries). It would be interesting if there were a characterization of 2isometries.
References [1] J. Araujo and J. J. Font, Linear isometries between subspaces of continuous functions, Trans. Amer. Math. Soc. 349 (1997), 413428. [2] W. B. Arveson, Subalgebms ofC*algebms, Acta Math. 123 (1969), 141224; II, 128 (1972), 271308. [3] D. P. Blecher, The Shilov boundary of an opemtor space, and the chamcterization theorems, J. Funct. An. 182 (2001),280343. [4] D. P. Blecher, E. G. Effros and V. Zarikian, Onesided MIdeals and multipliers in opemtor spaces. 1. To appear Pacific J. Math. [5] D. P. Blecher and D. Hay, Complete isometries into C* algebms, http://front.math.ucdavis.edu/math.OA/0203182, Preprint (March '02). [6] D. P. Blecher and L. E. Labuschagne, Logmodularity and isometries of opemtor algebms, To appear, Trans. Amer. Math. Soc .. [7] D. P. Blecher and C. Le Merdy, Opemtor algebms and their modules  an opemtor space approach, To appear, Oxford Univ. Press. [8] M. D. Choi and E.G. Effros, The completely positive lifting problem for C·algebms, Ann. Math. 104 (1976), 585609. [9] CH. Chu and NC. Wong, Isometries between C· algebms, Preprint, to appear Revista Matematica Iberoamericana. [10] J. B. Conway, A Course in Opemtor Theory, AMS, Providence, 2000. [11] E. G. Effros and Z. J. Ruan, Opemtor Spaces, Oxford University Press, Oxford (2000). [12] R. J. Fleming and J. E. Jamison, Isometries on Banach spaces: function spaces, Book to appear, CRC press. [13] M. Hamana, Injective envelopes of opemtor systems, Pub!. R.I.M.S. Kyoto Univ. 15 (1979), 773785.
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COMPLETE ISOMETRIES  AN ILLUSTRATION
[14] M. Hamana, Triple envelopes and Silov boundaries of operator spaces, Math. J. Toyama University 22 (1999), 7793. [15] W. Holsztynski, Continuous mappings induced by isometries of spaces of continuous functions, Studia Math. 24 (1966), 133136. [16] K. Jarosz and V. Pathak, Isometries and small bound peturbations of function spaces, In "Function Spaces", Lecture Notes in Pnre and Applied Math. Vol. 136, Marcel Dekker (1992). [17] R. V. Kadison, Isometries of operator algebras, Ann. of Math. 54 (1951), 325338. [18] E. Kirchberg, On restricted peturbations in inverse images and a description of normalizer algebras in C*algebras, J. Funct. An. 129 (1995), 134. [19] E. Kirchberg and S. Wassermann, C*algebras generated by operator systems, J. Funct. Analysis 155 (1998), 324351. [20] A. Matheson, Isometries into function algebras, To appear. [21] M. Nagasawa, Isomorphisms between commutative Banach algebras with an application to rings of analytic functions, Kodia Math. Sem. Rep. 11 (1959), 182188. [22] W. P. Novinger, Linear isometries of subspaces of continuous functions. Studia Math. 53 (1975), 273276. [23] V. I. Paulsen, Completely bounded maps and dilations, Pitman Research Notes in Math., Longman, London, 1986. [24] V. I. Paulsen, Completely bounded maps and operator algebras, To appear Cambridge University Press. [25] G. Pedersen, C*algebras and their automorphism groups, Academic Press (1979). [26] G. Pisier, Introduction to operator space theory, To appear Camb. Univ. Press. [27] M. Rieffel, Morita equivalence for C*algebras and W*algebras, J. Pure Appl. Algebra 5 (1974), 5196. [28] Z. J. Ruan, Subspaces ofC*algebras, Ph. D. thesis, U.C.L.A., 1987. [29] E. L. Stout, The theory of uniform algebras, Bogden and Quigley (1971). [30] C. Zhang, Representations of operator spaces, J. Oper. Th. 33 (1995), 327351. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF HOUSTON, HOUSTON.
Email address, David P. Blecher: dblecher~ath. uh. edu Email address, Damon Hay: dhayOmath.uh.edu
TX 772043008
Contemporary Mathematics Volume 328, 2003
Some Recent Trends and Advances in Certain Lattice Ordered Algebras Karim Boulabiar, Gerard Buskes, and Abdelmajid Triki ABSTRACT. In this paper we give a survey, intended as both a supplement as well as an update to a survey by Huijsmans [57], with results that have been obtained in the last ten years on Archimedean lattice ordered algebras. Special attention is paid to Ialgebras, almost Ialgebras and dalgebras and problems that were posed in the survey by Huijsmans about these special classes of lattice ordered algebras.
CONTENTS
l. Introduction 2. Definitions and elementary properties 3. ialgebra multiplications in C (X) 4. Multiplication by an element as an operator 5. Uniform completion and Dedekind completion 6. Powers in ialgebras 7. Functional Calculus on falgebras 8. Relationships between ialgebra multiplications 9. Connection between algebra and Riesz homomorphisms 10. Positive derivations 11. CauchySchwarz inequalities 12. Order biduals 13. Ideal theory 14. Representation of falgebras 15. Linear biseparating maps on falgebras References 1991 Mathematics Subject Classification. 06F25, 13J25, 16W80, 46A40, 46840, 46842, 46E25, 47L07, 47847, 47865. Key words and phrases. almost Ialgebra, algebra homomorphism, Ialgebra, dalgebra, lattice ordered algebra, order ideal, orthomorphism, representation theory, Riesz homomorphism, ring ideal, space of continuous functions, uniformly complete Riesz space. The second named author gratefully acknowledges support from an Office of Naval Research Grant with number N000140110322. Part of this survey was written while the first named author was visiting the University of Mississippi in the Spring of 2002. © 2003 American Mathematical Society
99
~!3QtJLA8lAR,
GERARD BUSKES, AND ABDELMAJID TRIKI
1. Introduction
ftehistory of lattice ordered vector spaces (so called Riesz spaces or vector
lattices) goes back to Riesz and the International Congress of Mathematicians in Bologna in 1928. A study of the most important class of lattice ordered algebras (but not their name), Jalgebras, was initiated by Nakano in [75] for aDedekind complete ordered vector space in 1951, subsequently in 1953 by Amemiya in [3], and finally with its present definition and name in 1956 by Birkhoff and Pierce in [22]. A precise date for the very first definition of lattice ordered algebras in general is very hard to pinpoint, but originated at around the same time as the previous three references. Indeed, in his review [44] of Birkhoff's 1950 address to the International Congress of Mathematicians in Cambridge, Massachusetts, Fl.·ink observed that a general study of lattice ordered rings seems to be needed to study what are now called averaging or Reynolds operators. A call for lattice ordered rings also had gone out by Birkhoff himself in the form of a listed problem at the end of his seminal 1942 paper [20] on lattice ordered groups. Thus lattice ordered algebras and Jalgebras seem to have multiple origins, including a study of averaging operators, which themselves sprang forth from problems in fluid mechanics. An appearance at about the same time of Jrings and Jalgebras has not resulted in an historical development on complete common ground for these objects. This is not unlike the development of lattice ordered groups versus the development of Riesz spaces. Where the latter have attracted attention from researchers in analysis, the former have been more widely investigated by algebraists. A similar divided attention from analysis versus algebra seems to underlie the connected but somewhat separate tracks of lattice ordered rings versus lattice ordered algebras. Though this separation of tracks is to some extent unavoidable, where each track does have ground that is truly its own, some overlap in results does exist, resulting in difficulties making accurate literature attributions in a survey like ours. We are grateful to two referees for pointing at some references that were missing in our manuscript, though we take full responsibility for possible remaining omissions in the reference list. This survey places itself almost completely on the track of lattice ordered algebras and our only apology for not linking algebra facts in a systematic way to ring results is that all three of us authors were trained as analysts. There is a natural back and forth between the two theories, in one direction by forgetting some of the structure, and in the other by finding, so to speak adjointly, an enveloping algebra. A nice survey on Jrings was written by Henriksen in 1995 (see [50]), to which we refer the interested reader for linkage to some of what follows in this survey. Historically, a lot of the credit for a revival of the theory of Jalgebras points to the highly motivating Arkansas Lecture Notes by Luxemburg [67] and the 1982 Ph.D. thesis of de Pagter [76], who systematically explored both the existing literature as well as new directions. Another impetus to research in the area of Jalgebras derived from the desire of Zaanen, who in the late seventies started to develop a program to prove many of the elementary results in the theory of Riesz spaces without using representation theorems for vector lattices. This desire is directly linked with a preference not to use the Axiom of Choice unnecessarily. The present survey is intended as an update to the one by Huijsmans [57]. We hasten to point out that we do not intend this survey to replace the one by Huijsmans, but rather that we think of it as augmenting part of it. Since [57] appeared, much progress has been made and several of the problems explicitly phrased in
LATTICE ORDERED ALGEBRAS
101
[57] have been solved. At the same time, some topics like ideal theory, connections between Riesz homomorphisms and algebra homomorphisms, and representations of falgebras were absent in [57]. Thus an update as well as a supplement was needed. However, some sections of [57] receive no attention at all in this survey. We have not included important topics like the role of falgebras in positive operator theory (e.g., we do not even include results on the previously mentioned averaging operators) and probability theory. We are rather focused on placing this update, as much as possible, in the setting of spaces of functions, hoping to interest as large an audience as possible via this approach. Moreover, we feel that the great source of inspiration was and continues to be the beautiful book by Gillman and Jerison [47], which certainly inspired and continues to inspire a large part of the research in lattice ordered algebras and rings. Last but not least we focus on what could be called distortions of falgebras, in particular on Archimedean almost falgebras and dalgebras. Though these distortions have the potential to be seen as aberrations by some, we believe they point the way to techniques that are needed for the broader theory of lattice ordered algebras, as well as for illumination of various aspects in the theory of falgebras. Note that the distortions disappear if the lattice ordered algebra under consideration has a multiplicative identity which is a weak order unit. Indeed, such algebras are automatically falgebras. It should be mentioned that classes of lattice ordered algebras other than the ones that appear in this survey have been studied. Notably, the papers [85], [86], [87], and [88] by Steinberg discuss lattice ordered algebras in which every square is positive, a class of algebras that includes all almost falgebras. We also pay no attention at all to nonArchimedean lattice ordered algebras. Finally, we point out that several results in this survey rely heavily on the (relative) uniform topology on Riesz spaces. In particular, since in Archimedean Riesz spaces uniform limits are unique, we shall include the 'Archimedean' property in the definition of uniformly complete Riesz spaces. A complete investigation of that topology can be found in Sections 16 and 63 of [69]. For terminology and concepts not explained or proved in this survey we refer the reader to the standards books [2], [47], [69], [72], [92] and [93].
2. Definitions and elementary properties A (real) Riesz space A is called a lattice ordered algebra (briefly, an ialgebra) if A also is an algebra and the positive cone A+ = {J E A : f ~ O} is closed under multiplication, that is, if f,g ~ 0 then fg ~ 0 (equivalently, if Ifgl :::; Ifllgl for all f,g E A).
We make the following blanket assumption: all Riesz spaces under consideration in this paper are assumed to be Archimedean (however, the latter blanket assumption has not stopped us to explicitly add the word Archimedean to the list of conditions in various results below). After B.irkhoff and Pierce (see [22, p. 55]), we define an ialgebra A to be an falgebra if for every f, 9 E A, the condition f /\g = 0 implies Uh) /\g = (hi) /\g = 0 for all h E A+
102
KARIM BOULABIAR. GERARD BUSKES, AND ABDELMA.lID TRIKI
holds. We call the Ealgebra A an almost falgebra after Birkhoff in [21, Section 6] if f /\ g
= 0 ill A implies
fg
= O.
An Ealgebra A for which f /\ 9 = 0 in A and h E A+ imply (.fh) /\ (gh)
=
(hf) /\ (hg) = 0
is called a dalgebra. The notion of dalgebra goes back to Kudlacek in [65]. Our focus in this survey on Ealgebras is almost exclusively on f algebras, almost falgebras and dalgebras. In this paragraph, we recall some properties of falgebras. Using the Axiom of Choice, Birkhoff and Pierce in [22, Theorem 13] proved that any falgebra is commutative. A constructive proof of this fact, due to Zaanen, can be found in [61, Theorem 2.1] or [92, Theorem 140.10]. All squares in an falgebra are positive. Also, Ifgl = Ifllgl for all f,g in an falgebra A. The multiplication by an element in the f algebra A is order continuous, i.e., if inf {fT : T} = 0 in A then inf {g fT : T} = o for all 9 E A +. Phrased more generally, the multiplication 7rf by an element f E A (7rf (g) = f 9 for all 9 E A) is an orthomorphism and all orthomorphisms are order continuous. Recall that an orthomorphism on a Riesz space L is an order bounded linear operator 7r such that 17r (f)I/\ Igl = 0 whenever Ifl/\ Igl = 0 in L (the reader is referred to [2] or [92] for elementary properties of orthomorphisms). There is another important relationship between orthomorphisms and falgebras, which we mention next. Indeed, let Orth (L) be the set of all orthomorphisms on a Riesz space L. Under the operations and the ordering inherited from £b (L), the ordered algebra of all order bounded operators on L, and under composition as multiplication, Orth (L) is an Archimedean falgebra with the identity map h on L as unit element. The details of the facts recalled above can all be found in [2], [76] or [92]. Next we present some properties of almost falgebras. Almost falgebras, like falgebras, are commutative too. The latter fundamental property was first established by Scheffold in [80, Theorem 2.1] for almost falgebras that are Banach lattices. Using both Scheffold's result and the Axiom of Choice, Basly and Triki were the first to prove commutativity for arbitrary almost falgebras [10, Thorme 1.1]. The first proof of the commutativity for almost falgebras within ZermeloFraenkel set theory was given by Bernau and Huijsmans in their paper [13, Theorem 2.15]. Recently, a shorter constructive proof was published in [34, Corollary 3] by Buskes and van Rooij. Another property of falgebras holds for almost falgebras, namely the positivity of squares. Also, if A is an almost falgebra then f2 = Ifl2 for all f E A. However, contrary to the order continuity of the multiplication in falgebra..'l, the multiplication by a fixed element in an almost falgebra is not always order continuous as is shown in the following example. EXAMPLE 2.1. Write A = C ([0,1]), the vector space of all realvalued continuous functions on [0, 1]. With respect to the pointwise ordering (i. e., f :::; 9 in A if an only if f (x) :::; 9 (x) for all x E [0,1]), A is an Archimedean Riesz space. Define a multiplication • in A by
(f. g)(x)
=
{
f(x)g(x)
f (1/2) 9 (1/2)
(0 :::; x :::; 1/2) ; (1/2 :::; x :::; 1)
LATTICE ORDERED ALGEBRAS
103
lor all I, 9 E A. Then A is an almost Ialgebm with respect to the multiplication •. For every natuml number n :::: 1, define the function In E A by (O :::; x :::; 1/2  l/n) ; (1/2  l/n :::; x :::; 1/2) ; (1/2:::; x :::; 1/2 + l/n) ; (1/2+1/n:::;x:::;1).
I f (:1") _ { n/2  n:1: " ,n/2 + nx 1
Then sup Un : n = 1,2, ... } exists in A and equals the function e defined bye (x) = 1 lor all x E [0,1]. On the other' hand,
(O :::; x :::; 1/2) ; (1/2:::;x:::;1)
lor all n E {I, 2, ... }, and clearly the set {e. In : n = 1,2, ... } does not have a supremum in A. We conclude that. is not order continuous. For more information about elementary theory of almost falgebras, the reader is encouraged to consult [13], [23], [34], and [35]. At this point, we turn our attention to some properties of dalgebras. It follows directly from the definition of dalgebras that an ialgebra A is a dalgebra if and only if the multiplication map induced by any fixed element in A + is a Riesz (or lattice) homomorphism. It follows that a necessary and sufficient condition for an ialgebra A to be a dalgebra is that the identity If gl = 1/IIgi holds for all f, 9 in A. Contrary to (almost) Ialgebras, dalgebras need not be commutative nor have positive squares. Next we give an example of a noncommutative dalgebra in which not all squares are positive. EXAMPLE
2.2. Let in this example A be the algebra of real {2 x 2)matrices of
the form
(~
g)
with the usual addition, scalar multiplication, matrix product and partial or·dering. It is not hard to see that A is an Archimedean dalgebra. But A is not commutative and not all squares in A are positive. Indeed, if p=
(~ ~)
and q =
(~ ~)
then pq
=q
and qp = O.
Moreover, the square
is not positive. As for almost falgebras, multiplication by a fixed element in a dalgebra is, in general, not order continuous. Point in case is the almost Ialgebra that we considered in Example 2.1, which also is a dalgebra. Our main reference about dalgebras is [13]. In what follows, we look at some of the connections between the three kinds of ialgebras that we consider in this paper. It is immediate that any Ialgebra is both, an almost falgebra and a dalgebra. Almost falgebras need not be dalgebras as we see in the next example.
KARIM BOULABIAR, GERARD BUSKES, AND ABDELMAJID TRIKI
104
EXAMPLE 2.3. Take A as in Example 2.1, and let () E A be the function () (x) = { 1/2  x x  1/2
(0::; x ::; 1/2) ; (1/2::; x ::; 1).
For f, 9 E A, define (f
9
)( ) _ {
x 
/,1X f(s)g(s)ds
(0::; x ::; 1/2);
1/2
(1/2::; x ::; 1).
() (x) f (x) 9 (x)
Then A is an almost f algebra under the multiplication . However, A is not a dalgebra. Indeed, let e, f in A be defined by e(x)=l and
={
f (x)
forallxE[O,l]
4x + 1 4x3
(0::; x ::; 1/2) ; (1/2::;x::;1).
If  el (0) = 0 and (If I e) (0) = If I Igl fails in A.
Observe that
If  gl =
/,1
If(s)1 ds ; O. Thus the property
1/2
Since every almost f algebra is commutative and has positive squares, Example 2.2 shows that dalgebras need not be almost falgebras. However, if a dalgebra A is commutative or has positive squares then A is automatically an almost falgebra [22, p. 60]. Summarizing part of the relations, we have the following diagram falgebra::::} commutative dalgebra ::::} almost falgebra. For more detail, see [22], [13] and [57]. The next lines deal with nilpotent element in ialgebras. The set of all nilpotent elements in the ialgebra A is denoted by N (A). In other words,
N (A)
= {f E A : r = 0 for some n =
1, 2, ... } .
Given a natural number p, we define Np (A)
= {f
E A : fP
= O} .
The ialgebra A is said to be semiprime if 0 is the only nilpotent element in A, that is, if N (A) = {O} . If A is an falgebra then the following equalities hold N (A)
= N2 (A) = {f E A
: fg
= 0 for all 9 E A}.
(see [76, Proposition 10.2] or [92, Theorem 142.5]). If A is an almost falgebra then N (A)
= N3 (A) = {J E A
: fg2
= 0 for all 9 E A}
= {f E A : fgh = 0 for all g, hE A}
(see [51, Theorem 3.11]) and, as for falgebras, N2 (A)
= {f
E A : fg
= 0 for
all 9 E A}
(see [23, Lemma 5.3]). If A is a dalgebra then N (A)
= N3 (A) = {f E A
: gfh
= 0 for all 9
• A}
(see [13, Theorem 5.5] or [29, Theorem 5]). However, and co we documented for falgebras and almost falgebras, in dalgebl
ry to what he equality
LATTICE ORDERED ALGEBRAS
105
N2 (A) = {J E A: fg = 0 for all 9 E A} does not necessarily hold as is illustrated by the following example.
EXAMPLE 2.4. Take A as in Example 2.1 and define the multiplication. in A by (f. g)(x) = f (0) 9 (1) for all f, 9 E A. Let f E A be the function defined by f (x) = 1  x for all x E [0,1]. Clearly, f. f = 0 but f. e i= 0 where e E A is defined bye (x) = 1 for all x E [0,1].
Finally, note that any falgebra with multiplicative identity is semiprime and any semiprime almost falgebra or semiprime dalgebra is automatically an falgebra (see Section 1 in [13]). As a final comment we remark that an lalgebra which has positive squares and has a multiplicative identity need not be an falgebra.
3. ialgebra multiplications in C (X) Let C (X) be the set of all realvalued continuous functions on a compact Hausdorff' topological space X. Under pointwise addition and scalar multiplication, C (X) is a real vector space. Moreover, C (X) is an Archimedean Riesz space with respect to the pointwise ordering (i.e., f ~ 9 in C (X) if and only if f (x) ~ 9 (x) for all x E X). By defining the multiplication in C (X) pointwise as well (Le., (fg) (x) = f(x)g(x) for all f,g E C(X) and all x E X), the space C(X) is easily seen to have the structure of an falgebra with e as unit element, where e (x) = 1 for all x E X. Now consider another associative multiplication. in C (X). The main topic of this section is to produce necessary and sufficient conditions for C (X) to be an falgebra (respectively, an almost falgebra, a dalgebra) with respect to this new multiplication •. The first theorem in this direction goes back to Conrad (see [38, Theorem 2.2]), who obtained the following. THEOREM 3.1. Let. be an associative multiplication in C (X). Then C (X) is an f algebra with respect to • if and only if there exists a positive function w E C (X) such that (f.g)(x) =w(x)f(x)g(x) for all f,g E C (X) and all x EX.
In fact, Conrad established the theorem above for any Archimedean fring with unit element. The representation formula given in Theorem 3.1 above was obtained in an alternative way by Scheff'old in [80, Korollar 1.4]. While Conrad's proof is purely algebraic and order theoretic, the proof presented by Scheff'old relies on analytic tools like the Riesz representation theorem. With the same analytic tools Scheff'old also obtained the following representation theorem for almost falgebra multiplications in C (X) (see [80, Theorem 1.2]). THEOREM 3.2. Let. be an associative multiplication in C (X). Then C (X) is an almost f algebra with respect to • if and only if there exists a family (/Jx : x E X) of positive measures such that (f. g)(x) for all f, 9 E C (X) and all x E
x.
=
Ix
f(s) 9 (s) d/Jx (s)
106
KARIM BOULABIAR, GERARD BUSKES. AND ABDELMAJID TRIKI
Since every commutative dalgebra is an almost Ialgebra, the previous theorem remains valid for commutative dalgebra multiplications in C (X) as well. Recently in [25, Corollary 3.2], Boulabiar proved the following representation formula for any (not necessarily commutative) dalgebra multiplication in C (X). THEOREM 3.3. Let. be an associative multiplication in C (X). Then C (X) is a dalgebra with respect to • if and only if there exist (i) a positive function wE C (X), and (ii) functions h,k: X + X (continuous on coz(w) = {x EX: w(x)"! O}) such that (J. g) (x) = w (x) I (h (x)) 9 (k (x)) for all I,g E C(X) and all x E X.
Notice that if C (X) is a dalgebra with respect to the multiplication. then • is commutative if and only if the functions hand k coincide on coz (w) (where 11" k and ware as in Theorem 3.3). The latter observation yields. in addition to the formula cited in Theorem 3.2 above, another representation for commutative dalgebra multiplications on C (X) . More abstract versions of the results above will be given in Section 8 below. 4. Multiplication by an element as an operator Let A be an (algebra and recall that C b (A) denotes the ordered algebra of all order bounded operators on A. For every f E A, we define the map 7rf on A by 7rf (g) = f 9 for all 9 E A. Clearly, 7rf is an order bounded operator on A for all lEA. The map p : A + Cb (A) defined by p (J) = 7rf for all I E A is obviously an algebra homomorphism, that is, p (J g) = p (J) p (g) for all I, 9 E A. Hence the range p (A) of p is a subalgebra of Cb (A). In this section, we will see that if A is an almost falgebra then p (A) can canonically be equipped with an ordering, under which p (A) is an Archimedean Ialgebra. A corresponding result will also be given for commutative dalgebras and falgebras. Let A be an almost Ialgebra. Since
N2 (A) = {f E A : Ig = 0 for all 9 E A}, = 7r9 if and only if f by putting
7rf

9 E N 2 (A). This allows us to define an ordering on p (A)
(0) The ordering defined by (0) coincides with the ordering inherited from Cb (A), namely, 7rf is positive with respect to (0) if and only if 7rf is a positive operator on A. Under the usual addition and composition of operators, and with the ordering defined by (0), p (A) is an Archimedean ordered subalgebra of Cb (A). In fact, we have the following theorem (see Theorem 4.2 and Theorem 4.4 in [23]). THEOREM 4.1. Let A be an Archimedean almost I algebra. Then p (A) is an Archimedean I algebra with respect to the addition and composition of operators, and the ordering inherited from Cb (A). The lattice operations in p (A) are given by 7r f
V 7r9
= 7rfV 9'
7r f 1\ 7r9
= 7rf /1.9
for all f, 9 EA.
In particular, (7rf)+
= 7rf+,
(7rf)
= 7rf'
l7rfl
= 7rlfl
for all lEA.
LATTICE ORDERED ALGEBRAS
107
In other words, p defines a surjective Riesz homomorphism from A onto p (A).
Theorem 4.1 of course holds in commutative dalgebras. Moreover, let A be a commutative dalgebra. For f E A and 9 E A + the equalities 17I'fl (g)
= 71'tft (g) = Iflg = sup{lfllhl : Ihl
::; g}
= sup {Ifhl : Ihl ::;g} = sup {17I'f (h)1 : Ihl ::;g} imply that 71'f has an absolute value in Lb (A), which coincides with its absolute value in p (A). We collect the latter observations for commutative dalgebras. THEOREM 4.2. Let A be an Archimedean commutative dalgebra. Then p (A) is an Ar'chimedean f algebra when equipped with the addition and composition of operators, and the ordering inherited from Lb (A). Moreover, the absolute value 71'tft of 71'f in p (A) coincides with the absolute value of 71' f in Lb (A) for all f E A. that is, 17I'fl (g) = 71'tft (g) = sup {17I'f (h)1 : Ihl ::; g} for all 9 E A+.
We obtain the falgebra case as a corollary. COROLLARY 4.3. Let A be an Archimedean falgebra. Then p (A) is an fsubalgebra of the Archimedean falgebra Orth (A) of all orthomoTphisms of A.
The fact that the range of p in Corollary 4.3 is an f algebra was first proved in [22, Corollary 3, p. 57] by Birkhoff and Pierce, while the fact that Orth (A) itself is an falgebra has been proved in [18] by Bigard and Keimel and in [39] by Conrad and Diem. This topic was also discussed in great detail by de Pagter in his thesis [76, Proposition 12.1]. Note that if A is an falgebra then A is semiprime if and only if p is onetoone as a map from A into Orth (A). In this case, A and p (A) are isomorphic a.'! falgebras. Also, if A is an falgebra then A has a multiplicative identity if and only if the map p is onetoone and onto as a map A > Orth (A), and consequently A and Orth (A) are isomorphic as falgebra.'!.
5. Uniform completion and Dedekind completion Let A be an Archimedean ialgebra. The closure A1'U of A in its Dedekind completion A8 with respect to the uniform topology is a uniformly complete Riesz space. Using Quinn's Definition 2.12 in [79], A1'U is the uniform completion of A. The following theorem was obtained by n'iki in [91]. THEOREM 5.1. Let A be an Archimedean falgebra (respectively, almost falgebra, dalgebra, f algebra). Then the multiplication in A extends uniquely to a multiplication in A1'U such that A1'U is a uniformly complete ialgebra (respectively, almost f algebra, dalgebra, f algebra) with respect to this extended multiplication. Moreover, if A is semiprime (respectively, has a unit element e) then ATU is semiprime (respectively, has e as unit element).
We now turn our attention to the Dedekind completion of the ialgebra A. Johnson in his paper [64] proved that if A is an falgebra (or even an Archimedean fring), then the multiplication in A extends uniquely to an falgebra multiplication in A8. The uniqueness of such an extended multiplication in A8 of course arises from the order continuity of the multiplication in the falgebra A. Alternative proofs of this extension can be found in [76, pp. 6667] and [59, p. 166].
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KARIM BOULABIAR. GERARD BUSKES. AND ABDELMAJID TRIKI
THEOREM 5.2. Let A be an Archimedean f algebra. Then the multiplication in A extends uniquely to a multiplication in A" such that Ad 'is a Dedekind complete falgebra with respect to this extended multiplication. FUrthermore, if A is semiprime (respectively, A has a unit element e) then A" is semiprime (respectively. has e as unU element).
The corresponding results for (Ialgebras in general, or almost falgebras and dalgebras in particular, is much harder because of the absence of order continuity of the multiplication. Nonetheless, extensions of the multiplication to the Dedekind completion often exist, though such extensions are no longer necessarily unique. For almost falgebras Buskes and van Rooij proved the following (see [35, Theorem 10]). THEOREM 5.3. Let A be an Archimedean almost falgebra. Then the multiplication in A extends to a multiplication 'in A" such that A" is a Dedekind complete almost f algebra with respect to that extended m'ultiplication.
Using the previous result as a starting point, Boulabiar and Chil in [28, Corollary 3] proved that from amongst the extensions provided, A" can be equipped with a commutative dalgebra multiplication whenever A is a commutative dalgebra. Then in [37, Theorem 7], Chil wa.c; able to drop the commutativity condition and prove the following theorem. THEOREM 5.4. Let A be an Archimedean dalgebra. Then the multiplication in A extends to a multiplication in Ad such that A" is a Dedekind complete dalgebra with respect to that extended multiplication.
In summary, all but one of the problems concerning Dedekind completions that Huijsmans raised in his survey paper [57] have now been solved. The remaining problem, though admittedly outside the scope of this survey, is the following. PROBLEM 5.5. Let A be an Ar'chimedean (Ialgebra. Does the multiplication in A extend to a multiplication in A" so that A" is a Dedekind complete (Ialgebra?
6. Powers in ialgebras Let A be a uniformly complete ialgebra and let P E lR+ [Xl, ... , Xn] be a homogeneous polynomial of degree a non zero natural number p. In their paper [16]' Beukers and Huijsmans considered the following problem: does there exist in A a 'pth root' of P(iI, ... ,fn) for iI, ... ,fn in A+? They gave an affirmative answer in the case where A is a semiprime falgebra. More precisely, they prowd the following theorem (see [16, Theorem 5]). THEOREM
6.1. Let A be a uniformly complete semiprime falgebra and let
P E lR+ [Xl, ... , Xn] be a homogeneo1Ls polynomial of degree a non zero natural numbe1'p. Thenfo1' every fl, ... ,fn E A+ there exists a 'unique f E A+ such that fP = P (iI, ... , fn). As a consequence, one has the following corollary (see Corollary 6 in [16]). COROLLARY 6.2. Let A be a uniformly complete f algebra with unit element and p E {I, 2, ... }. Then for each f E A +, there exists a unique 9 E A + such that
gl'
= f.
LATTICE ORDERED ALGEBRAS
109
Note that the previous result was first proved for p = 2 in [17, Theorem 4.2 and Cororllary 4.3] by Beukers, Huijsmans and de Pagter. Also, it should be noted that Theorem 6.1 above is proved alternatively by Buskes, de Pagter, and van Rooij in [31, Corollary 4.11], a paper that deals with a more general functional calculus on Riesz spaces and falgebrar; to which we will return in the next section. The problem corresponding to Theorem 6.1 for almost falgebras was considered by Boulabiar and follows next (see Theorem 3 in [24]). jR+
THEOREM 6.3. Let A be a uniformly complete almost f algebra and let P E [Xl, ... , Xn] be a homogeneous polynomial of degree a natural number p. Then
for every II, ... , fn E A+ there exists a (not necessarily unique) f E A+ such that fP = P (II, ... , fn)·
Observe that, where roots are unique in semiprime falgebras, this is no longer always the case for almost falgebras. We illustrate this with an example. EXAMPLE 6.4. Let A = C ([1,1]) be the uniformly complete Riesz space of all realvalued continuous functions on [1, 1] and define w E A by
W(x)={ Ox
(1::;x::;0); (0::;x::;1).
For every f, g E A, we put
(f. g) (x) =
{
w(x)f(x)g(x)
(1::; x::; 0);
lO/(S)g(S)dS
(0::; x::; 1).
Then A is an almost f algebra with respect to the multiplication.. h, g, Ct, {3 E A defined by g(x) =
Ixl
h (x)
Consider
= exp (x),
and Ct
(x)
= Jx 2 + exp (2x)
, and {3 (x)
for all x E [1,1]' where X[O,I] (x) Then Ct • Ct
=1
= X.X[O,I] (x) + Jx 2 + exp (2x)
if x E [0,1] and X[O,I] (x)
=0
if x E [1,0).
= {3 • {3 = 9 • g + h • h.
At this point, we define for each non zero natural number p, Ap =
{II ... fp : II, ... , fp
E A}.
In what follows, we will investigate the order structure as well as the algebra structure of Ap (since Al = A, we suppose that p ~ 2). The sets A2 and A3 were first considered in [35] by Buskes and van Rooij and then in great detail by Boulabiar in [24] from which we summarize the results in the following theorem (see Theorems 4, 5, and 6 in [24]). THEOREM 6.5. Let A be a uniformly complete almost f algebra and let p ~ 3 be a natural number. Then Ap is a uniformly complete semiprime f algebra under the ordering and multiplication inherited from A. The positive cone At of Ap is defined by
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KARIM BOULABIAR, GERARD BUSKES, AND ABDELMAJID TRIKI
The lattice operations I\p and V p in Ap are given by
fP I\p gP = (f 1\ g)P and the absolute value
1.l p
and
fP
Vp gP
= (f V g)P
for all 0 ~ f,g E A,
in Ap is defined by If Pip
=
Ifl P
for all f E A.
Contrary to Ap (p 2: 3), A2 need not be a Riesz space under the ordering inherited from A as is proved by the next example. EXAMPLE 6.6. Consider A = C ([0,1]) with the pO'intwise addition, scalar rrmltiplication and partial ordering. For f, 9 E A, define
l
(f _ g)(x) = {
(0 x  1/2
~
x ~ 1/2);
f(s)g(s)ds (1/2 < x ~ 1). o Then A is a uniformly complete almost f algebra under the multiplication  and h is an element of A2 if and only if h(x) = 0 for all x E [0,1/2] and the restriction of h to [1/2,1] belong to C 1 ([1/2,1]). Hence A2 is not a Riesz space under the order inherited from A.
The following example proves that though Ap (p 2: 3) is a Riesz space, in general it is not a Riesz subspace of A. EXAMPLE 6.7. Take A = C ([1, 1]) with the pointwise addition, scalar multiplication and ordering, and define w EA· by
w (x)
=
{ x 0'
(l~x~O);
(0 ~ x ~ 1) .
For f, 9 E A, define
(f  g) (x) =
{
w(x)f(x)g(x)
(l~x~O);
10/(S)g(S)ds
(O~x~l).
Clearly, A is a uniformly complete almost f algebra under the multiplication _. Define 0 E A by o (x)
= 2x + 1
for all x E [1, 1].
It follows that
10  0  01 (1) =
1/10
=110  0  ob (1) = (101101101) (1) =
1/8.
If, however, A is a commutative dalgebra then some of the unpleasantness of the preceding example disappears. COROLLARY 6.8. Let A be a uniformly complete commutative dalgebra and p 2: 2 be a natural number. Then Ap is a uniformly complete f subalgebra of A. If in addition p 2: 3 then Ap is semiprime.
In spite of the improvement in the conclusion of Corollary 6.8 over the conclusion for the more general situation of almost falgebras, A2 still need not be semiprime. This is illustrated in the next example.
LATTICE ORDERED ALGEBRAS
111
EXAMPLE 6.9. Let A be the coordinatewise ordered vector space R3 with the multiplication defined by:
n; R}
Then A is a uniformly complete commutative dalgebm and
A,
~{(
X,Y E
Dbuiou,ly, A, is an f""balg,bm of A. FUrth,""ore, (
~ ) ' ~ 0 and A, is not
semiprime. For f algebras we have the following corollary. COROLLARY 6.10. Let A be a uniformly complete f algebm and let p 2: 2 be a natuml number. Then Ap is a uniformly complete semiprime f subalgebm of A.
There is a universal way in which A 2 , or more generally Ap for any p 2: 2 can be described. We provide the details of that description for A2 next (see [36]). Let E and F be Riesz spaces. A bilinear map q> : E x E ~ F is called orthosymmetric if whenever f I\g = 0 for f,g E E we have q>(f,g) = 0 (the notion of orthosymmetric bilinear map was introduced by Buskes and van Rooij in [34]). The bilinear map q> is a Riesz bimorphism if it is a Riesz homomorphism in each variable separately (more about Riesz bimorphisms can be found in [29]). Let E be a Riesz space. The pair (E8, 8) is called a square of E, if E8 is a Riesz space and if (1) 8: E x E ~ E8 is an orthosymmetric Riesz bimorphism, and (2) for every Riesz space F, whenever q> : E x E ~ F is an orthosymmetric Riesz bimorphism there exists a unique Riesz homomorphism q>8 : E8 ~ F such that q>8 08 = q>. The existence and uniqueness of squares for any Riesz space follows easily from the Riesz space tensor product as constructed by Fremlin in [43]. To understand the structure of the square of a Riesz space is best not done via this tensor product. The set A2 described above is often more helpful. The connection between semi prime falgebras and squares of uniformly complete Riesz spaces is described in the next theorem. After reading that theorem the reader might feel like moving the lower index 2 in A2 to an upper index. THEOREM 6.11. Let E be a uniformly complete Riesz subspace of an Archimedean semiprime f algebm G whose multiplication is indicated by a period e. Put E2 := {x e y : x,y E E} as before. Then E2 is a Riesz subspace of G and (E2,e) is a square of E.
7. Functional Calculus on falgebras The theorem that we presented in Section 6 on the existence of pth roots of homogeneous polynomials in falgebras is a very special case of a rich functional calculus on uniformly complete falgebras. The idea behind functional calculus
112
KARIM BOULABIAR, GERARD BUSKES, AND ABDELMAJID TRIKI
for Riesz spaces in general is straightforward. For elementary functions on JRN one ought to be able to simply substitute elements of the Riesz space into these functions and get elements of the Riesz space as output. The idea of how to execute this substitution of elements in sufficiently simple functions essentially goes back to Yudin and in the form that we represent it to Lozanovsky [70]. The technical problem surmounts to what the class of sufficiently simple functions really looks like. Let n EN. We denote by 1i(JRN) the Riesz space of all continuous functions r.p : JRN + JR for which
r.p(tx) = tr.p(x) for all x E JRN and all t ~ O. Let E be a Riesz space, r.p E 1i(JRN) and
iI, ... , fn
E E. We say that
r.p(iI, ... '/n) exists in E if there is an element 9 of E such that
w(g) = r.p(w(iI), ... ,w(fn)) for every realvalued Riesz homomorphism w on the Riesz subspace of E generated by iI, ... , f n, g. For any given E, r.p and iI, ... ,/n there exists at most one 9 with this property. This 9 is also indicated by
r.p(iI, ... , fn). In this situation we have the following theorem (see Lozanovsky [70]). THEOREM 7.1. Let E be a uniformly complete Riesz space and Then r.p(iI, ... , fn) exists fOT every r.p E 1i(JRN). The map
iI, ... , fn
E
E.
r.p(iI, ... , fn) (r.p E 1i(JRN)) is a Riesz homomorphism from 1i(JRN) into E. r.p
+
Remark. In a way, r.p(iI, ... , fn) is independent of E. Indeed, if D is any Riesz subspace of E that is uniformly complete and contains iI, ... , fn then r.p(iI, ... , fn) relative to D means the same as r.p(iI, ... , fn) relative to E. In particular, every Riesz subspace of E that is uniformly complete and contains iI, ... , fn must also contain r.p(fl, ... , fn). By A(JR N) we denote the set of all continuous functions r.p : JRN + JR that are of polynomial growth and for which limt!o rlr.p(tx) exists uniformly on bounded subsets of JRN (the latter condition is equivalent to the existence of a 't/J E 1i(JRN) such that r.p(x) = 't/J(x) + 0(11 x II) (x + 0)). Observe that A(JRN) is an falgebra. Let E be a semiprime falgebra, r.p E A(JRN) and iI, ... , fn E E. We say that
r.p(iI, ... ,fn)exists in E if there is agE E with
w(g) = r.p(w(iI), ... ,w(fn)) for every realvalued multiplicative Riesz homomorphism w defined on the fsubalgebra of E generated by iI, ... , fn,g. There exists only one such g, which is then called r.p(iI, ... , fn). This definition is in accordance with the one we gave for 1i(JRN) if r.p E 1i(JRN). For 1i(JRN) we have the following theorem (see [31, Theorem 4.10]).
LATTICE ORDERED ALGEBRAS THEOREM
ft, ... , fn
113
7.2. Let E be a uniformly complete semiprime falgebra and let
E E. Then cp(ft, ... , fn) exists for every cp E A(~JII). The map
cp+cp(ft, ... ,fn) (cpEA(~JII)) is a multiplicative Riesz homomorphism from A(~JII) into E.
8. Relationships between ialgebra multiplications
Let A be an ialgebra with multiplication denoted by juxtaposition, and assume that A is equipped with another associative multiplication e. In the first theorem of this section, we present a relationship between the two multiplications in A, under the conditions that A is a unital falgebra with respect to the initial multiplication and an (almost) falgebra with respect to the other multiplication e. For proofs, see [38, Theorem 2.2] and [23, Theorem 5.2]. THEOREM 8.1. Let A be an Archimedean f algebra with identity element e and assume that A is furnished with another associative multiplication e. Then (i) A is an falgebra under e if and only if
feg=(eee)fg
for allf,g E A, and
(ii) A is an almost falgebra under e if and only 'if f e9
= e e (I g)
for all
J, 9 E A.
The corresponding problem in the case where A is dalgebra with respect to e is rather more difficult. Indeed, since then e need not be commutative, one cannot write the product f e 9 as a function of the product fg (I, 9 E A). However, there exists another way (involving f, 9 and the initial multiplication in A) to express the product f e g. This is the subject of the next result. First recall that the maximal ring of quotients Q (A) of the Archimedean f algebra A with unit element e is again an Archimedean Jalgebra with the same e as multiplicative identity. Moreover, A is an fsubalgebra of Q (A), a fact proved by Anderson in [4] (see also the recent paper [71, Cororllary 2.7.1] by Martinez). For the definition of the maximal ring of quotients of a ring, the reader can consult e.g. [66]. The proof of the following theorem can be found in [25, Theorem 4.3]. THEOREM 8.2. Let A be an Archimedean falgebra with 'identity element e and let e be another associat'ive multiplication in A. Then A is a dalgebra with respect to e if and only if there exist two algebra and Riesz homomorphisms cp and 'l/J from A into its maximal ring of quotients Q (A) such that
f e 9 = (e e e) cp (I) 'l/J (g)
for all f, 9 E A.
As mentioned at the end of Section 3, the two preceding theorems are abstract versions of the corresponding results, given in that section, for the C (X)case. In the second part of this section, we are interested in A being a commutative dalgebra with respect to the initial multiplication rather than an falgebra with unit element. However, we will impose the additional assumption that A is uniformly complete. Uniform completeness is not needed for all our results but we will use the set A2 and remind the reader of the special nature of that set under the extra condition of uniform completeness (see Corollary 6.8). If there exists a positive operator T OIl A2 such that
(T)
f e9 =
T (lg)
for all
f, 9 E
A
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KARIM BOULABIAR, GERARD BUSKES. AND ABDELMA.HD TRIKI
then e is an almost falgebra multiplication and N2 (A) c N; (A), where
N; (A)
= {f
E A :
f e f = O} .
In what follows, we show in detail what happens if we assume that A with the initial multiplication is a (uniformly complete) commutative dalgebra, and the inclusion N2 (A) c N; (A) holds. Under those circumstances, we then relate a necessary and sufficient condition for the new multiplication to be an almost falgebra, a dalgebra or an falgebra to the existence of some posit.ive operator T satisfying the relation (T). The details follow in the next theorem, the proof of which can be found in [23, Theorems 5.4 and 5.5]. THEOREM 8.3. Let A be a uniformly complete commutative dalgebra and assume that A is an I!algebra with respect to another associative multiplication e such that p = 0 implies f e f = O. Then the following statements hold.
(i) A is an almost f algebra under
e if and only if there exists a positive operator T from A2 into A such that
feg=T(fg)
forallf,gEA.
(ii) A is a commutative dalgebra under
e if and only if there exists a Riesz homomorphism T from A2 into A such that
f e 9 = T (fg)
for all f, 9 E A.
(iii) A is an falgebra under e if and only if there exists an operator T from A2 into A such that To 7rf E Ort.h (A) for all f E A+, where 7rf (g) = fg for all 9 E A, and f e9
=T
(f g)
for all f, 9 E A.
We remark that A2 in the previous theorem is a Riesz space (see Section 6). Next, we produce an example which shows that in Theorem 8.3 above, the hypothesis 'A is a commutative dalgebra' cannot be replaced by 'A is an almost falgebra'. EXAMPLE 8.4. Take A = C ([1,1]) with the usual operations and order and define a, (3 E A by
(4x+l) a(x)= { 0 4x 1
(I~x~1/4); ~ x ~ 1/4); (1/4 ~ x ~ 1)
(1/4
and (3 (x) = {
~4x + 1
(1 ~x~ 1/4); (1/4 ~ x ~ 1).
For f, 9 E A, define a(x)f(x)g(x) (f x g)(x) =
{
(1~x~1/4); ~ 3/4);
(1/4 ~ x °J(X3/4)
f(s)g(s)ds
(3/4~x~1)
(x3/4)
and (feg) (x) =f3(x)f(x)g(x)
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115
for all x E [1,1]. Then A is an almost falgebra (respectively, an falgebra) with respect to the multiplication x (respectively, .). It follows that N; (A)
= N 2x
(A)
= {f
E A : f (x)
= 0 for
all x E [1, I/4]}.
Consider <.p, w E A, defined by
"'(X)~{ and oJ (x)
~
3xI x 3x+I
(1 ::=;x::=; 1/4); (I/4::=; x::=; 1/4); (i/4::=;x::=;I)
0 4x+ 1 { 4x+I 0
(I::=;x::=;I/4); (I/4::=; x::=; 0); (0::=;x::=;I/4); (i/4::=;x::=;I).
Then and 'Pxw=O 'P. W =f:. O. Therefore, there is no operator T satisfying the condition f • g = T (f x g)
for all f, 9 E A.
We notice that the kind of relationships between two falgebra multiplications via an operator T as above were first studied in [35] by Buskes and van Rooij. In particular part (i) of the theorem above has its origins in [35, Theorem 1], where a general representation theorem for almost falgebras (without any extra conditions) was proved by first dividing out the radical and then showing the existence of an operator T a.'l above on the square (see the end of Section 6 above) of the resulting falgebra, thus giving quantified credence to the almost part in the name almost falgebras. They also used this representation to discover that the Dedekind completion of an almost falgebra is an almost falgebra and to prove (CSP) for almost falgebras. Finally, we should point out that the Buskesvan Rooij representation theorem for almost falgebras itself is, in a way, an abstract reformulation of Theorem 3.2 by Scheffold that we discussed earlier. We end this section with the following problem, left open in the above discussion. PROBLEM 8.5. Let A be a uniformly complete commutative dalgebra and assume that A is a non comm'utative dalgebra with respect to another multiplication •. Does there exists a relationship between the two multiplications in A?
9. Connection between algebra and Riesz homomorphisms Since in any falgebra there is an order structure as well as an algebra structure, it is natural to compare the operators that preserve the lattice operations, namely the Riesz homomorphisms, with those that preserve the algebra structure, the algebra homomorphisms. Such a comparison was initiated by Ellis [41], who considered the problem for operators between spaces of continuous functions on compact Hausdorff spaces, and his central result is then one of equivalence: a Markov operator (Le., a positive operator preserving the identity) between such spaces is a Riesz homomorphism if and only if it is an algebra homomorphism. Some years. after Ellis's paper was published, Hager and Robertson presented in [48] a more abstract version of Ellis's theorem. More precisely, Hager and Robertson proved that any
116
KARIM BOULABIAR, GERARD BUSKES. AND ABDELMAJID TRIKI
Riesz homomorphism between two Archimedean falgebras with unit elements that preserves the identity is an algebra homomorphism. Later, van Putten established the converse of the result by Hager and Robertson in his thesis [78]. The aforementioned results were generalized by Huijsmans and de Pagter in [59, Theorem 5.4] as follows. THEOREM 9.1. Let A be an Archimedean falgebra with unit element e, B be an Archimedean semiprime falgebra. and T : A > B be a positive operator. Then the following are equivalent (i) T is an algebra homomorphism. (ii) T is Riesz homomorphism with (Te)2 = Te.
Notice that the previous theorem generalizes all of the facts cited above. In the same paper [59], Huijsmans and de Pagter proved that., if A is an Archimedean falgebra with unit element and B is an Archimedean semiprime falgebra then any order bounded algebra homomorphism from A into B is automatically a Riesz homomorphism [59, Theorem 5.3]. Very recently in [91, Theorem 4.3], Triki obtained this result in the more general setting of almost falgebras. THEOREM 9.2. Let A be an Archimedean almost falgebra and let B be an Archimedean semiprime f algebra. Then any order bounded algebra homomorphism from A into B is a Riesz homomorphism.
Since any conmmtative dalgebra is automatically an almost falgebra, Theorem 9.2 holds if one replaces 'A is an Archimedean almost falgebra' by 'A is an Archimedean commutative dalgebra'. However, in the recent. work [89], Toumi proved that we have the same conclusion even if the dalgebra under consideration is not commutative. THEOREM 9.3. Let A be an arbitrary Archimedean dalgebra and let B be an Archimedean semiprime falgebra. Then any order bounded algebra homomorphism from A into B is a Riesz homomorphism.
Even for falgebras, the condition of order boundedness in Theorems 9.2 and 9.3 cannot be dropped as is shown in the following example. EXAMPLE 9.4. Consider the set A of all real sequences u = {un} n>l for which there exists a polynomial Pu E IR [X] and a natural number N such that Un = Pu (n) for all n 2: N. Under the usual operations and partial ordering, A is an Archimedean falgebra (and thus an almost falgebra and a dalgebra) and A is not relatively uniformly complete. Define the algebra homomorphism T : A > IR by T(u) = Pu (1) (u E A). For every A E [l,+po[, we define u), = {U),.n}n~l E A by
u)',n =
{ 0An
(n ~ A);
(n> A) .
lfu = {n 2 } n>l then 0 ~ u), ~ u for all A E [1, +00[. Observe now that T (1L),) Therefore T 1:s not order bounded and hence not a Riesz homomorphism.
= A.
In [59, Theorem 5.1], Huijsmans and de Pagter further illustrated the close ties between Riesz homomorphisms and algebra homomorphisms on falgebras as follows. If A and Bare semiprime falgebras and A is uniformly complete, then any algebra homomorphism from A into B is a Riesz homomorphism. Their theorem was generalized by Boulabiar in [27, Theorem 3] to the setting of almost falgebras.
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117
THEOREM 9.5. Let A be a uniformly complete almost f algebra and let B be an Archimedean semiprime falgebra. Then any algebra homomorphism from A into B is a Riesz homomorphism.
Trivially, one can replace in Theorem 9.5 above 'almost falgebra' by'commutative dalgebra'. The case of dalgebras that are not necessarily commutative was considered by Toumi in [89, Theorem 3]. THEOREM 9.6. Let A be an arbitrary uniformly complete dalgebra and let B be an Archimedean semiprime falgebra. Then any algebra homomorphism from A into B is a Riesz homomorphism.
The condition that A is uniformly complete is not redundant in Theorems 9.5 and 9.6, even if A is an falgebra. Indeed, A in Example 9.4 above is not uniformly complete. The following theorem by Triki in [91, Theorem 4.4] is an falgebra version of the well known Nagasawa's theorem (see [74, Theorem 1]). First, recall that an operator T between two unital falgebras A and B is said to be contractive if IT (J)I ~ eB in B whenever If I ~ eA in A, where eA and eB are the unit elements of A and B, respectively. If moreover T is bijective and the inverse T 1 of T is also contractive then we say that T is bicontractive. THEOREM 9.7. Let A and B be Archimedean falgebras with identity elements eA and eB, respectively. For an order bounded bijection T : A t B such that T (eA) = eB, the following are equivalent. (i) T is an algebra homomorphism. (ii) T is a Riesz homomorphism. (iii) T is bicontractive.
The last result of this section again deals with operators between two unital falgebras that preserve identities, so called Markov operators. Denote by M (A, B) the set of all Markov operators from an falgebra A with unit element eA into an falgebra B with unit element eB, that is, M (A,B)
= {T:
A
t
B: T linear positive with T(eA)
= eB}.
Obviously, M (A, B) is a convex set. In [77, Theorem 2.1], Phelps proved that a Markov operator T from C (X) into C (Y) (where X and Yare compact Hausdorff topological spaces) is an algebra homomorphism if and only if T is an extremal point in M (C (X) ,C (Y)). This connection to extreme points predates the paper by Ellis and we refer the reader to Arens and Kelley [8] and A. and C. Ionescu Thlcea [62]. Combining the aforementioned Phelps's theorem and Ellis's result cited above we get that for T in M (C (X), C (Y)) the following are equivalent. (i) T is a Riesz homomorphism. (ii) T is an algebra homomorphism. (iii) T is an extremal point in M (C (X) , C (Y)). As a generalization, van Putten in his thesis obtained the following result [78, Theorem 18.8]. THEORSM 9.8. Let A and B be Archimedean falgebras with unit elements, and let T E M (A, B). Then the following are equivalent. (i) T is a Riesz homomorphism.
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KARIM BOULABIAR, GERARD BUSKES, AND ABDELMAJID TRIKI
(ii) T is an algebra homomorphism, (iii) T is an extremal point in M (A, B). An elementary proof of van Putten's theorem above, due to Huijsmans and de Pagter, can be found in [59, Theorem 5.7].
10. Positive derivations We recall that an operator D on a commutative algebra A is said to be a derivation if for all f,g EA. D (1g) = f D (g) + gD(1) In this section, we investigate positive derivations on falgebras as well as on almost falgebras. Positive derivations on falgebras were first considered in great detail by Colville, Davis and Keimel in [40]. Their main result provides the following necessary and sufficient condition for a positive operator on an falgebra to be a derivation (see Theorem 5 in [40]), where we recall that if p is a non zero natural number, and A is an algebra then Ap = {11 ... fp : 11, ... , fp E A}. THEOREM 10.1. Let A be an falgebra and D be a positive operator on A. Then D is a derivation if and only if
D (1) = 0 for all f E A2
and
D (1)2 = 0 for all f E A.
Positive derivations on frings were considered by Henriksen and Smith in [54]. It straightforwardly follows from Theorem 10.1 above that, if A is in addition semiprime, then there exist no non trivial positive derivations on A. We turn our attention now to positive derivations on almost falgebras. The result corresponding to Theorem 10.1 for almost falgebras is the following (for a proof, see [26, Theorem 3]). THEOREM
10.2. Let A be an almost falgebra and D be a positive derivation
on A. Then D (1)
= 0 for
all f E A3
and
D (1)3
= 0 for
all f E A.
Contrary to falgebras, Theorem 10.2 does not produce a characterization of positive derivations on almost falgebras. Also, the third power in Theorem 10.2 is the best possible. The next example illustrates these facts. EXAMPLE 10.3. Consider A the Cartesian product IR x IR with coordinatewise addition, scalar multiplication and ordering. Define the multiplication. on A by
for all 0,/3,0',/3' E R
(0,/1). (0',/3') = (0,00')
Then A is an Archimedean almost f algebra with respect to.. Observe that A3 fA satisfies
=
{o}. Hence the identity map fA
(1) = 0 for all f E A3
and
fA
(1) 3 = 0 for all f E A,
but fA is not a derivation on A. On the other hand, let D be the positive operator defined on A by for all a, /3 E R D(o,/3) = (0,2/3) Clearly, D is a derivation on A. However, D ((1, 0) • (1,0))
= (0,2)
=1=
(0,0)
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119
and
D (1, 0) • D (1, 0) = (0,1) =I (0,0) . Theorem 10.2 above of course holds for commutative dalgebras as well. Moreover, for commutative dalgebras the third power is the best possible too. Indeed, the almost falgebra considered in Example 10.3 is in fact a commutative dalgebra.
11. CauchySchwarz inequalities We say that an ialgebra A possesses the CauchySchwarz property (abbreviated as (CSP)), if for every vector space V and every bilinear map 'l/J : V x V > A such that (i) 'l/J is symmetric, and (ii) 'l/J (I, f) E A+ for all f E V, we have (CSI)
'l/J(I,g)2 S'l/J(I,f)'l/J(g,g)
for all f,g E V
In [61, Corollary 3.5], Huijsmans and de Pagter proved that any Archimedean semiprime falgebra has (CSP). Later, Bernau and Huijsmans in [15, Theorem 2.6] generalized (CSP) to arbitrary Archimedean falgebras. THEOREM 11.1. Let A be an Archimedean falgebra, V a vector space and 'l/J : V x V > A a bilinear map such that (i) 'l/J is symmetric, and (ii) 'l/J (I, f) E A+ for all f E V . Then
'l/J (I, g)2 S 'l/J (I, f) 'l/J(g, g)
for all f,g E V.
Some years after Theorem 11.1 was published, Buskes and van Rooij established the corresponding inequality for Archimedean almost falgebras and therefore for Archimedean commutative dalgebras (see [34, Corollary 4]). THEOREM 11.2. Let A be an Archimedean almost falgebra, V a vector space and'l/J : V x V > A a bilinear map such that (i) 'l/J is symmetric, and (ii) 't/J (I, f) E A + fOT all f E V . Then
'1/) (I, g)2 S .t/J (I, f) ,¢'(g, g)
for all f, g E V.
An alternative proof of the previous inequality was given by Boulabiar in [23, Theorem 3.9]. Not every dalgebra A has (CSP). An example illustrating this situation is the following. EXAMPLE 11.3. Let A be the set of all realvalued functions defined on [0,1] equipped with the usual operations and order. Consider the multiplication. defined in A by (I )()  { f (0) g (1) (0 S x S 1/2) ; .g x f(l)g(l) (1/2SxS1)
for all f, g E A. Then A is an Archimedean noncommutative dalgebra with respect to the multiplication •. At this point, let V be the falgebra C ([0,1]) of all realvalued continuous functions on [0, 1] , provided with the pointwise addition, multiplication, scalar multiplication and partial ordering. We define a positive operator T from V into A by T (I) (x) = f (1  x) for all f E V and x E [0,1]. Finally, let f E V such that f (0) = 2, f (1) = 1 and e E B such that e (x) = 1 for all x E [0,1]. Then (T (Ie) • T (Ie)) (0)
= (T (I) • T
(I)) (0)
=f
(0) f (1)
=2
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KARIM BOULABIAR, GERARD BUSKES. AND ABDELMAJID TRIKI
and Hence the inequality
T (fe). T (fe) :s T (J2) • T (e 2) does not hold in A. By defining '!/J(f,g) = T(fg) (f,g E V), we find a bilinear map V x V t A for which 1jJ (f, e)2 :s 1jJ (f, f) 1jJ(e, e) does not hold. Though (CSP) does not hold for all non commutative dalgebras, Boulabiar and Toumi proved, in essence, the following variant of (CSP) in such algebras (see [29, Theorem 6]). THEOREM 11.4. Let V be a vector space and let A be an Archimedean dalgebra. Consider a bilinear map '!/J : V x V t A such that (i) 1jJ is symmetric, and (ii) 1jJ (f, f) E A+ for all f E V. Then
11,b(f,g)21:s
~[(1jJ(f,f)1jJ(g,g))+('!/J(g,g)1jJ(f,f))]
and hence 11jJ(f,g)21:s (1jJ(f,f)'!/J(g,g)) V (l/J(g,g)'!/J(f,f)) for all f,g E V. 12. Order biduals We refer the reader to [2] for terminology and notations not explained below. For an Archimedean Calgebra A, the order dual is denoted by A~ and the order bidual is denoted by A~~. Recall here that the order dual A~ of A is the Dedekind complete Riesz space of all realvalued order bounded functionals on A. A multiplication can be introduced in A~~ in three steps as follows: for all u, v E A, f E A~ and cp, '!/J E A~~, we define f.u E A~, '!/J.f E A~ and cp.1jJ E A~~ by the following equations
(1)
(f:u) (v) = f (uv)
(2)
('!/J.f) (u) = '!/J (f.u)
(3)
(cp.'!/J) (f) = cp ('!/J.f)
The multiplication defined by the equation (3) is called the Arens multiplication in A~~ (see [6] and [7]). The order continuous order bidual (A~);: of A is the projection band of all order continuous elements in A~~. Moreover, (A~);: is closed under the Arens multiplication, that is, cp ..l/J E (A~);: whenever cp, '!/J E (A~);: . The following theorem was established by Huijsmans and de Pagter (see [60, Theorem 4.1]). 12.1. Let A be an Archimedean Calgebra. Then A~~ (and hence is a Dedekind complete Calgebra with respect to the Arens multiplication.
THEOREM
(A~);:)
e
If the falgebra A has, in addition, a unit element e then E (A~);: (defined by e(f) = f (e) for all f E A~) is the unit element of A~~. Generally, A~~ need not be commutative even if A is commutative. An example was provided by Arens in [7]. However, commutativity does carryover from A to (A~);:, which was first established for normed Calgebras by Scheffold in his paper [82] and generalized by Grobler in [46, Theorem 4] to the more general setting of Archimedean Calgebras.
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THEOREM 12.2. Let A be an Archimedean ialgebra. Then the order continuous (A~);: , provided with the Arens multiplication, is commutative.
orde1' bidual
The first results concerning the order bidual of falgebras are due to Huijsmans and de Pagter (see [60]). They proved, among other results, that if A is an falgebra with unit element such that A~ separates the points of A (i.e., if u E A and f (u) = 0 for all f E A~ then u = 0) then A~~ = (A~);:. They deduced that A~~ then again is an falgebra (of course, with unit element) with respect to the Arens multiplication (note that if A~ separates the points of A then A is Archimedean). Later Huijsmans was able to do away with the condition that A have a unit element, and he showed that the order bidual of an falgebra with separating order dual is also an falgebra with respect to the Arens multiplication (see [56, Theorem 2.8]). Then, without assuming any additional 'separation' condition, Bernau and Huijsmans proved in [15] that the order bidual of an Archimedean falgebra, equipped with the Arens multiplication, is again an falgebra (see Theorem 3.2 and Theorem 3.5 in [15]). THEOREM 12.3. Let A be an Archimedean falgebra. The order bidual A~~ of A, equipped with the Arens multiplication is an falgebra and hence 'is commutative.
In the paper [14], Bernau and Huijsmans also dealt with the case of almost falgebras. THEOREM 12.4. Let A be an Archimedean almost f algebra. Furnished with the Arens multiplication, the order bidual A~~ of A is an almost falgebra and hence is commutative.
Combining Theorem 12.4 and the fact that commutative dalgebras are almost falgebras, Bernau and Huijsmans deduced the following result (see [14, Theorem 4.2]). COROLLARY
order bidual plication.
A~~
12.5. Let A be an Archimedean commutative dalgebra. then the of A is a commutative dalgebra with respect to the Arens multi
It is interesting to observe that the square of the singular part of the bidual (the orthogonal complement of the order continuous part) vanishes in the case of an almost falgebra. Contrary to the case of almost falgebras, the corresponding problem for non commutative dalgebras remains open and only a partial result has been obtained, again by Bernau and Huijsmans in [14, Theorem 4.1]. Their result is the following. PROPOSITION 12.6. Let A be an Archimedean dalgebra. Then the order continuous order bidual (A~);: is a dalgebra with respect to the Arens multiplication.
As just observed, the question whether the full order bidual of a non commutative dalgebra is a dalgebra with respect to the Arens multiplication is still open. PROBLEM 12.7. Let A be an Archimedean dalgebra. Is the order bidual of A a dalgebra with respect to the Arens multiplication?
A~~
At the end of this section, we notice for the sake of completeness that all of the results above were obtained independently by Scheffold under the additional . assumption of A being a Banach lattice (including the result on dalgebras) (see [81] and [83]).
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13. Ideal theory In this section, we follow [69J in our terminology and notations. Recall that an order ideal in a Riesz space L is a vector subspace I of L with the extra condition III ::; Igl in Land gEl imply I E I. We call a ring ideal I in an ialgebra A an iideal if I is simultaneously an order ideal in A. Ideal theory in lattice ordered algebras comprises an investigation of ring ideals, order ideals and the connection between them. In the first part of this section, we focus on the following problem: (P 1) Under what conditions is any order ideal in an I algebra a ring ideal? In the second part we consider the converse problem: (P2) Under what conditions is any ring ideal in an I algebra an order ideal? The first problem was initiated by Henriksen in [49J, who studied the ideal theory in Irings by means of representations. In their work [58J on ideal theory in Ialgebras, Huijsmans and de Pagter proved the next result by a representationfree approach (see [58, Proposition 3.1]). PROPOSITION 13.1. Let A be an Archimedean Ialgebra. Then any unilormly closed order ideal is a ring ideal.
The problem (PI) for non uniformly closed order ideals was considered cxtensively by Basly and Triki in [l1J and [12J. In what follows, we present the major results they obtained in that direction. First, we need to recall some prerequisites. A ring ideal I in a commutative algebra A is said to be modular whenever there exists h E A such that I  I h E I for all I E A. In particular, any ring ideal in A is modular if A is in addition unital. From now on A is an Ialgebra. We know that the map p: A ~ Orth(A)
I
I>
7rf
(where 7rf (g) = I g for all I, g E A) is an algebra and Riesz homomorphism (see Section 4 above). An element I in A is said to be bounded if p (a) = 7ry. E Z (A), where Z(A) = {7r E Orth(A): 17r1::; >.IA for some real number A} is the centre of A (Le., the principal order ideal generated by IA in Orth (A)). We denote, after Triki (see [90]), the set of all bounded elements in A by Ab. Note that rather than A b , Henrikscn and Johnson use A* in [53J after a similar usage in Gillman and Jerison's book. Clearly, Ab is an Isubalgehra of A. The Ialgebra A is said to be bounded if A = Ab, that is, if p (A) is a subset of Z (A). In particular, a unital Ialgebra A is bounded if and only if its multiplicative identity is also a strong order unit, and in this situation A and Z (A) are isomorphic as falgebras. Here we recall that the identity element in an Archimedean unital falgebra is a weak order unit (see [22, p. 60]). The equivalence of (i) and (ii) in the next theorem is an immediate consequence of the definitions of an order ideal and a bounded Archimedean Ialgebra, while for the proof of the rest of tllis theorem, we refer to Theorem 3 in [l1J and Theorem 4 in [12J. THEOREM 13.2. Let A be an Archimedean Ialgebra. Then the following are equivalent. (i) Every order ideal is a ring ideal. (ii) A is bounded.
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123
If A is in addition uniforrn,ly complete then (i) above is equivalent to each of the following.
(iii) Every maximal modular ring ideal in A is uniformly closed. (iv) Every maximal modular ring ideal is the kernel of a Hiesz and algebra homomorphism from A onto JR. (v) Every maximal modular ring ideal is a maximal or'der ideal. We turn our attention to semiprime falgebras. Let L be a Riesz space. A norm on a L is called a Riesz norm if IIfil ::; IIgil whenever If I ::; Igl in L. If a Riesz norm on L exists then L is said to be a normed Riesz space. If the normed Riesz space L is a Banach space as well, we say that L is a Banach lattice. We call the Riesz norm 11.11 on L an AInorm if IIf V gil = max {IIfil , IIgll} for all f,g E L. An Aispace is an Atnormed Banach lattice. We can now state the following theorem (see [11, Theorem 5] and [12, Corollary 5]). 11.11
THEOREM 13.3. Let A be an Archimedean semiprime f algebra. The following are equivalent.
(i) (ii) (iii) (iv)
Every order ideal in A is a ring ideal in A. A is a isomorphic as an f algebra to a subalgebra of Z (A). There exists an M norm in A. There exists a Riesz norm in A.
If, in addition, A is uniformly complete then each of (i), (ii), (iii) and (iv) above is equivalent to
(v) Every maximal modular ring ideal in A is uniformly closed. We note that Problem (PI) was extensively studied by Henriksen, Larson and Smith [52] in the context of frings. We move on to discuss Problem (P2). To this end, we need the notion of a normal Riesz space. The Riesz space L is said to be normal if L = {f+} d+{f} d for allf E L, where {f+}d = {g E L: Igl.l\f+ = O} and {f_}d = {g E L: Igl.l\f = O}. For a completely regular topological space X, the Riesz space C (X) is normal if and only if the sets P(f) = {x EX: f(x) > O} and N(f) = {x EX: f(x) < O} are completely separated for every fEe (X). For spaces of the type C (X), Problem (P2) has the following solution (see Theorem 14.24 in [47]). THEOREM 13.4. Let X be a completely regular (Hausdorff) topological space. Then the following are equivalent. (i) Every ring ideal in C (X) is an order ideal. (ii) Every finitely generated ring ideal in C (X) is a principal ring ideal (i.e., X is an F space). (iii) C (X) is normal. Huijsmans and de Pagter considered (P2) in falgebras. They only considered the unital case and their central result is the following generalization of Theorem 13.4 above (see [58, Theorem 6.6] for the proof). THEOREM 13.5. Consider the following conditions for an Archimedean f algebra A with unit element.
(i) Every ring ideal is an order ideal. (ii) Every finitely generated ring ideal is a principal ring ideal.
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KARIM BOULABIAR, GERARD BUSKES, AND ABDELMAJID TRIKI
(iii) A is normal. (ii) (iii). If, in addition A is uniformly complete then (i) {:} (ii) {:} Then (i) (iii) .
'*
'*
Next we focus on the non unital case, which was considered by Triki in [90]. First we define the notion of a stable falgebra. The falgebra A is said to be stable if 1f (f) E (f) (where (f) is the principal ring ideal generated by f in A) for all 1f E Z (A). It is clear that A is stable if and only if 1f (1) c I for all ring ideals I in A and all 1f E Z (A). We now are in a position to present the result corresponding to Theorem 13.5 for semiprime falgebras (see [90, Theorem 5.4]). THEOREM 13.6. Consider the following conditions for an Archimedean scmiprime f algebra A. (i) Every ring ideal is an order ideal. (ii) Every finitely generated ring ideal is a principal ring ideal. (iii) A is stable and normal, and {f+} d or {f} d is a modular ring ideal for all f E A. Then (i) (ii) (iii). Furthermore, if A is in addition uniformly complete then (i) ¢:> (ii) {:} (iii).
'*
'*
We proceed to the non semiprime case (see Theorem 5.5 in [90]). THEOREM 13.7. Let A be a non semiprime Archimedean falgebra. Then the following are equivalent. (i) Every ring ideal is an orner zdeal. (ii) There exist a semiprime falgebra B such that (a) every 7ing ideal in B is an order ideal, and (b) {b} d is a modular ring ideal in B for every b E H, so that A is isomorphic to the f algebra B x JR endowed with the multiplication defined by (f, 0:) (g, 13) = (fg,O) for all i, g E B; 0:, 13 E JR.
Finally, note that the only ialgebras that we have considered in this section are ialgebras. Indeed, a Problem (PI) for more general ialgebras is futile since an ialgebra in which every order ideal is a ring ideal automatically is an falgebra (see Page 144 in the classical book [42] or Proposition 1 in [12]). The situation for Problem (P2) is less clear. Indeed, in matrix algebras and algebras of formal power series in one variable, when ordered coordinatewise, every ring ideal is an order ideal. Thus we phrase Problem (P2) for lattice ordered algebras in general. PROBLEM
13.8. Study Problem
(P2)
for Archimedean ialgebras other than
ialgebras. 14. Representation of Ialgebras
Let A be an falgebra and recall that Ab denotes the isubalgebra of all bounded elements in A (see Section 13 above). Several properties are satisfied by A if and only if they are satisfied by Ab and vice versa (see Sections 3,4 and 5 in [90]). Thus we can study some aspects of A via an investigation of Ab. For instance, if we assume that every ring ideal in A is an order ideal then A b has the same property and the converse holds if A in addition is uniformly complete (see [84] by Steinberg). In particular, if every ring ideal in A is an order ideal then order
LATTICE ORDERED ALGEBRAS
125
ideals and ring ideals coincide in Ab (see Theorem 13.2). It turns out that under the latter hypothesis, A b has a nice representation as a space of functions. This kind of a representation then is precisely the topic of this section. First assume that A has an identity element (Le., A is a
k (D) = n {M : M E D} (where it is understood that k (0)
= A). The
hull h (I) of an (ideal I in A is
h(I) = {M E M (A): I
c M}. = h (k (D)).
The subset D of M (A) is said to be closed if D One thus defines the hullkernel (or Stone) topology on M (A). It turns out that M (A) with respect to this hullkernel topology is a compact Hausdorff space (see [53, Theorem 2.3]). Suppose at this point that A is in addition uniformly complete (instead of 'uniformly complete', Henriksen and Johnson in [53] use 'uniformly closed') and denote the identity element of A bye. Since A is unital, Ab is precisely the principal order ideal generated bye. Also recall that A and Orth (A) are isomorphic as falgebra under the given condition. As usual, the falgebra of all realvalued continuous functions on the compact Hausdorff space M (A) is denoted by C (M (A)). Henriksen and Johnson in [53, 3.2, p. 84] proved, using the StoneWeierstrass theorem, the following variant of Stone's representation theorem. THEOREM 14.1. Let A be a uniformly complete falgebra with identity element e. Then Ab and C (M (A)) are isomorphic as falgebras. In particular, if e is a strong order unit in A then A and C (M (A)) are isomorphic as falgebras.
The latter result can of course also be obtained via Kakutani's representation theorem. Indeed, being the principal order ideal in A generated bye, Ab is a uniformly complete Riesz space with e as a strong order unit. It follows that Ab is an Mspace with respect to the Mnorm 11.ll e defined by IIflle = inf {A > 0 : If I :'S Ae}
for all
f
E A
(see [72, Proposition 1.2.13]). Kakutani's representation theorem (see, for instance, [72, Theorem 2.1.3]) guarantees the existence of a compact Hausdorff space 0 so that Ab and C (0) are isomorphic as falgebras and isometric as Mspaces. From an investigation of the cited proof of Kakutani's theorem, we see that 0 is the set of all algebra homomorphisms from Ab onto JR, or, equivalently, the set of all real valued Riesz homomorphisms that send e to 1. With respect to the weak topology a ((Abf ,Ab), where (Ab)'" is the order dual of Ab, the set 0 is a a ((Ab)'" ,Ab)compact Hausdorff space. In view of Theorem 13.2, we observe that o is homeomorphic to M (Ab) and then to M (A) since M (Ab) and M (A) are also homeomorphic (see [53, Corollary 2.8]). We thus recover Theorem 13.1 above. The reader should also compare the above with Gelfand theory for Banach algebras
[94]. We proceed to representation theorems for non unital falgebras. For the terminology and notations concerning the topological spaces under consideration, we refer the reader to the book [47] by Gillman and Jerison. The following two theorems deal with semiprime falgebras (see Theorems 7.2 and 7.9 in [90]).
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KARIM BOULABIAR, GERARD BUSKES, AND ABDELMAJID TRIKI
THEOREM 14,2, Let A be a uniformly complete semiprime falgebra with a weak order unit, Then the following are equivalent. (i) Every ring ideal in A is an order ideal in A. (ii) There exists an almost compact F space 0 such that A b and Co (0) are isomorphic as f algebras.
For almost compact. Fspaces, we refer to [47, 6J]. THEOREM 14.3. Let A be a semiprime Dedekind acomplete f algebra. Then the following are equivalent. (i) Every ring ideal in A is an order ideal in A. (ii) There exist a basically disconnected locally compact Hausdorff space 0 1 and a basically disconnected almost compact Hausdorff space O2 so that Ab is isomorphic as an falgebra to one of the algebras
CK (Ot) ,Co (0 2 ) or CK (Od EB Co (0 2 )
.
Our next representation theorem represents non semiprime falgebras (see [90, Theorem 7.10]). THEOREM 14.4. Let A be a uniformly complete non semiprime falgebra A. Then the following are equivalent. (i) Every ring ideal in A is an order ideal in A. (ii) There exists a locally compact F'space 0 such that Ab is 'isomorphic as an f algebra to the Cartesian product Co (0) x JR, where the multiplication in the falgebra Co (0) x JR is defined by
(1,.x) (g, J.t)
= (1g,O)
for all f, g E Co (0) and.x, J.t E JR.
Note that the topological spaces 0 in Theorem 14.2 and in Theorem 14.4 are constructed in the same way as follows: for the uniformly complete falgebra A (semiprime or not), the centre Z (A) is a uniformly complete falgebra and its multiplicative identity IA is a strong order unit. According to Theorem 14.1, Z (A) is isomorphic as an falgebra to the falgebra C (M (Z (A))) of all realvalued continuous functions on the compact Hausdorff topological space M (Z (A)) (the set of all maximal fideals in Z (A)). It turns out that 0=
u {coz (1) : f
E C (M (Z (A)))).
In a similar way, the topological spaces 0 1 and O2 in Theorem 14.3 are constructed from a certain uniformly complete fsubalgebra of A (for more information, we refer to [90]). We end this section with two comments. Observe that representation theorems listed in this section apply to Ab and not to the whole falgebra A. What one can say about A itself is the following. If A is unital then A can be embedded as an falgebra into an algebra of extended functions on M (A) [53, Theorem 2.3], and if A is semiprime then A can be considered as an fsubalgebra of M (Orth (A)) (recall that if A is semiprime then A can be considered as an fsubalgebra of Orth (A)). In addition, the representation theorems that we presented in this section are not necessarily excessively restrictive due to the previously mentioned fact about transference of various properties of Ab to A. Our final comment about representing lattice ordered algebras deals with a matter of set theory. Zaanen started an ambitious program at around 1980, intending to prove all available material in vector lattices in as far as possible in an
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elementary way, i.e., without using representation theorems. A commonly quoted reason to adhere to Zaanen's program is the need to not use the Axiom of Choice unnecessarily. It is therefore interesting to know that one can avoid the Axiom of Choice and still use representation theorems, as long as the constructs that one has in mind depend on say countably many elements of a Riesz space. For falgebras one of the most useful theorems in that direction is the following (combine Theorem 2.2 in [33] and Corollary 2.7 in [36]). THEOREM 14.5. Let A be a semiprime falgebra. If D is a countable subset of A then the f subalgebra generated by D in A can be represented within ZermaloF'raenkel set theory as an f subalgebra of the space of continuous functions on a metric space.
It should be noted at that the connection between the Axiom of Choice and representations of frings in terms of sub direct products of totally ordered algebras was discussed in the works [42] by Feldman and Henriksen, [67] by Luxemburg, and [9] by Banaschewski. Contrary to the possibility of locally representing countably many elements of any semiprime falgebra as continuous functions without any choice, the global representation of falgebras as sub direct products of totally ordered algebras can not be obtained without appealing to some transcendent tool from set theory. The latter kind of representation theorem is important for more than historical reasons. First of all, many researchers define falgebras as subdirect products of totally ordered algebras. Secondly, Birkhoff and Pierce in their seminal paper [22] observed that the Axiom of Choice seems to be involved if one wishes to obtain (with the definition for falgebras as in this survey) a representation theorem for falgebras as a sub direct product of totally ordered algebras, which for convenience we will now name the BirkhoffPierce Representation Theorem. The three papers [42], [67], and [9] independently prove that the Boolean Prime Ideal Theorem is both sufficient as well as needed for the BirkhoffPierce Representation Theorem. Thus Stone's Representation Theorem for Boolean algebras is constructively equivalent to the BirkhoffPierce Representation Theorem for falgebras. In turn, each of the latter representation theorems is constructively equivalent to the Kakutani Representation Theorem for vector lattices with a strong order unit. In about that same direction, we observe that it is still unknown whether the Boolean Prime Ideal Theorem suffices for that other main representation tool for vector lattices as vector sublattices of extended real valued continuous functions, the socalled MaedaOgasawara Representation Theorem (see [32]).
15. Linear biseparating maps on Ialgebras A linear map T between two algebras A and B is said to be separating if T (I) T (g) = 0 in B whenever f 9 = 0 in A. If in addition T is bijective and its inverse T 1 is separating as well then T is said to be biseparating. Clearly, if T is onetoone and onto then T is biseparating if and only if
f9=
0 in A <=> T (I) T (g) = 0 in B.
If A and B are assumed to be falgebras with unit elements, then T is separating if and only if Tis disjointness preserving, that is, If I II Igl = 0 in A implies IT (1)1 II IT (g) I = 0 in B. This follows directly from the equivalence fg = 0 <=> If I II Igl = 0,
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which holds in any semiprime Ialgebra. For the same reason, the linear map T between two I algebras with multiplicative identity is biseparating if and only if T is adisomorphism (Le., T is bijective and both T and T 1 are disjointness preserving) The reader is encouraged to consult the beautiful memoir [1 J by Abramovich and Kitover for the theory of disjointness preserving linear maps on Riesz spaces. The study of when linear biseparating maps on algebras of real or complex valued continuous functions are weighted isomorphisms started in 1990 with the paper [63J by Jarosz and culminated in the work [5J by Araujo, Beckenstein and Narici with the following result. Let C (X) and C (Y) be the algebras of real or complex valued continuous functions on completely regular topological spaces X and Y, respectively. If T is a linear biseparating map then there exist a nonvanishing wEe (Y) and an homeomorphism h from the realcompactification vX of X onto v Y, such that T (f) (y) = w (y) I (h (y))
for all lEe (X) and y E Y.
Henriksen and Smith in [55J explored the aforementioned result by Araujo, Beckenstein and Narici in the more general setting of unital Ialgebras. They proved that every positive linear biseparating map T between two unital Ialgebras A and B closed under inversion is a weighted isomorphism, that is, there exist an invertible wEB and a Riesz isomorphism S : A > B which is simultaneously an algebra isomorphism, such that
T (f) = wS (f)
for all lEA.
Very recently in [30J, Boulabiar, Buskes, and Henriksen extended the latter result to all order bounded linear biseparating maps on arbitrary (not necessarily closed under inversion) unital Ialgebras over the reals as well as over the complex numbers (for the theory of complex Ialgebras, we refer to [17]). The theorem they obtained is the following. THEOREM 15.1. Let A and B be (real or complex) Ialgebras with unit elements, and let T : A > B be an order bounded linear biseparating map. Then T is a weighted isomorph'ism.
In the previously mentioned memoir by Abramovich and Kitover, we find the following theorem. A disomorphism between two uniformly complete Riesz spaces A and Jyf is automatically order bounded as soon as every universally acomplete projection band in A is essentially onedimensional (see [1, Corollary 15.3]). This theorem is used in [30], under the same condit.ions on A, to show that every linear biseparating map between two uniformly complete Ialgebras A and B is a weighted isomorphism. We point out that a complex Ialgebra is by definition uniformly complete. Thus the phrase 'uniformly complete unital Ialgebra' is understood to mean either a uniformly complete unital I algebra over the reals, or simply a unital Ialgebra over the complex numbers. For t.he proof of the next theorem, see Proposition 5.1 and Theorem 5.2 in [30J. THEOREM 15.2. Let A and B be unilormly complete unital I algebras and assume that every universally acomplete projection band in A is essentially onedimensional. Then every linear biseparating map from A onto B is order bounded and then a weighted 'isomorphism.
To obtain the result by Araujo, Beckenstein and Narici cited above as a consequence of the preceding theorem, Boulabiar, Buskes and Henriksen proved t.hat
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every algebra of all scalarvalued continuous functions on a completely regular topological space has the property that every universally acomplete projection band is essentially onedimensional (see [30, Theorem 5.5]). THEOREM 15.3. Let X be a completely regular topological space X. Then every universally acomplete projection band in the Riesz space C (X) is essentially onedimensional.
Combining Theorems 15.2 and 15.3, we arrive at the next result. COROLLARY 15.4. Let X and Y be completely regular topological spaces. Then every biseparating linear map T : C (X) > C (Y) is a weighted isomorphism. In particular, C (X) and C (Y) are isomorphic as f algebras if and only if there exists a linear biseparating map from C (X) onto C (Y).
It is well known that if X and Yare completely regular topological spaces and S is an isomorphism from C (X) into C (Y) then there exists an homeomorphism h from vY into vX such that S (f) = f 0 h, where vX and vY denote the realcompactifications of X and Y, respectively (see Section 10 in [47]). The latter fact, together with Corollary 15.4 above, directly leads to the next corollary, which was proved earlier in an alternative way by Araujo, Beckenstein and Narici in [5, Proposition 3]. COROLLARY 15.5. Let X and Y be completely regular topological. Then for every l'inear biseparating map T : C (X) > C (Y) there exist a nonvanishing function wE C (Y) and an homeomorphism h : vY > vX such that
T (f) (y) = w (y)
f
(h (y))
for all f E C(X) and y E Y.
It is shown in [47, Theorem 8.3] that two realcompact X and Yare homeomorphic if and only if C (X) and C (Y) are isomorphic as falgebras. Another classical result of rings of continuous functions theory is that if X is a completely regular topological space then C (X) and C (vX) are isomorphic as falgebras (see Remark 8 (a) in [47]). It follows immediately that if X and Y are two completely regular topological spaces, and C (X) and C (Y) are isomorphic as falgebras, then vX and v Yare homeomorphic. The latter implies that, without further assumptions, the conclusion that vX is homeomorphic to vY in Corollary 15.5 is best possible. Under additional assumptions, however, X and Y may themselves be homeomorphic as is shown in the next Corollary. COROLLARY 15.6. Let X and Y be completely regular topological spaces and assume either (i) or (ii) below. (i) X and Yare realcompact. (ii) The points of X, as well as those ofY, are Gopoints. If there exists a linear biseparating map from C (X) onto C (Y) then X and Yare homeomorphic.
Under the condition (i), the result in Corollary 15.6 above follows straightforwardly from Corollary 15.5 while under the condition (ii), it stems directly from [73] by Misra. Finally, it should be noted that the result of Theorem 15.2 above is false if the universally acomplete projection bands in A fail to be essentially onedimensional. Indeed, Example 7.13 in [1] produces a linear biseparating map T on the Dedekind
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complete (and then uniformly complete) unital ialgebra Lo ([0, 1]) of all equivalence classes of measurable functions on [0,1)' which is not order bounded and thus cannot be a weighted isomorphism.
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[61] Huijsmans, C. B. and B. de Pagter, Averaging operators and positive contractive projections, J. Math. Ana. Appl., 113 (1986), 163184. [62] Ionesco Tulcea, A. and C. Ionesco Tulcea, On the lifting property I, J. Math. Anal. Appl., 3 (1961), 537546. [63] Jarosz, K., Automatic continuity of separating linear isomorphisms, Bull. Canadian Math. Soc., 33 (1990), 139144. [64] Johnson, D. G., The completion of an archimedean Iring, J. London Math. Soc, 40 (1965), 493493. [65] Kudhicek, V., On some types of Rrings. Sb. Vysoke. Ucen{ Tech. Bmo 12 (1962), 179181. [66] Lambek, J. Lectures on Rings and Modules, Blaisdell, 1966. [67] Luxemburg, W. A. J., Some Aspects 01 the Theory 01 Riesz Spaces, Univ. Arkansas Lecture Notes Math. 4, Fayetteville, 1979 [68] Luxemburg, W. A. J., A remark on a paper by D. Feldman and M. Henriksen concerning the definition of Irings, Nederl. Akad. Wetensch. Indag. Math., 50 (1998), 127130. [69] Luxemburg, W. A. J. and A. C. Zaanen, Riesz spaces I, NorthHolland, Amsterdam, 1971. [70] Lozanovsky, G., The functions of elements of vector lattices, Izv. Vyssh. Uchebn. Zaved. Mat., 4 (1973), 4554. [71] Martinez, J. The maximal ring of quotient Iring. Algebra Univ .• 33 (1995), 355369. [72] MeyerNieberg, P., Banach Lattices, Springer Verlag, Berlin Heidelberg New York, 1991. [73] Misra, P. R., On isomorphism theorems for C(X), Acta. Math. Acad. Sci. Hungar., 39 (1982), 179180. [74] Nagasawa, M., Isomorphisms between commutative Banach algebras with an application to rings of analytic functions, Kodai Math. Semin. Rep., 11 (1959), 182188. [75] Nakano, H., Modern Spectral Theory, Tokyo Math. Book Series II, Maruzen, Tokyo, 1950. [76] de Pagter, B., Ialgebras and orthomorphisms (Thesis, Leiden, 1981). [77] Phelps, R. R., Extremal operators and homomorphisms, Trans. Amer. Math. Soc., 108 (1963), 265274. [78] van Putten, B., Disjunctive linear operators and partial multiplication in Riesz spaces, (Thesis, Wageningen, 1980). [79] Quinn, J., Intermediate Riesz spaces, Pacific J. Math., 56 (1975), 225263. [80] Scheffold, E., FFBanachverbandsalgebren, Math. Z., 177 (1981),193205. [81] Scheffold, E., Der Bidual von FBanachverbandsalgebren, Acta Sci. Math., 55 (1991), 167179. [82] Scheffold, E., Uber Bimorphismen und das ArensProdukt bei kommutativen DBanachverbandsalgebren, Preprint. [83] Scheffold, E., Uber den ordnungsstetigen Bidual von FFBanachverbandsalgebren, Arch. Math., 60 (1993), 473477. [84] Steinberg, S. A., Quotient rings of a class of latticeordered rings, Can. J. Math., 25 (1973), 627645. [85] Steinberg, Stuart A. On the unitability of a class of partially ordered rings that have squares positive. J. Algebra 100 (1986), no. 2, 325343. [86] Steinberg, Stuart A. On latticeordered algebras that satisfy polynomial identities. Ordered algebraic structures (Cincinnati, Ohio, 1982), 179187, Lecture Notes in Pure and Appl. Math., 99, Dekker, New York, 1985. [87] Steinberg, Stuart A. Unital $l$prime latticeordered rings with polynomial constraints are domains. Trans. Amer. Math. Soc. 276 (1983), no. 1, 145164. [88] Steinberg, Stuart A. On latticeordered rings in which the square of every element is positive. J. Austral. Math. Soc. Ser. A 22 (1976), no. 3, 362370. [89] Toumi, M. A., On some Isubalgebras of dalgebras, To appear in Math. Reports [90] Triki, A., Stable Ialgebras, Algebra Univ., 44 (2000) 6586. [91] Triki, A., On algebra homomorphisms in complex almost Ialgebras, Comment. Math. Univ. Carolinae, 43 (2002), 2331. [92] Zaanen, A. C., Riesz Spaces II, North Holland, AmsterdamNew YorkOxford, 1983. [93] Zaanen, A. C., Introduction to Operator Theory in Riesz Spaces, Springer Verlag, Berlin, 1997. [94] Zelazko, W., Banach algebras, Elsevier, AmsterdamLondonNew York, 1973.
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DEPARTEMENT DU CYCLE AGREGATIF, INSTITUT PREPARATOIRE AUX ETUDES SCIENTIFIQUES ET TECHNIQUES, UNIVERSITE DU 7 NOVEMBRE
A CARTHAGE,
BP 51, 2070LA MARSA, TUNISIA
Email address:karim.boulabiar«lipest.rnu.tn DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MISSISSIPPI, UNIVERSITY, MS38677, USA
Email address: mmbuskes«lsunset. olemiss. edu DEPARTEMENT DE MATHEMATIQUES, FACULTE DES SCIENCES DE TUNIS, UNIVERSITE TUNIS EL MANAR, 1060TUNIS, TUNISIA
Email address:abdelmajid.triki«lfst.rnu.tn
Contemporary Mathematics Volume 328, 2003
An extension of a theorem of Wermer, Bernard, Sidney and Hatori to algebras of functions on locally compact spaces Eggert Briem
ABSTRACT. It follows from a theorem of J. Wermer that if A is a uniformly closed algebra of continuous complexvalued functions vanishing at infinity on a locally compact Hausdorff space X, the property that b2 E Re A for all bE Re A implies A = Co(X). In other words, if the function h(t) = t 2 operates by composition on ReA then A = Co(X). This result was generalized by O. Hatori. He proved that in place of the function h(t) = t 2 one can put any realvalued function defined in a neighbourhood of 0, thus extending a similar result for the compact case. Here a simple alternative proof is given for the locally compact case.
1. Introduction
Let X be a compact Hausdorff space and B a uniformly closed subspace of CIR(X), the space of all continuous realvalued functions on X, which separates the points of X and contains the constant functions. A version of the StoneWeierstrass theorem says that if b2 E B for all b E B then B = CIR(X). Clearly this result does not hold if, instead of assuming that B is uniformly closed, one assumes that B is a Banach space in some norm which dominates the supnorm, as the example of any nontrivial real Banach function algebra shows. However, if B is the real part of a uniform algebra, a theorem of J. Wermer says that if b2 E B for all b E B then
B = CIR(X). Let us say that a realvalued function h, defined on an interval I of the real line, E B whenever b E Band b maps X into I. Thus b2 E B for all bE B means that h(t) = t 2 operates on B. The StoneWeierstrass theorem W8.', generalized by K. de Leeuw and Y. Katznelson (see [4], theorem 4.21), they showed that h(t) = t 2 can be replaced by any continuous nonaffine function (Le. a function not ofthe type h(t) = o:t+!3) defined on an interval, and the theorem of J. Wermer was similarily generalized by A. Bernard, S. Sidney and O. Hatori [1], [5] and [8]. Here one can even do without the continuity assumption, a function operating on the real part of a uniform algebra of infinite dimension is automatically continuous.
operates on B if hob
1991 Mathematics Subject Classification. Primary 46JlOj Secondary 46E15. Key words and phmses. uniform algebra, locally compact space, functional calculus. © 2003 American Mathematical Society
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EGGERT BRIEM
In the case where X is locally compact, the functional calculus for a uniformly closed subspace of Co(X, JR), the space of all continuous realvalued functions vanishing at infinity on X, may be nontrivial (cf. [2]). What then about the real part of a uniformly closed subalgebra of Co(X), the space of all continuous complexvalued functions vanishing at infinity on X? Wermer's theorem, [10], clearly applies to this situation. It turns out that the functional calculus for the real part of a uniformly closed subalgebra, which is not a C* algebra, of Co(X) is trivial. This result, which is due to O. Hatori, is to be found in [7]. To prove this one has to do more than just adapt the proofs for the compact case to the locally compact situation. The purpose of this paper is to give a simple alternative proof using antisymmetric decomposition of uniform algebras. 2. Proofs and results Let X be a locally compact Hausdorff space and let Xl denote the onepoint compactification of X, with Xoo denoting the point at infinity. The functions in Co(X) have a natural extension to functions in C(Xl)' the space of all continuous complexvalued functions on Xl. If A is a uniformly closed subalgebra of Co(X), that separates the points of X and does not vanish identically at any point of X, then Al = AEI1C is a uniformly closed subalgebra of C(Xd that separates the points of Xl and contains the contant functions and A = {a E All a(x oo ) = O}. Thus, we can assume that there is a uniformly closed subalgebra Al of C(Xd containing the constant functions, where Xl is compact, and a point Xoo in Xl, such that A is obtained by restricting the functions in the set {a E Al Ia(x co ) = O} to the set X = Xl \ {xoo}. We need some results from the theory of uniform algebras. A subset E of Xl is a set of antisymmetry for Al if the only functions in Al that are realvalued on E are constant functions. Maximal sets of antisymmetry are closed generalized peak sets (intersections of peak sets), they form a partition of Xl and a continuous complexvalued function f on Xl is in Al if and only if its restriction to each maximal set of antisymmetry E is in the space AIlE of restrictions of functions in Al to E. Thus Al(Xd = C(Xd if and only if each maximal set of antisymmetry is a singleton. Also, AIlE is uniformly closed for each maximal set of antisymmetry, there is actually for each a in AIlE an element a' in Al with a' = a on E and II a' 1100=11 a llco,(E). (The latter norm is the supnorm w.r.t E). This material can f. ex. be found in [3]. If Al is the disc algebra on the closed unit disc in the complex plane and A is the algebra of functions vanishing at the origin, then any nonzero function in A takes real values of opposite signs in any neighbourhood of O. Thus any realvalued function, defined on an interval which is not a neighbourhood of 0, operates trivially on Re A. We shall therefore always assume that the interval on which an operating function is defined is a neighbourhood of O. We can now state the main result of this note. THEOREM 1. Let A be a uniformly closed subalgebra of Co(X) which separates the points of X and does not vanish identically at any point in X and let Re A be the space of real parts of functions in A. If Re A has a nonaffine operating function h: I ........ JR where I, an interval, is a neighbourhood of 0, then Re A = Co(X, JR) and thus also A = Co(X).
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The first step in the proof of Theorem 1 is to show that operating functions are continuous. To prove this we need the following lemma. LEMMA 1. Let E be a maximal set of antisymmetry for A1 and let b E Re A. Then b(E) is either an interval or a singleton.
Proof. Suppose b(E) is not connected, and that b = Rea for some a in A 1. Then we can find two disjoint closed rectangles Ro and R1 in the complex plane such that a(E) is a subset of Ro u R1 but not a subset of anyone of the two rectangles. By Runge's theorem there is sequence of polynomials (Pn) converging uniformly to o on Ro and to 1 on R 1. Since AdE is uniformly closed, the sequence (Pn 0 a) converges on E to an element of A1IE. But the sequence converges on E to a function which takes only the values 0 and 1, contradicting the fact that E is a set of antisymmetry. 0 LEMMA 2. Let E be a maximal set of antisymmetry for A 1 such that E\ {xoo } =I 0. Then there is an Xo E E such that for every to E I (resp. to in the interior of I), there exists bo E ReA with bo(xo) = to and bo(X) C I (resp. bo(X) is a subset of the interior of I).
Proof. If to = 0, the existence of the desired function is easily proved. We consider the case where to > O. (The proof for the case to < 0 is similar.) Let Xo be a generalized peak point in E, other than x oo , for A 1 • Simple calculations show that we can take 1 E A with I(xo) = 11/1100 = 1. Let c: be a positive number such that (2c:, to] C I (resp. (2c:, to] is a subset of the interior of I), and R the closed rectangle with corners c: ± i and to ± i. Let
~
be an operating function for Re A. Then, unless
Proof. If A =I Co(X,~), there is a maximal set of antisymmetry E for A1 which contains more than one point. Thus E \ {xoo} =I 0. Let t be an arbitrary point in I. Then by Lemma 2 there exists and x E E and bEReA such that b(x) = t and b(X) C I. Since hob is in ReA, hob is continuous. It follows that h is continuous at t. We conclude that h is continuous on its domain of definition. If A = Co(X), h need only be continuous at 0, f. ex., if A = Co. 0
In the compact case, where A contains the constant functions, Re A is dense in CR(X) (cf. Proposition 4 below). Here we obtain a similar result. PROPOSITION 2. Let h : I + IR be an operating function for ReA which is nonaffine in every neighbourhood of an interior point to for I. Let Xo E X and suppose that there is a function bo E ReA which maps X into the interior of I and satisfies bo(xo) = to· Then there is an open neighbourhood Xo of Xo such that every f E Co (X, ~) which vanishes outside Xo can be approximated uniformly on X by elements of ReA.
EGGERT BRIEM
138
Proof. We are assuming that X is locally compact and not compact so that
h(O) = o. We may assume that h is continuous by Proposition 1. Let us first look at the case where h is a polynomial of degree at most three in some neighbourhood of 0 but not affine there. Then bed E Re A for all b, e, d E Re A and hence fb E cl(Re A) for all b E cl(Re A) and all f in the algebra generated by Re A. Here cl(Re A) denotes the uniform closure of Re A. It follows that
Co(X,JR)· cl(ReA)
~
cl(ReA)
and hence ReA is dense in Co(X,JR). Suppose first that h is affine in some neighbourhood ofO. Replacing h by h(t),t we may assume that h = 0 in a neighbourhood of o. Put e = tC; l bo. For bEReA, r,t E JR and '{) E CO'(JR) the function
J
h 0 (rb o + tb  se)'{)(s)ds
is in cl(Re A) if r is close to 1, t is close to 0 and the support of'{) is contained in a small neighbourhood of 0, so that the expression above is defined. We differentiate twice w.r.t. 0, put t = 0 to obtain
b2 e 2
J
h 0 (rbo  se)'{)"(s)ds E cl(ReA),
for all bEReA. (If le(x)1 is small the expression is 0.) Put
d= We have
d(xo) =
J
h 0 (rb o  se)'{)"(s)ds.
J
h 0 (rto  s)'{)"(s)ds = (h
* '{))"(rto).
Since h is not affine in any neighbourhood of to we can choose '{) and r, where the support of'{) is contained an arbitrarily small neighbourhood of 0 and r is arbitrarily close to 1 such that d(xo) '" O. Let
Xo
=
{x
E
Xld(x) '" O}.
Since b2 e 2 d E cl(Re A) for all b E Re A it follows that
b1 ,b2 e 2 d E cl(ReA) for all b1 , b2 E Re A. Put
M(ReA) = {J
E
Co(X,JR) If· cl(ReA)
~
cl(ReA)}.
be 2 d
Above we have E M(ReA) for bEReA. Since these functions separate the points of Xo and since M(ReA) is an algebra we deduce that cl(ReA) contains every f E Co(X,JR) which vanishes outside Xo. Suppose now that h is not affine in any neighbourhood of o. For e E Re A let
B(e) = {u E cl(ReA) 13>' E JR+ s.t. lui::::; >'Iel}. If b ERe A and b + ie E A then be and be(b2  e2 ) belong to B(e). Let also
M(e) = {g
E
Co(X, JR) Ig. cl(B(e))
~
cl(B(e))},
a subalgebra of Co(X, JR). We note that since h is continuous h also operates on
cl(ReA).
ALGEBRAS ON LOCALLY COMPACT SPACES
139
For bl ERe A, for u E B(e), r, t E lR and
J
(1)
h 0 (rb l
+ tu 
se)
is in cl(Re A), if rand t are sufficiently small and the support of
uie i
J
ho (rb l  se)
Put
J
d= With i
= 2 and
h 0 (rb l  se)
instead of
u 2 e 2 d E cl(Re A) for all u E B(e) and hence
(2) for all
Ul, U2
E B(e). With i
= 3 and u = be E B(e)
in (1) we get
b3 d E cl(ReA), and with i
=
1, u
= beW  e2 )
E B(e) and
b(b2

e2 )d E cl(Re A)
and thus
If we put
U2
be2 d E cl(ReA). = be2 d in (2) we find that ubd2 E cl(Re A)
for all u E B(e). Since u E B(e) implies ubd2 E B(e) we deduce that
bd 2 E M(e). Let us determine how well the functions bd2 separate the points of the set
Xo = {x
E
X Ib(x)e(x) i a}.
Let x, y E Xo. Suppose that does not separate x and y for every d. Since A is an algebra we can choose bl E ReA such that bl(x) = 0 and bl(y) = 1. Then
bd 2
d2 (y)
= (b(x)/b(y))d 2 (x).
The right hand side is independant of r so that
d(y)
=
J
h 0 (r  se(y))
a differentiable function of r, is constant in for r in some neighbourhood of O. It follows that h 0 (r  se(y))
J
for all r in a neighbourhood of O. Since this holds for all
EGGERT BRlEM
140
dense in Co(X,JR). Otherwise, the functions bd2 separate the points of X o, and since these functions vanish outside X o, it follows that
{f E Co(X, JR) I I = 0 on Xo \ Xo}
~
M(c).
Thus by a simple calculation we have
{f
E
Co(X,JR) I I = 0 on Xo \ Xo}
~
cl(ReA).
o To proceed to the proof of Theorem 1 we need a local version of the so called Bernard's Lemma [1]. Let A be an infinite set and let C be a Banach space with norm I . II. Put
eOO(C) = {{c.d ICA E C and sup
AEA
I CA 11< oo},
where {cA} denotes a function from A into C such that {cA}(a) = a E A. Then eOO(C) is a Banach space in the norm
I
{cA }
Co
E C for each
11= sup II CA I . AEA
The space Re A is a Banach space in the norm
lib 11= inf{11 a 1100 Ia E A and clearly lib 1100::;11 b II. We thus have eOO(ReA)
~
and b = Rea},
eOO(Co(X,JR)).
Bernard's lemma (see [1]) says that if eOO(ReA) is dense in eOO(Co(X,JR)) then ReA = Co(X,JR). To prove that eOO(ReA) is dense in eOO(Co(X,JR)), we represent eOO(Co(X,JR)) as a space of functions on a compact Hausdorff space. We identify Co(X, JR) with a subspace of CIR(Xd in the obvious manner. Let A have the discrete topology and let (3(A x Xd) be the StoneCech compactification of Ax Xl' The space eOO(C(xd) can be identified with C({3(A x Xd) in a natural way
{fA}
FUll} where F{!lI}b,x) = 1'Y(x) for each b,x) in A x Xl' subspace of CIR ({3(A x Xl)) and we have the inclusions
eOO(ReA)
~
+
eOO(Co(X,JR))
~
Thus eOO(Co(X,JR)) is a
CIR({3(A x Xd).
The subsets of the StoneCech compactification may be complicated but the subsets we are interested in have a simple description. For each I in C(Xl ) let {f} denote the net {fA} where IA = I for each A Let now Po be in {3(A x Xd. The map
I
+
{f }(Po)
is a homomorphism of C(X l ) and is thus given by pointevaluation at some point in Xl' Put
Xo
5: 0
= {p E (3(A x Xd I {f}(p)
=
I(xo)
VI E CIR(Xl )}.
ALGEBRAS ON LOCALLY COMPACT SPACES
141
The sets x form a partition of {3(A x Xt} into closed sets. Their significance stems from the following local version of Bernard's lemma, similar to the one given by O. Hatori [6].
Bernard's lemma (local version) Let A denote the unit ball of Co(X,JR.), let x be in X and suppose that the restriction space lOO(ReA)lx is dense in CIR(x). Then there is a compact neighbourhood Kx of x such that ReAIKx = CIR(Kx )' Proof. For each A in A let 1>.. = A. By assumption there is an element {b A } in lOO(ReA) such that we have the inequality
I{fA}  {bA}1 < 1/2 on x and hence also on some open subset U of {3(A x Xt} containing X. By a simple calculation we have
Xo = {p E (3(A x Xl) I I{f}(p)  f(xo)1
:=; ~
Vf E CIR(Xl )}.
Thus there is a function fo in CIR(Xt} such that the set
{p
E
{3(A x Xt}
I I{fo}(p) 
fo(x)1
:=; 1/2}
is a subset of U. Put
Kx = {y E Xd Ifo(Y)  fo(x)1 :=; 1/2}, a compact neighbourhood of x. Then A x Kx is contained in U so that IfA  bA I :=; 1/2 on Kx for all A in A. Let M = SUPA II bA II, a finite quantity because (b A ) is in lOO(Re A). Take any f in Co(X, JR.) with II f lloo:=; 1. By induction we construct a sequence (b n ) of elements from ReA, with II bn II:=; M for all n, such that nl
12n( f  L
2 i
bi )  bnl < ~
i=O
on Kx. The function b =
E:o 2 b is in Band b = f on Kx. 0 i i
PROPOSITION 3. Let Xo E X and suppose that there is an open neighbourhood Xo of Xo such that every function in Co(X,JR.), which vanishes outside Xo can be unifromly approximated on X by elements from ReA. Then lOO(ReA) sepamtes the points of xo.
Proof. Suppose p,q are in xo. There is an element {fA} in lOO(Co(X,JR.», where each 1>.. vanishes outside X o, such that {fA}(P) = 0 and {fA}(q) = 1. By assumption we can for each A E A find a function bA in ReA with II fA  bA lloo:=; ~. Thus we have {bA}(p) :=; ~ and {bA}(q) ~ ~ since II{fA}  {bA}lIoo,{3(Axxt} :=; ~. Let a A = bA + iC A be in A for each A. Now, the Anet {e a ...  1} is in lOO(A) and it separates p from q since {e an } does. Since
I{e a ... }(p)1 =
{e b... }(p)
:=; e l / 4
and I{e a ... }(q)1 = {e b... }(q) ~ e3 / 4
,
we see that lOO(A) separates p from q and thus lOO(ReA) does so as well. 0
EGGERT BRIEM
142
We also need the aforementioned extension of the StoneWeierstrass theorem due to de Leeuw and Katznelson (see [4], Theorem 4.21). We can prove a version of this result, Proposition 4, in the same way as the original one, the proof is omitted. PROPOSITION 4. Let Y be a compact Hausdorff space and B a subspace ofCR(Y) which separates the points of Y and contains the constant functions. Suppose that B is also a normed space with the norm II· liB which dominates the uniform norm. Let h be a continuous function defined on an interval I. Suppose that h is nonaffine in every neighbourhood of an interior point to in 1. Suppose that there exists a positive real number 8 > 0 with (to  8, to + 8) ~ I such that h 0 (to + u) E B for every u E B with Ilull < 8. Then B is uniformly dense in CIR(Y).
Proof of Theorem 1. We are going to use the local version of Bernard's Lemma. We thus have to show that the conditions stated there are satisfied. For this we use the result of de Leeuw and Katznelson above. The main obstacle is that we can not conclude that if {b.>.} is in [00 (Re A) then h e {b.>.} is in [00 (Re A) although the composite function is defined, i.e. we can not conclude that composition with h maps bounded nets to bounded nets. To overcome this difficulty we use a method of Sidney, [8], to obtain local boundedness for composition with h. Suppose A =I Co(X) and thus also Al =I C(X 1 ). Let E be a maximal set of antisymmetry for Al containing more than one point. Let further to be an interior point of I such that h is not affine in any neighbourhood of to and let bo be a function in Re A which maps X into the interior I for which to is an interior point of bo(E). By Lemma 2 such a function exists. We now choose E > 0 such that if b is in the Eball, (ReA)., of ReA then ho(bo+b) is defined and to is an interior point of (b o + b)(E). We write (ReA).
= U{b E (ReA). I I h 0 (bo + b) II::; n}. n
The Baire Category Theorem shows that the closure of one of the sets on the right hand side has an interior point and thus there is a function b1 E Re A and positive numbers 8, lvI, with I b1 I +8 < E, such that
I he (bo +b 1 +c) II::; lvI for c in a dense subset of the 8ball of Re A. Let b = bo + b1 , and let [00 (Re A) be as in the local version of Bernard's Lemma. We then have
h 0 {b + c.>.} E cl([OO(Re A)), the uniform closure of [OO(ReA), if {c.>.} E [OO(ReA) and such that b(xo) = to, an interior point of b(E). We restrict to Xo and deduce that
h 0 {to
I
{c.>.}
11< 8.
Take Xo E E
+ c.>.} E cl([OO(ReA)lxo),
for every {c.>.} E [OO(ReA)lxo whose quotient norm satisfies II {c.>.} 11< 8. Since {c} is constant on xo for any c E ReA, the space [OO(ReA)lxo contains the constant functions, it also separates the points of Xo by Proposition 2 and 3. Then Proposition 4 shows that [OO(ReA)lxo is dense in CIR(xo) and thus, by the local version of Bernard's Lemma, ReAIK = CIR(K) for some compact neighbourhood K of Xo. The theorem of Sidney and Stout, [9], then shows that AIK = C(K). By Proposition 2, there exists an open neighbourhood Xo of Xo such that every f E Co(X, 1R) which vanishes outside Xo can be approximated uniformly on X by
ALGEBRAS ON LOCALLY COMPACT SPACES
143
elements of ReA. We may assume K c Xo. Let Kl be a compact neighbourhood of Xo such that K 1 is in the interior of K. Since E is antisymmetric and contains more than one point, EnK l contains more than one point. On the other hand, we will show that {f E Co(X, IR) II = 0 outside Kd CAl. It will follow that En Kl = {xo}, which will be a contradiction proving A = Co(X, IR). Let 1 E Co(X, IR) with 1 = 0 outside K 1 . A function u E Co(X, IR) with lui::; 1 on X, u = 1 on K l , and u = 0 outside K can be uniformly approximated by functions in ReA. Thus, for every positive integer n, there exists bn E ReA such that 1  ,& ::; bn ::; 1 on K 1 and bn ::; ,& outside K. Without loss of generality we may assume that Ibnl ::; 1 on X. Take Cn E A with Recn = bn and put an = (eCnl)n. Then an E AI, e1. ::; lanl ::; Ion Kl and lanl ::; e n+1. outside K. Since AllK = C(K) and thus AllKl = C(Kd, there exists a positive real number M such that for every positive integer n there exists gn E Al such that gnan = 1 on Kl with IIgnlloo ::; M. Since AllK = C(K), there is a function af E Al with af = 1 on K. Then afgnan E Al and by a simple calculation lIafgnan  11100 + 0 as n + 00, that is, 1 E AI' 0 References [lJ A. Bernard, Espaces des parties relies des Iments d'une algebre de Banach de fonctions, J. Funct. Anal. 10 (1972), 387409. [2J E. Briem, Approximations from Subspaces of Co(X), J. Approx. Theory 112 (2001), 279294. [3J A. Browder, Introduction to Function Algebras, W. A. Benjamin, Inc. !969. [4J R.B. Burckel, Characterizations of C(X) among its subalgebras (Lecture Notes in Pure and Appl. Math. 6). Marcel Dekker, New York 1972. [5J O. Hatori, Functions which operate on the real part of a uniform algebra, Proc. Amer. Math. Soc. 83 (1981), 565568. [6J O. Hatori, Separation properties and operating functions on a space of continuous functions, lntemat. J. Math. 4 (1993), 551600. [7J O. hatori, Range transformations on a Banach function algebra. IV, Proc. Amer. Math. Soc. 116 (1992), 149156. [8J S. J. Sidney, Functions which operate on the real part of a uniform algebra, Pac. J. Math. 80 (1979), 265272. [9J S. J. Sidney and E. L. Stout, A note on interpolation, Proc. Amer. Math. Soc. 19 (1968), 380382. [10J J. Wermer, The space of real parts of a function algebra, Pac. J. Math. 13 (1963), 14231426. SCIENCE INSTITUTE, UNIVERSITY OF ICELAND. REYKJAVIK, ICELAND
Email address:briemillhi.is
Contemporary Mathematics Volume 328, 2003
Some mapping properties of psumming operators with Hilbertian domain Qingying Bu
ABSTRACT. We prove that if H is a Hilbert space, Y is a Banach space and u : H + Y is absolutely psumming for some p :::: 1, then for any 1 < q < 00, u takes absolutely qsummable sequences in H into members of iq®Y, the projective tensor product of lq and Y.
Given a real or complex Banach space X and 1 ::; p < 00, we denote by and f;eak(X) the Banach spaces of sequences in X with norms II(xn)nlle;trong(X) = 11(llxnll)nllep and II(xn)nlle;eak(X) = sUP",*eB x * II(x*xn)nllep, respectively (cf. [4, pp. 3236]). For 1 < p < 00, let fp(X) denote the space of all (strongly psummable) sequences in X such that E~=l Ix~(xn)1 < 00 for each (X~)n E f't},eak(x*), normed by f~trong(x)
II(xn)nlltp(X) = sup
{I ~ x~(xn)1
:
II(x~)nlle~~eak(X.) ::; 1} ,
where pI is the conjugate of p, i.e., lip + 11pl = 1. With this norm fp(X) is a Banach space (cf. [1, 3]). Note: In [2] it was shown that fp(X) is exactly fp®X, the projective tensor product of fp and X. In this note we use this identification of fp(X) with fp®X to deduce a surprising mapping property of absolutely psumming operators that have a Hilbert space domain. While the main result of this note can be derived from some bynow famous results of Kwapien, it was discovered because of the identification of fp(X) with fp®X, moreover, this identification leads itself to a proof that is a clean and clear application of Khinchin's inequality, Kahane's inequality, and Pietsch's Domination theorem  all fundamental aspects of the theory of psumming operators. From the definitions, we have for 1 < p < 00, fp(X) ~ f;trong(x) ~ f;eak(x), and 11·lIe~..ak(X) ::; 1I'lIl~trong(X) ::; 11·llep(x)· Moreover, in case dimX = 00, all the containments are proper. For Banach spaces X and Y and a continuous linear operator 'U : X + Y, define ft. : XN + yN by (xn)n 1+ (uxn)n. Then ft. is a linear operator. Thanks 2000 Mathematics Subject Classification. 46B28. © 145
2003 American Mathematical Society
146
QINGYING BU
to the Closed Graph Theorem, each of
it: f;eak(x)
>
f;eak(y);
it:
f~trong(x) > f~trong(y);
it: fp(X)
>
fp(Y)
is a continuous operator with
Ilitll£;:,eak(x)_e;:,eak(Y) = Ilitll£~trong(x)_e~trong(y) =
Ilitllep(x)£p(Y) = Ilull·
We should mention here Khinchin's inequality (cf. [4, p. 10]) and Kahane's inequality (cf. [4, p. 211]) each of which plays a critical role in this paper. Let rn(t) denote the Rademacher functions (cf. [4, p. 10]), namely, rn : [O,IJ > ~, n E Pi! defined by rn(t) := sign(sin2n7rt).
< p < 00, there are positive constants Ap, Bp such that for any scalars a1, a2, ... ,an, we have
Khinchin's Inequality. For any 0
Kahane's Inequality. If 0 < p, q < which
([II t, r,('lx, I ,d')
1/, ,;
00,
then there is a constant Kp,q > 0 for
K",' ([II
t,
r,('lx,lI' dt )
1/,
regardless of the choice of a Banach space X and of finitely m.any vectors Xl, X2, ... , Xn from.X.
In 1967, A. Pietsch [5J introduced psumming operators between Banach spaces, namely, a Banach space operator u : X > Y is called psumming operator, 1 :::; p < 00, if it takes f~eak(x) into f~trong(y). Let IIp(X, Y) denote the space of all psumming operators from a Banach space X to a Banach space Y. If u : X > Y is a psumming operator then, thanks to the Closed Graph Theorem, it : f~eak(x) > f~trong(y) is continuous. So we define the psumming norm 7rp(u) on IIp (X, Y) to be 7rp (u) = Ilitll£;:,eak(x)_£~trong(y). With this norm IIp (X, Y) is a Banach space. There is an equivalent definition of psumming operators, namely, a Banach space operator u : X > Y is psumming if and only if there is a constant c > 0 such that for any XI,X2,'" ,Xn E X, (1)
In this case, 7rp (u)
= inf{c
> 0: for all possible c in (I)}.
In 1973, J. S. Cohen [3J introduced strongly psumming operators between Banach spaces, namely, a Banach space operator u : X > Y is called strongly psumming operator, 1 < p < 00, if it takes f~trong(x) into fp(Y). Let Dp(X, Y) denote the space of all strongly psumming operators from a Banach space X to a Banach space Y. If u : X > Y is a strongly psumming operator then, thanks to the Closed Graph Theorem, it : f~trong(x) > fp(Y) is continuous. So we define a
PROPERTIES OF pSUMMING OPERATORS
147
strongly psumming norm Dp(u) on Dp(X, Y) to be Dp(u) = lIulll~trong(X)+lp(Y)' With this norm Dp(X, Y) is a Banach space. There is an equivalent definition of strongly psumming operators, namely, a Banach space operator U ': X ~ Y is strongly psumming if and only if there is a constant c > 0 such that for any Xll X2,'" , Xn E X, and any yi, Y2,'" , y~ E Y*,
(2)
In this case, Dp(u) = inf{c > 0: for all possible c in (2)}.
Note: Actually, U E Dp(X, Y) means that u takes absolutely psummable sequences in X into members of ip®Y.
Main Theorem. Let 1 < p, q < 00; and let H be a Hilbert space and Y be a Banach space. Then IIp(H, Y) ~ Dq(H, Y), i.e., if u : H ~ Y is absolutely psumming, then u takes absolutely qsummable sequences in H into members of
lq®Y.
PROOF. First consider H = i2 for n E N. Let u E IIp(l2' Y). By Pietsch's Domination Theorem [4, p. 44], there is a regular probability measure J.L on Bl'.J: such that for any x E i 2, lip (
Iluxll ::; 1Tp (U)' Lt'.J: I(x, z)IP dJ.L(z) Now for
Xl,X2,'"
,Xm
Ei
)
(3)
2, and Yi'Y2"" ,Y;" E Y*, we have n
Xk =
~Xk ·e· L.J ,'&"',
k = 1,2,··· ,m.
i=l
Then m
m
L
I(UXk,
yZ)1
n
L I(Lxk,iuei, yZ)1 k=l
k=l
< <
i=1
t. (t, lx,.,1 (t,1 t.IlX,II' 1" ' (1' 1t, 2) ,/, ,
(He;, y;)I' ) II'
r,(t)(ue" Yk)
I" II"~ dt)
148
QINGYING BU
by Khinchin's inequality,
L· (t." (t.[1 t, I"
<
x."') II, .
(x.);"
<
r,(t)(ue, , yk)
",(t)"""
(X) • ( [
(x,);"
~
11,·
dt)
r,(t)"e,
r,(t)ue,
dt)
(4) Now by Kahane's inequality and (3),
([II t, r,(t)"" II" dt) 'I,' "K,.,.· ([II t, r,(t)ue,II'
(L'r I(t,
q ([
u) .
<
~
r,(t)e;, z)I' dp(z) )
r,(t)e;, z) dt) dP(Z)),,'
z)
u) .
B, .
dp(z) )'/'
r
dp(z)
B, .
(5) Combining (4) and (5),
So
U
E
Dq(H, Y) with 1
Dq(u) :::; A . Bp· Kp,ql· 7rp(u). ql Now consider a general Hilbert space H. Let
Xl, X2,··· , Xm E
H and
Yi,y2,··· ,y:n, E Y*. Then there is an n E N such that span{xk}r is isometrically isomorphic to i 2. Let P be the orthogonal projection from H onto span{xk}r.
PROPERTIES OF pSUMMING OPERATORS
149
Then by (6), m
m
L I(UPXk, Yk)1
k=1
k=1
<
:
ql
. Bp' Kp,ql' ll"p(U) . II (PXk)i"lI egtron 9 (X)
. Bp' Kp,ql' ll"p(U) ·11(Xk)i"lIe~tr.on9(X) ql This shows us that U E Dq(H, Y) and again 1 Dq(u) :::; A . Bp' Kp,ql' ll"p(u). ql The proof is complete. :
·11(Yk)i"lle~,.ak(X.)
·11(Yk)i"lle~,eak(X.).
o
ACKNOWLEDGEMENT. I am grateful to my advisor, Professor Joe Diestel, for his good suggestions for this paper.
References 1. H. Apiola, Duality between spaces of psummable sequences, (p, q)summing operators and characterization of nuclearity, Math. Ann. 219 (1976), 5364. 2. Q. Bu and J. Diestel, Observations about the projective tensor product of Banach spaces, I  ip®X, 1 < p < 00, Quaestiones Math. 24 (2001),519533. 3. J. S. Cohen, Absolutely psumming, pnuclear operators, and their conjugates, Math. Ann. 201 (1973), 177200. 4. J. Diestel, H. Jarchow, and A. Tonge, Absolutely Summing Operators, Cambridge Univ. Press, Cambridge, 1995. 5. A. Pietsch, Absolut psummierende abbildungen in normierten riiumen, Studia Math. 28 (1967), 333353. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MISSISSIPPI, UNIVERSITY, MS
Email address: qbu(llolemiss. edu
38677
Contemporary Ma.thema.tics Volume 328, 2003
The Unique Decomposition Property and the BanachStone Theorem Audrey Curnock, John Howroyd, and NgaiChing Wong* ABSTRACT. We show an affine version of the BanachStone theorem. Given compact convex sets K and S with unique decomposition property, we show that every surjective linear isometry T between the affine function spaces A(K) and A(S) induces an affine homeomorphism between K and S. Furthermore, T can be written as a weighted composition operator in this case.
1. Introduction
The celebrated BanachStone Theorem states that two compact Hausdorff spaces X and Y are homeomorphic if and only if the corresponding real continuous function spaces C(X) and C(Y) are linearly isometric (see, for example, [3, Chapter 7]). As is well known, see for example [2, Theorem 1.4.9], C(X) can be identified with the Banach space A(K) of real continuous affine functions on K = {'P E C(X)* : II'PII = 1 = 'P(l)}, the state space of C(X), where the norm in A(K) is the usual supremum norm. Here K is a Bauer simplex and consequently there is a natural reformulation of the BanachStone theorem in the context of affine geometry as follows. Two Bauer simplexes K and S are affinely homeomorphic if and only if their corresponding affine function spaces A(K) and A(S) are linearly isometric. Clearly an affine homeomorphism between any two compact convex sets K and S induces a linear isometry between A(K) and A(S). However, an example of J.T. Chan given in [9] shows that the converse of this cannot hold for arbitrary compact convex sets in locally convex (Hausdorff) spaces, even in finite dimensions. Thus it is natural to ask what conditions on K and S imply that this converse does hold. Lazar [11] showed that the converse holds if both K and S are Choquet simplexes. Ellis and So [9] extended this to the case when both K and S have the property that every pair of complementary closed faces is split, which applies to the state spaces of function algebras. In this paper we use another geometric condition on K and S, namely the unique decomposition property of Ellis [6, 7, 8], under which K and S are affinely homeomorphic whenever there is a linear isometry T between A(K) and A(S). 2000 Mathematics Subject Classification. Primary 46A55; Secondary 46B04. Key words and phrases. Affine function space, isometry, unique decomposition property. *Partially supported by Taiwan National Science Council Grants: 892115Mll0009,37128F. 151
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AUDREY CURNOCK, JOHN HOWROYD, AND NGAICHING WONG
Moreover, we show that in this case T can be written as a weighted composition operator. This includes Lazar's result as a special case and, more generally, the state spaces of C*algebras. Further results by the same authors relating to skewsymmetries are to be found in [4]. We would like to express our deepest gratitude to Professor ChoHo Chu for many useful discussions. 2. Preliminaries We note that every compact convex set K (in a locally convex space) can be embedded into A(K)* as the state space {cp E A(K)* : Ilcpll = 1 = cp(l)} of A(K) with the weak* topology, where x E K is identified with the linear functional f
1+
(j, x) = f(x)
for all f in A(K). If Y is a subset of K then we denote the convex hull of Y by co (Y). A point x in K is called an extreme point of K if K \ {x} is a convex set. The set of extreme points of K is denoted by 8K. In the above setting the closed unit ball BA(K)' of A(K)* is the convex hull of K and K; namely co (K U K) where K denotes {k: k E K}. Thus 8B A(K)' is contained in 8KU8(K). The reverse inclusion follows easily from the fact that for cp E BA(K)" \ (K U K) we have 1 < cp(l) < 1. Therefore, 8B A(K)" is equal to 8KU8(K). A convex subset F of K is called a face of K if whenever x E F with x = >..y + (1  >..)z for some y, z E K and >.. E (0,1), then both y and z are in F. A pair of faces (FI, F2) is said to be complementary whenever FI n F2 = 0 and K = co (FI U F 2 ). Thus, in this case, each x in K has a decomposition relative to (FI' F2); namely x = >..y + (1  >..)z for some y E F I , Z E F2 and>" E [0,1]. If >.. is unique, for each x E K \ (FI U F 2 ) but independent of y and z, then FI and F2 are said to be parallel faces; parallel faces are automatically norm closed. If in addition y and z are unique then FI and F2 are called split faces. See, for example, [1, 2] for the general theory of compact convex sets and related topics. Let K be a compact convex set. Recall that the facial topology on 8K is given by defining
{F n 8K : F is a closed split face of K} to be the family of all closed sets. The facial topology is weaker than the relative topology on 8K. The centre Z(A(K)) of A(K) is the set of all those functions in A(K) whose restriction to 8K is facially continuous. The central functions h E Z(A(K)) are characterised by the following property (see, for example, [1, Corollary 11.7.4] or [2, Theorem 3.1.4]): for all f E A(K), there exists 9 E A(K) such that g(x) = h(x)f(x) for all x in 8K. The uniqueness of the continuous affine function 9 is clear, since a continuous affine function on a compact convex set is completely determined by its values on the extreme boundary, and consequently we may write 9 = h . f. In this way it is useful to think of the central functions as the multipliers of A(K). A compact convex set K is a Choquet simplex whenever for all bounded (real) linear functionals cp in A(K)* and 0 > 0 the set K n (cp + oK) is either empty or of the form 'IjJ + (3K for some '¢ in A(K)* and {3 ~ 0 (see, for example, [12]); note that {3 = 0 allows K n (cp + oK) to be a singleton. In a Choquet simplex every closed face is split (see [2, Theorem 2.7.2]).
THE UNIQUE DECOMPOSITION PROPERTY AND BANACHSTONE THEOREM
153
In [6, 7, 8], Ellis defined the unique decomposition property of K by the condition that for every
0 the set Kn(
O. In particular, every Choquet simplex has the unique decomposition property. A result of Grothendieck [10], see also [5, p. 272], shows that the state space of a (unital) C*algebra has the unique decomposition property. Thus the results of this paper apply, giving an affine homeomorphism between the state spaces K and S of two C*algebras whenever the associated (real) affine function spaces A(K) and A(S) are linearly isometric. We give some examples below, the second and third of which show that the unique decomposition property and the condition of Ellis and So are independent geometric properties of a compact convex set. EXAMPLE 2.1. Let K be the state space of the C*algebra M 2 , of all 2 x 2 matrices over C. Then, by Grothendieck's result, K has the unique decomposition property. Also, K is affinely homeomorphic to a closed ball in 1R3 (see [2, p. 241]) and satisfies the condition of Ellis and So since it has no proper complementary faces. EXAMPLE 2.2. Let K be a triangular bisimplex in 3dimensional space as in the figure below, Figure 1. Then no proper face is (geometrically) parallel to any other and hence K n (
o. It follows, by the geometric characterisation of Ellis, that K has the unique decomposition property. However (F, F') is a pair of complementary faces of K, but not split, and hence K does not satisfy the condition of Ellis and So. EXAMPLE 2.3. Let K be an icosahedron in 3dimensional space. Then K satisfies the condition of Ellis and So because it has no proper complementary faces. However it does not have t.he unique decomposition property because it has
F
FIGURE 1. The bisimplex of Example 2.2
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AUDREY CURNOCK, JOHN HOWROYD, AND NGAICHING WONG
faces (geometrically) parallel to each other and hence K n (cp + aK) can have empty interior for some cp E A(K)* and a > O. 3. Results
Throughout this section K and 8 will denote compact convex sets of locally convex (Hausdorff) spaces. We will also let T: A(K) ~ A(8) denote a surjective linear isometry. Notice that if T1 = 1 then for cp E A(8)* with Ilcpli = 1 = cp(1), we have IIT*cpll = licp 0 Til = licpli = 1 = cp(1) = cp(T1) = T*cp(1); here T*: A(8)* ~ A(K)* denotes the dual map of T. Consequently T*(8) = K and hence T* induces an affine homeomorphism a: 8 ~ K such that Tf(s) = T*s(f) = f(a(s)) for all s E 8; that is, T is a composition operator f f+ f 0 a. In Proposition 3.3 we see that T is a weighted composition operator if and only if T1 is central. To do this we 'decompose' 8 by defining (3.1)
81
= {s E 8: (Tl)(s) = I}
and
82 = {s E 8 : (T1)(s)
= I}.
It is clear that 8 1 and 8 2 are closed faces of 8. LEMMA 3.1. Let 8 1 and 8 2 be as in (3.1). Then 88 are closed parallel faces of 8.
~
8 1 U 8 2, and 8 1 and 8 2
PROOF. Observe that the dual map T* is a linear isometry from A(8)* onto A(K)*. Hence T* maps the extreme points of the closed unit ball of A(8)* onto the extreme points of the closed unit ball of A(K)*. Thus
T*(88 U 8( 8)) = 8K U 8( K). Consequently, for each s E 88 we have T* s is in 8K or 8(  K) and hence
Tl(s) = T*s(l) = ±1. Therefore 88 ~ 8 1 U 8 2 and thus by the KreinMilman Theorem 8 = co (81 U 8 2), since 8, 8 1 and 82 are all (weak*) compact. Thus 8 1 and 8 2 are complementary faces since they are clearly disjoint. For each s in 8 with s = >.x + (1  >')y, where x E 8 1 , Y E 8 2, and>' E (0,1), we have T* s = >'T*x + (1  >')T*y. Thus,
= T*s(1) = >'T*x(1) + (1 >')T*y(1) = >.  (1 >.) = 2>'  1. This establishes the uniqueness of>. = ((Tl)(s) + 1)/2 and the result follows. Tl(s)
We now specialise to the case when K and 8 have the unique decomposition property. LEMMA 3.2. Let 8 1 and 8 2 be as in (3.1). 8uppose that K has the unique decomposition property. Then 8 1 and 8 2 are complementary split faces of 8. PROOF. By Lemma 3.1, for each s E 8\(81 U 8 2) we may write s = >.x + (1  >.)y where x E 8 1 , Y E 8 2 and 0 < >. < 1, and>' is unique. We consider the decomposition T* s = >'T*x  (>. 1)T*y. Since x E 8 1 we have T*x E K and hence >'T*x is positive. Similarly, T*y E K and hence (>.  I)T*y is positive. Also
IIT*sll = 1 = >. + (1 >.) = Ii>'T*xli + 11(>' 
I)T*yli· Thus, by the unique decomposition property, >'T*x and (>. 1)T*y are unique. By the uniqueness of >., we have T*x and T*y are unique. Since T is surjective, T* is injective and the result follows. 0
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155
By replacing T by T 1 in (3.1) we may 'decompose' K by defining (3.2) K1 = {k E K : T 11(k) = I} and K2 = {k E K : T 11(k) = I}. Applying Lemmas 3.1 and 3.2 to T 1 we see that K1 and K2 are complementary split faces of K whenever S has the unique decomposition property. We say that T is a weighted composition operator whenever there exists a central function h in A(S) and a continuous affine mapping (1: S + K such that Tf = h· f 0 (1 for all f E A(K); that is, Tf(s) = h(s)f((1(s)) for all sEaS. The following proposition asserts that the linear isometry T is a weighted composition operator, with Tl(s) = ±1 on as, if and only if Tl is central. PROPOSITION 3.3. Let T: A(K) + A(S) be a linear mapping. Then the following are equivalent: a) T is an isometry and Tl is central; b) T is a weighted composition operator of the form T f = h . f 0 (1 for all f E A(K) where (1 is an affine homeomorphism and h(s) = ±1 for all sEas. PROOF. See [4, Theorem 3.3] or [13].
o
We now apply the above decompositions of K and S to prove our main theorem. THEOREM 3.4. Suppose that K and S have the unique decomposition property. Then the real affine function spaces A(K) and A(S) are linearly isometric if and only if K and S are affinely homeomorphic. Moreover, every linear isometry from A(K) onto A(S) may be written as a weighted composition operator. PROOF. It suffices to show necessity. Suppose that T is a linear isometry from A(K) onto A(S). We 'decompose'S into the complementary split faces Sl and S2 of (3.1) and, similarly, K into the complementary split faces K1 and K2 of (3.2). Since (T1)* = (T*)l, we have T*(Sd = K1 and T*(S2) = K2' and hence we may define (1: S + K by (1(,xx + (1  ,x)y) = ,xT*(x)  (1  ,x)T*(y)
whenever x E Sl, Y E S2 and 0 :5 ,x :5 1. We see that (1 is an affine homeomorphism from S = co (Sl U S2) onto K = co (K1 U K 2). Moreover, to show that T is a weighted composition operator it suffices, by Proposition 3.3, to show that h = Tl is central. Let f E A(S) then, since (1: S + K is an affine homeomorphism, we may write f = gO(1 for some g E A(K). Note that for x E Sl we have Tg(x) = T*x(g) = g((1(x)) = h(x)f(x). Similarly for x E S2 we have Tg(x) = T*x(g) = g((1(x)) = h(x)f(x). Therefore, for all x E as £;; Sl U S2 we have Tg(x) = h(x)f(x), and the result follows. 0 References [1] E.M. Alfsen, Compact convex sets and boundary Integmls, Ergebnisse der Mathematik, 57, (SpringerVerlag, BerlinHeidelbergNew York, 1971). [21 L. Asimow and A.J. Ellis, Convexity theory and its applications in functional analysis, London Math. Soc. Monograph, 16 (Academic Press, London, 1980). [3] E. Behrends, M Structure and the BanachStone Theorem, Lecture Notes in Mathematics 736, (SpringerVerlag, BerlinHeidelbergNew York, 1979). [4] A. Curnock, J. Howroyd, and N.C. Wong, Isometries of affine function spaces, preprint. [5] J. Dixmier, C*Algebms (NorthHolland Publishing Co., AmsterdemNew YorkOxford, 1982).
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[6] A.J. Ellis, An intersection property for state spaces, J. London Math. Soc., 43 (1968), 173176. [7] A.J. Ellis; Minimal decompositions in partially ordered normed spaces, Proc. Camb. Phil. Soc., 64 (1968),9891000. [8] A.J. Ellis, On partial orderings of normed spaces, Math. Scand., 23 (1968), 123132. [9] A.J. Ellis and W.S. So, Isometries and the complex state spaces of uniform algebras, Math. Z., 195 (1987), 119125. [10] A. Grothendieck, Un result at sur Ie dual d'une C*algebre, J. Math. Pures Appl., 36 (1957), 97108. [11] A.J. Lazar, Affine products of simplexes, Math. Scand., 22 (1968), 165175. [12] R.R. Phelps, Lectures on Choquet's Theorem, Second Edition, Lecture notes in Mathematics 1757 (SpringerVerlag, Berlin, 2001). [13] T.S.R.K. Rao, Isometries of Ac(K), Proc. Amer. Math. Soc., 85 (1982), 544546. SCHOOL OF COMPUTING, INFORMATION SYSTEMS AND MATHEMATICS, SOUTH BANK UNIVERSITY, LONDON SE1 OAA, ENGLAND. Email address:curnocaOsbu.ac.uk DEPARTMENT OF MATHEMATICAL SCIENCES, GOLDSMITHS COLLEGE, UNIVERSITY OF LONDON, LONDON SE14 6NW, ENGLAND. Email address:masOljdhOgold.ac.uk DEPARTMENT OF ApPLIED MATHEMATICS, NATIONAL SUN YATSEN UNIVERSITY, KAOHSJUNG 80424, TAIWAN, R.O.C.
address:wong~ath.nsysu.edu.tw
Contemporary Mathematics Volume 328, 2003
A Survey of Algebraic Extensions of Commutative, Unital Normed Algebras Thomas Dawson ABSTRACT. We describe the role of algebraic extensions in the theory of commutative, unital normed algebras, with special attention to uniform algebras. We shall also compare these constructions and show how they are related to each other.
Introduction Algebraic extensions have had striking applications in the theory of uniform algebras ever since Cole used them (in [5]) to construct a counterexample to the peakpoint conjecture. Apart from this, their main use has been in (a) the construction of examples of general, normed algebras with special properties and (b) the Galois theory of Banach algebras. We shall not discuss (b) here; a summary of some of this work is included in [29]. In the first section of this article we shall introduce the types of extensions and relate their applications. The section ends by giving the exact relationship between the types of extensions. Section 2 contains a table summarising what is known about the extensions' properties. A theme lying behind all the work to be discussed is the following question:
(Q) Suppose the normed algebra B is related to a subalgebra A by some specific property or construction. (For example, B might be integral over A: every element b E B satisfies ao + ... +an_1bn1 +bn = 0 for some ao, ... ,an 1 EA.) What properties of A (for example, completeness or semisimplicity) must be shared by B? This is a natural question, and interesting in its own right. Many special cases of it have been studied in the literature. We shall review the related body of work in which B is constructed from A by adjoining roots of monic polynomial equations. Throughout this article, A denotes a commutative, unital normed algebra, and A its completion. The fundamental construction of [1] applies to this class of 1991 Mathematics Subject Classification. Primary 46J05, 46J10. This research was supported by the EPSRC. © 2003 American Mathematical Society 157
THOMAS DAWSON
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algebras. Algebraic extensions of more general types of topological algebras have received limited attention in the literature (see [19], [21]). If E is a subset of a ring then (E) will stand for the the ideal generated by E. 1. Types of Algebraic Extensions and their Applications 1.1. ArensHoffman Extensions. Let a(x) = ao + ... + an_lx n l + xn be a monic polynomial over the algebra A. The basic construction arising from A and a(x) is the ArensHoffman extension, Ao. This was introduced in [1]. Most of the obvious questions of the type (Q) for ArensHoffman extensions were dealt with in this paper and in the subsequent work of Lindberg ([18]' [20], [13]). See columns two and three of Table 2.2. All the constructions we shall meet are built out of ArensHoffman extensions. DEFINITION 1.1.1. A mapping 0: A  B between algebras A and B is called unital if it sends the identity of A to the identity of B. An extension of A is a commutative, unital normed algebra, B, together with a unital, isometric monomorphism
O:AB.
The ArensHoffman extension of A with respect to a(x) is the algebra Ao := A[x]/(a(x)) under a certain norm; the embedding is given by the map v: a t+ (a(x)) + a.
To simplify notation, we shall let x denote the coset of x and often omit the indeterminate when using a polynomial as an index. It is a purely algebraic fact that each element of Ao has a unique representative of degree less than n, the degree of a(x). Arens and Hoffman proved that, provided the positive number t satisfies the inequality t n ~ ~~:~ lIakll tk, then
I~
bkX
k
=
~ IIbkll t
k
(bo, ... , bn 
l
E
A)
defines an algebra norm on Ao. The first proposition shows that ArensHoffman extensions satisfy a certain universal property which is very useful when investigating algebraic extensions. It is not specially stated anywhere in the literature; it seems to be taken as obvious. 1.1.2. Let A(l) be a normed algebra and let 0: A(l) _ B(2) be a unital homomorphism of normed algebras. Let al (x) = ao + ... + an_lXn  l + xn E A(l) [x] and B(1) = A~l}. Let y E B(2) be a root of the polynomial a2(x) := O(ad(x) := O(ao) + ... +O(an_l)X n  l +xn. Then there is a unique homomorphism 1>: B(l) _ B(2) such that PROPOSITION
B(1)
~
II
/9
r
B(2)
is commutative and 1>(x) = y.
A(l)
The map 1> is continuous if and only if 0 is continuous. PROOF.
This is elementary; see [7]
o
ALGEBRAIC EXTENSIONS OF COMMUTATIVE NORMED ALGEBRAS
159
1.2. Incomplete Normed Algebras. A minor source of applications of ArensHoffman extensions fits in nicely with our thematic question (Q): these extensions are useful in constructing examples to show that taking the completion of A need not preserve certain properties of A. The method uses the fact that the actions of forming completions and ArensHoffman extensions commute in a natural sense. A special case of this is stated in [17]; the general case is proved in [7), Theorem 3.13, and follows easily from Proposition 1.1.2. It is convenient to introduce some more notation and terminology here. Let O(A) denote the space of continuous epimorphisms A + Cj when n appears on its own it will refer to A. As discussed in [1], this space, with the weak *topology relative to the topological dual of A, generalises the notion of the maximal ideal space of a Banach algebra. In fact, it is easy to check that 0 is homeomorphic to 0(..4), the maximal ideal space of the completion of A. The Gelfand transform of an element a E A is defined by
a: n + C; W
1+
w(a)
and the map sending a to a is a homomorphism, r, of A into the algebra, C(O), of all continuous, complexvalued functions on the compact, Hausdorff space O. We denote the image of r by A. A good reference for Gelfand theory is Chapter three of [24]. DEFINITION
1. 2.1 ([1)). The algebra A is called topologically semisimple if
r
is injective. If A is a Banach algebra then this condition is equivalent to the usual notion of semisimplicity. The precise conditions under which Aa is topologically semisimple if A is are determined in [1]. In [17] Lindberg shows that the completion of a topologically semisimple algebra need not be semisimple. In order to illustrate Lindberg's strategy we recall two standard properties of normed algebras.
1.2.2. The normed algebra A is called regular if for each closed subset E ~ 0 and wE OE there exists a E A such that a(E) ~ {O} and a(w) = 1. The algebra is called local if A contains every complex function, f, on 0 such that every wE 0 has a neighbourhood, V, and an element a E A such that flv = alv. DEFINITION
It is a standard fact that regularity is stronger than localness; see Lemma 7.2.8 of [24). EXAMPLE 1.2.3. Let A be the algebra of all continuous, piecewise polynomial functions on the unit interval, I, and a(x) = x 2  id/ E A[x]. Let A have the supremum norm. By the StoneWeierstrass theorem, A = C(I) and hence n is identifiable with I. Clearly A is regular. We leave it as an exercise for the reader to find examples to show that Aa is not local. This is not hardj it may be helpful to know that in this example the space O(Aa) is homeomorphic to {(s, oX) E I xC: oX 2 = s}. This follows from facts in [1]. In the present example, neither localness nor regularity is preserved by (incomplete) ArensHoffman extensions.
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Finally we can explain the method for showing that some properties of normed algebras are not shared by their completions because, in the above, 'nonregularity' is not preserved by completion of Ao (nor is 'nonlocalness'). To see this, note that Ais clearly regular if A is and so by a theorem of Lindberg (see Table 2.2) the ArensHoffman extension (A)o is regular. But, by a result of [17] referred to above, this algebra is isometrically isomorphic to the completion of Aa. Of course Lindberg's original application was much more significant; there are simpler examples of the present result: for example the algebra of polynomials on I.
1.3. Uniform Algebras. It is curious that the application of ArensHoffman extensions to the construction of integrally closed extensions of normed algebras did not appear in the literature for some time after [1]. It was seventeen years later until a construction was given in [22]. Even then the author acknowledges that the constuction was prompted by the work of Cole, [5], in the theory of uniform algebras. Cole invented a method of adjoining square roots of elements to uniform algebras. He used it to extend uniform algebras to ones which contain square roots for all of their elements. Apart from feeding back into the general theory of commutative Banach algebras (mainly accomplished in [22] and [23]) his construction provided important examples in the theory of uniform algebras. We shall describe these after recalling some basic definitions. DEFINITION 1.3.1. A uniform algebra, A, is a subalgebra of C(X) for some compact, Hausdorff space X such that A is closed with respect to the supremum norm, separates the points of X, and contains the constant functions. We speak of 'the uniform algebra (A, X)'. The uniform algebra is natural if all of its homomorphisms wEn are given by evaluation at points of X, and it is called trivial if A = C(X).
Introductions to uniform algebras can be found in [4], [11], [26], and [16]. An important question in this area is which properties of (A, X) force A to be trivial. For example it is sufficient that A be selfadjoint, by the StoneWeierstrass theorem. In [5] an example is given of a nontrivial uniform algebra, (B,X), which is natural and such that every point of X is a 'peakpoint'. It had previously been conjectured that no such algebra existed. We shall describe the use of Cole's construction in the next section, but now we reveal some of the detail. PROPOSITION 1.3.2 ([5],[7]). Let U be a set of monic polynomials over the uniform algebra (A, X). There exists a uniform algebra (AU, XU) and a continuous, open surjection 7r: XU > X such that (i) the adjoint map 7r*: C(X) > C(XU) induces an isometric, unital monomorphism A > AU, and (ii) for every a E U the polynomial 7r*(a)(x) E AU[x] has a root Po E AU. PROOF. We let XU be the subset of X x such that for all a E U f(a)(K)
JO
cU consisting of the elements (K, >.)
+ ... + /(0) (K)>.n(0)1 + >.n(o) n(a)1 a 0
= 0
ALGEBRAIC EXTENSIONS OF COMMUTATIVE NORM ED ALGEBRAS
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where o:(x) = f~Ol) + ... + f~(l)_lXn(Ol)l + xn(Ol) E U. The reader can easily check that XU is a compact, Hausdorff space in the relative product topology and so the following functions are continuous: 11":
XU + X; (~, >.) ...... ~
POI: XU + C; (~, >.) ...... >'01
(0:
E U).
The extension AU is defined to be the closed subalgebra of C(XU) generated by 1I"*(A) U {POI: 0: E U} where 11"* is the adjoint map C(X) + C(XU) ; g ...... go 11". It is not hard to check that AU is a uniform algebra on XU with the required properties. 0 We shall call AU the Cole extension of A by U. Cole gave the construction for the case in which every element of U is of the form x 2  f for some f E A. It is remarked in [22] that similar methods can be used for the general case; these were independently, explicitly given in [7]. By repeating this construction, using transfinite induction, one can generate uniform algebras which are integrally closed extensions of A. Full details of this, including references and the required facts on ordinal numbers and direct limits of normed algebras, can be found in [7]. Again this closely follows [5]. Informally the construction is as follows. Let v be a nonzero ordinal number. Set (Ao,Xo) = (A,X). For ordinal numbers T with 0 < T ~ V we define (A~" , X!:" )
(A.,, X T )
if T = a
+1
and
= { l~ . (A u,Xu)u<.,, (* 1I"p,u, 1I"p,u ) p~U
if T is a limit ordinal.
The construction requires sets of monic polynomials, Uu ~ Au [x], to be chosen inductively. The notation (Au, Xu )u<.,, (1I";,u, 1I"p,u )P~U
(A, X).

Thus (A1' Xl) is just a Cole extension of (Ao, Xo). When U1 is a singleton we call Al a simple extension of Ao; the same adjective can be applied to ArensHoffman extensions. An integrally closed extension, (Av, Xv), is obtained by taking v to be the first uncountable ordinal. At the successor ordinals the whole set of monic polynomials is frequently used to extend the algebra, but this set is larger than necessary. The same procedure is used to obtain the integrally closed extensions in other categories (to be discussed in Section 1.6).
THOMAS DAWSON
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1.4. Some Applications of Cole's Construction. Cole's method has been developed by others, including Karahanjan and Feinstein, to produce examples of nontrivial uniform algebras with interesting combinations of properties. We cite the following example of Karahanjan. THEOREM 1.4.1 (from [15], Theorem 4). There is a nontrivial, antisymmetric uniform algebra, A, such that (1) A is integrally closed, (2) A is regular, (3) n is hereditarily unicoherent, (4) G(A) is dense in A, and (5) the set of peakpoints of A is equal to n. In the above, G(A) is our notation for the invertible group of A. We refer the reader to [15] and the literature on uniform algebras for the definitions of other terms we have not defined here. A further example in [15] also strengthens Cole's original counterexample. Both examples (of nontrivial, natural uniform algebras on compact, metriseable spaces, every point of which is a 'peakpoint') are regular. Feinstein has varied the construction to obtain such an example which is not regular in [10]. The same author also used Cole extensions in [9] to answer a question of Wilken by constructing a nontrivial, 'strongly regular', uniform algebra on a compact, metriseable space. Returning to the sample theorem quoted above, note that some of these properties (for example the topological property of 'hereditary unicoherence') are consequences of the combination of other properties of the final algebra. By contrast, (2) and (4) hold because they are true for the base algebra on which the example is constructed. It is therefore very useful to know exactly when specific properties of a uniform algebra are transferred to those in a system of Cole extensions of it. The known results on this problem are summarised in the first column of Table 2.2. Determining if an algebra's property is shared by its algebraic extensions has led to some interesting devices. We shall elaborate on this topic in the next section. We remark in passing that the methods used in [15] to show that the final algebra has a dense invertible group have been simplified in [8]; in particular there is no need to develop the theory of 'dense thin systems' in [15].
1.5. A Further Remark on Cole Extensions. The reader will notice from Table 2.2 that virtually all properties of uniform algebras are preserved by Cole extensions. The key to obtaining most of these results is the following result, originating with Cole. PROPOSITION 1.5.1 ([5]'[23]). Let (Ar,Xrk:;v be a system of Cole extensions of (A, X). There exists a family of unital contractions (T.,.,r: C(Xr) + C(X.,.)).,.::;r::;v such that for all a ~ T ~ v (i) T.,.,r(A r ) ~ A.,., and (ii) T.,.,r 07r;,r = idc(X u )'
o
PROOF. See [23]. For example, it is easy to see from the existence of T: C(Xu) Cole extension AU is nontrivial if A i= C(X).
+
C(X) that the
ALGEBRAIC EXTENSIONS OF COMMUTATIVE NORMED ALGEBRAS
163
The operator T was constructed in [5] for extensions by squareroots. In the case of a simple Cole extension, (A{o}, X{o}), there are at most two points y±(x:) in the fibre 7r l (x:) for each x: E X and they correspond to the roots of the equation x 2  f(x:) = 0 where o(x) = x 2  f. The operator is then defined by
(g E C(X{o}), x: EX). For other sorts of monic polynomials it was not so obvious how to construct T. The basic techniques appeared in [22] (see the proof of Theorem 3.5) for simple extensions, and were further developed in the proof of Theorem 4 of [15], but it was not until [23] that a comprehensive construction was given. We must also mention the role of E. A. Gorin: he appears to have paved the way for [15] and [23].
1.6. Algebraic Extensions of Normed and Banach Algebras. As we have seen, algebraic extensions have had striking applications in the theory of uniform algebras. They have long been used as auxiliary constructions in the general theory of Banach algebras. Notable examples of this are in [14] and [25]; the latter explicitly uses ArensHoffman extensions. However algebraic extensions for Hormed algebras were apparently only studied in their own right in order to generalise the work of Cole and Karahanjan. We now turn to these generalisations. The basic extension generalising ArensHoffman extensions is called a standard normed extension. It is defined in the following theorem of Lindberg. THEOREM 1.6.1 ([22]). Let A be a normed algebra and U a set of monic polynomials over A. Let ~ be a wellordering on U with least element 00' Then there exists a normed algebra, B u , with a family of subalgebras, (Bo)oEU, such that: (i) for all 0, (3 E U, Bo ~ Bf3 if 0 ~ (3, and, (ii) for all (3 E U, Bf3 is isometrically isomorphic to an ArensHoffmBll extension of B
PROOF. See [22].
o
Lindberg shows how this leads to the construction of Banach algebras with interesting combinations of properties, one of which is integral closedness. Let the isometric isomorphism B
THOMAS DAWSON
164
Let ~Ot be the standard root of a E U, witll associated norm parameter tOt, and suppose (T/Ot)OtEU ~ B(2) is such that £I(a)(T/Ot) = 0 for all a E U. Tllen there is a unique, unital homomorphism ¢: B(l) + B(2) such that the following diagram is commutative B(1)
r~
~
B(2)
/9
A(1)
(Note added in proof: The map ¢ is continuous if and only if £I is continuous and
L OtEU
(n(a) l)log+
(11~OtII) < +00 Ot
where log+ denotes the positive part of the logarithm, max(log,O).) PROOF. A simple application of transfinite methods and Proposition 1.1.2. 0 Purely algebraic standard extensions are defined in [22] and the main content of Lemma 1.6.2 is a statement about these. Narmania gives ([23]) an alternative construction for integrally closed extensions of a commutative, unital Banach algebra, A. His method is rather more conventional than the one used to define standard extensions. If U is a set of monic polynomials over A then the Narrnania extension of A by U is equal to the Banachalgebra direct limit of (As: 8 is a finite subset of U) where each As is isometrically isomorphic to A extended finitely many times by the ArensHoffman construction. As this paper is not readily available in English and we shall refer to the explicit construction of Narmania's extensions in the next result, we stop to report the precise details of this. If E is a set, the set of all finite subsets of E will be written E<wo. Let 8 = {ai, ... , am} ~ U and let tOt (a E U) be a valid choice of ArensHoffman normparameters (see Section 1.1). It is important to insist that distinct elements a, /3 E U are associated with distinct indeterminates XOt , x{3. Thus 8 is an abbreviation for {a1(xoJ, ... ,am(xo,J}. It is proved carefully in [23] that for q = Ls qsx~ll ... x~;;, E A[x U1 ' · •• ,xo: m ], the algebra of polynomials in m commuting indeterminates over A (s is a mult.iindex in No where No = {O}UN), then (8) +q has a unique representative whose degree in 3:O: j is less than than n(aj), the degree of aj(:ro: j ) (j = 1, ... ,m). For convenience we shall call such representatives minimal. Then if q is the minimal representative of (8) +q, 11(8) + qll := Ls IIqsll t~ll ••• t~: defines an algebra norm on As. The index set, U<wo is a directed set, directed by ~. The connecting homomorphisms VS.T (for 8 ~ T E U<wo) are the natural maps; they are isometries. Thus (sec [24] Section 1.3) Au is the completion of the normed direct limit, D := UsEU<wo As / "', where '" is an equivalence relation given by (8) + q '" (T) + r if and only if q  r E (8 UT) for 8, T E U<wo. Furthermore, the canonical map, Vs, which sends an element of As to its equivalence class in D, is an isometry. Note that A0 is defined to be A. We can now show how the types of extensions we have been considering are related. Many of the idea.'3 behind Proposition 1.6.3 are due to Narmania but we take the step of linking them to Cole and standard extensions.
ALGEBRAIC EXTENSIONS OF COMMUTATIVE NORMED ALGEBRAS
165
1.6.3. Let A be a commutative, unital Banach algebra and U a set of monic polynomials over A. Then, up to isometric isomorphism, Au = Bu. If A is a uniform algebra then we have PROPOSITION
A
u


= (Au)" = (Bur,
where the closures are taken with respect to the supremum norm. PROOF.
It is easily checked that if B is a normed algebra then the homeomor
phism 12(B) t O(B) induces an isometric isomorphism B" t (B)". It is therefore sufficient to prove that Au = Bu and that AU = (Bu)". The last equality follows very quickly from the universal property of standard extensions mentioned above and the simplicity of the definition of AU. We shall only prove the first identification; the second can be proved by a similar approach. Although what follows is routine, we hope that it will help to clarify the details of standard and Narmania extensions. As before let to (0 E U) be a valid choice of ArensHoffman normparameters for the respective extensions Ao. We shall show that there is then an isometric isomorphism between Bu and D (when defined by these parameters); the result then follows from the uniqueness of completions. For each a E U let Yo be the equivalence cla..<;s [({o(:c o )}) + xo] E D. Since Yo is a root of V0(0)(X) in D there exists, by the universal property of standard extensions, a (unique) homomorphism ¢: Bu t D such that ¢I A = V0 and for all a E U, ¢(~o) = Yo' Here, ~o is the the element of Bu associated with x by the isometric isomorphism 1/)0 : B
L,j!!j111¢(bj)11 tb.
tb
Since the algebras vs(As) are directed there exists 8 E U<W(I such that ¢(bj ) E vs(As) (j = 0, ... ,n(,6) 1). We can assume that 8 = {al,'" ,am} and ol(X) = ,6(x). Let qo,·.· ,qn((3)IE A[XQ1"" ,XOm] be the minimal representatives such that ¢(bj ) = [(8) + qj] (j = 0, ... ,n(,6) 1). So Ilbjll = Ilqjll (j = 0, ... ,n(,6)  1). A routine exercise in the transfinite induction theorem shows that for all 'Y E U, ¢(B"() ~ UTE[O,"(]<w" vT(AT). It follows that the
THOMAS DAWSON
166
degree of qj in
XO
is zero. Hence n({3)l
11¢(b)11 =
L
reS) + qjX~l]
=
j=O n(tJ)l
(S) +
L
qj.T~l
j=O n({3)l
=
L
IIqj
II t~l
=
Ilbll ,
j=O
from above. The penultimate equality above follows from noting that the representative of the coset is minimal and then expanding and collecting terms. By the transfinite induction theorem, .:J = U as required. 0
2. A Survey of Properties Preserved by Algebraic Extensions 2.1. Introduction. We summarise in Table 2.2 what is currently known about the behaviour of certain properties of normed algebras with respect to the types of extensions we have been considering. Some preliminary explanation of the entries is in order first. Extra information about the polynomial(s) generating an algebraic extension can help to determine whether certain properties are preserved or not. For example if a(x) has degree nand factorises completely over A with distinct roots AI, ... ,An E A such that for all W E fl, .x:(w) =I :X;(w) if i =I j then n(Ao<) decomposes into n disjoint homeomorphs of fl in which case very many properties of A, for example localness, are shared by Ao<' This property, referred to as 'complete solvability', is investigated in [12]. The condition on a(x) most frequently encountered in the literature is that it should be 'separable'. This means that its 'discriminant', which is a certain polynomial in the coefficients of a(x), is invertible in A. It is interesting to compare columns two and three. Of course one can make additional assumptions on the algebra (for example that A be regular and semisimple) but the resulting table would become too large and we have restricted it to three popular categories. References to the results follow the table. We should mention that some of the entries have trivial explanations. For example Sheinberg's theorem, that a uniform algebra is amenable if and only if it is trivial, explains the entries for amenability in column one. Also, applying the ArensHoffman construction to a uniform algebra need not result in a uniform algebra so not all the entries make sense. We have already met most of the properties listed in the table. We end this section by discussing the ones which have not yet been specially mentioned. 1. Denseness of the invertible group. Although this property is selfexplanatory it might not be obvious why it is listed. However, the condition G(A) = A appears in the literature in various contexts; see for example [8].
ALGEBRAIC EXTENSIONS OF COMMUTATIVE NORMED ALGEBRAS
167
2. The Banach algebra, A, is called supnorm closed if A is uniformly closed in C(f2) (and therefore a uniform algebra). It is called symmetric if A is selfadjoint. 3. For the definitions of 'amenability' and 'weak amenability' we refer the reader to section 2.8 of [6]. All the properties in the table are preserved by forming the standard unitisation of a normed algebra. Most of these results are standard facts or ea'iy exercises; some are true by definition. However this question does not fit into our scheme because the embedding is not unital in this case.
2.2. Table. Cole extensions have only been defined for uniform algebra'l; the algebra is therefore assumed to be a uniform algebra throughout column one. Colulllns two and three, a'l mentioned above, refer to ArensHoffman extensions of a normed algebra, A, by a monic polynomial a(x); in column three it is given that a(:1:) is separable. Type of Extension: Property:
Cole
A.H.
Ac.
A.H. standard Narmania a sep.
for normed algebras
complete topologically semisimple nonlocal local regular
1.
2. 3. 4. 5.
• • • ?
• 0
•
• • •
•
? '?
?
'?
'!
• • • • •
• •
• • • •
•
•
•
•
'?
?
for Banach algebras
6. 7. S. 9. 10.
local regular dense invertible group supnorm closed symmetric 11. amenable 12. weakly amenable
?
0 0 0 0
for uniform algebms
13. nontrivial 14. trivial 15. natural
• • •
Key • property is always preserved o property is sometimes, but not always preserved
? ?
? ?
?
•
• •
168
THOMAS DAWSON
? not yet determined  it doesn't always make sense to consider this property here References for the Entries. If we do not mention an entry here, it can be taken that the result is an immediate consequence of the definition or was proved in the same paper in which the relevant extension was introduced (that is in [5], [1], [22], or [23]). The results of row three are not hard to obtain, using appropriate versions of Proposition 1.5.1. Localness and regularity were discussed in Section 1.2. The main result about this is due to Lindberg in [18]; the same section of his paper also deals with the results on the symmetry of ArensHoffman extensions. That regularity passes to direct limits of such extensions has been widely noted by many authors, for example in [15]. Results of row eight follow from [8]; the case of Cole extensions was partially covered in [15], but the reasoning is not clear. The property of being supnorm closed was investigated in [13]; this work was generalised in [28]. Finally, examples of amenable Banach algebras which do not have even weakly amenable ArensHoffman extensions have been known for a long time. For example, the algebra C E9 C under the multiplication (a, b) (e, d) = (ae, be + ad) is realisable as an ArensHoffman extension of C. Examples with both A and Ao semisimple have been found by the author. However the entries marked "?' in rows eleven and twelve represent intriguing open problems. 3. Conclusion The table in Section 2.2 still has gaps, and there are many more rows which could be added. For example it would be interesting to be able to estimate various types of 'stable ranks' (see [2]) of the extensions in terms of the stable ranks of the original algebras. (The condition G(A) = A is equivalent to the 'topological stable rank' of A not exceeding 1.) Remember too that there are many more questions which can be asked, of the form: 'if n has the topological property P, does n(Ao) have property P?' By way of a conclusion we repeat that algebraic extensions have proved immensely useful in the construction of examples of uniform algebras. There is therefore great scope for and potential usefulness in augmenting Table 2.2. It might also be valuable to reexamine the techniques used to obtain the entries to produce more general results (of the kind in [28] for example) in the context of question (Q).
References 1. Arens, R. and Hoffman, K., Algebraic Extension of Normed Algebras., Proc. Am. Math. Soc. 7 (1956), 203210. 2. Badea, C., The Stable Rank of Topological Algebras and a Problem of R. G. Swan., J. Funet. Anal. 160 (1998), 4278. 3. Batikyan, B. T., Point Derivations on Algebraic Extension of Banach Algebra., Lobachevskii J. Math. 6 (2000), 337. 4. Browder, A., Introduction to Function Algebras., W. A. Benjamin, Inc., New York, 1969.
ALGEBRAIC EXTENSIONS OF COMMUTATIVE NORMED ALGEBRAS
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5. Cole, B. J., OnePoint Parts and the PeakPoint Conjecture., Ph.D. Thesis, Yale University, 1968. 6. Dales, H. G., Banach Algebms and Automatic Continuity., Oxford University Press Inc., New York,2000. 7. Dawson, T. W .. Algebmic Extensions of Normed Algebms., M.Math. Dissertation, accessible from the web at: http://xxx.lanl.gov/abs/math.FA/0102131, University of Nottingham, 2000. 8. Dawson, T. W., and Feinstein, J. F., On the Denseness of the Invertible Group in Banach Algebms., Proc. Am. Math. Soc. (to appear). 9. Feinstein, J. F., A NonTrivial, Strongly Regular Unif01m Algebm., J. Lond. Math. Soc. 45 (1992), 288300. 10. Feinstein, J. F., Trivial Jensen Measures Without Regularity., Studia Math. 148 (2001).6774. 11. Gamelin, T. W., Uniform Algebms., PrenticeHall Inc., Engelwood Cliffs, N. J., 1969. 12. Gorin, E. A., and Lin, V ..J., Algebmic Equations with Cont'inuous Coefficients and Some Problems of the Algebmic Theory of Bmids., Math. USSR Sb. 7 (1969), 569596. 13. Heuer, G. A., and Lindberg, J. A., Algebmic Extensions of Continuous Function Algebms., Proc. Am. Math. Soc. 14 (1963),337342. 14. Johnson, B. E., Norming C(O) and Related Algebms., Trans. Am. Math. Soc. 220 (1976), 3758. 15. Karahanjan, M. I., Some Algebmic Chamcterizations of the Algebm of All Continuous Functions on a Locally Connected Compactum., Math. USSR Sb. 35 (1979),681696. 16. Leibowitz, G. M., Lectures on Complex Function Algebm,~., Scott, Foresman and Company, Glenview, Illinois, 1970. 17. Lindberg, J. A .• On the Completion of Tractable Normed Algebms., Proc. Am. Math. Soc. 14 (1963),319321. 18. Lindberg, J. A., Algebmic Extensions of Commutative Banach Algebms., Pacif. J. Math. 14 (1964), 559583. 19. Lindberg, J. A., On Singly Genemted Topological Algebras, Function Algebras (ed. Birtel, F. T.), ScottForesman, Chicago, 1966, pp. 334340. 20. Lindberg, J. A., A Class of Commutative Banach Algebms with Unique Complete Norm Topology and Continuous Derivations., Proc. Am. Math. Soc. 29 (1971), 516520. 21. Lindberg, J. A .. Polynomials over Complete l.m.c. Algebms and Simple Integml Extensions., Rev. Roumaine Math. Pures Appl. 17 (1972), 4763. 22. Lindberg, J. A., lntegml Extensions of Commutative Banach Algebms., Can. ,/. Math. 25 (1973), 673686. 23. Narmaniya, V. G., The Construction of Algebmically Closed Exten,~ions of Commutative Banach Algebms., Trudy Tbiliss. Mat. Inst. Razmadze Akad. 69 (1982), 154162. 24. Palmer, T. W., Banach Algebras and the Geneml Theory of *Algebms. Vol. 1, Cambridge University Press, Cambridge, 1994. 25. Read, C. J., Commutative, Radical Amenable Banach Algebms., Studia Math. 140 (2000), 199212. 26. Stout, E. L., The Theory of Uniform Algebms., Bogden and Quigley Inc., TarrytownonHudson, New York, 1973. 27. Taylor, J. L., Banach Algebms and Topology, Algebras in Analysis. (ed. Williamson, J. H.), Academic Press Inc. (London) Ltd., Norwich:, 1975, pp. 118186. 28. Verdera, J., On Finitely Genemted and Projective Extensions of Banach Algebms., Proc. Am. Math. Soc. 80 (1980), 614620. 29. Zame, W. R., Covering Spaces and the Galois Theory of Commutative Banach Algebms., J. Funet. Anal. 27 (1984), 151171.
Acknowledgements The author would like to thank the Division of Pure Mathematics and the Graduate School at the University of Nottingham for paying for his expenses in
170
THOMAS DAWSON
order to attend the 4th Conference on FUnction Spaces (2002) at the Southern Illinois University at Edwardsville. The author is grateful to Mr. Brian Lockett who provided him with a translation of the paper [23]. Special thanks are due to Dr. J. F. Feinstein who offered much valuable advice and encouragement and also proofread the article. DIVISION OF PURE MATHEMATICS, SCHOOL OF MATHEMATICAL SCIENCES, UNIVERSITY OF NOTTINGHAM, UNIVERSITY PARK, NOTTINGHAM, NC7 2RD, UK.
Email address:pmxtwdlDnottingham.ac.uk
Contemporary Mat.hematics Volume 328, 200a
Some more examples of subsets of Co and Ll[O, 1] failing the fixed point property P.N. Dowling, C.J. Lennard, and B. Thrett We give examples of closed, bounded. convex, nonweakly compact subsets of Co on which the right shift is expansive, and we construct two nonexpansive selfmappings (one affine and one nonaffine) on these sets which fail to have a fixed point. We also prove that every closed, bounded, convex subset of L1 [0,1] with a nonempty interior fails the fixed point property for nonexpansive mappings. Finally, we extend this result by showing that every closed, bounded, convex subset of L1 [0,1] that contains a nontrivial order interval must fail the fixed point property. ABSTRACT.
1. Introduction
In [5], LlorensFuster and Sims construct examples of closed, bounded, convex subsets of Co that are not weakly compact but are compact in a topology that is slightly coarser than the weak topology, and they nonetheless fail the fixed point property for nonexpansive mappings. These examples led LlorensFuster and Sims to conjecture that closed, bounded, convex nonempty subsets of Co have the fixed point property if and only if they are weakly compact. This conjecture has been recently settled in the affirmative [2, 3]. All the examples constructed in [5] had a common feature  they all support a nonexpansive right shift. In the first part of this short note we produce a collection of sets of the type considered by LlorensFuster and Sims, but which do not support a nonexpansive right shift and yet they fail the fixed point property for nonexpansive mappings. In fact, we will produce two nonexpansive fixed point free mappings: one affine and the other nonaffine. Variations on the themes of these examples are important in the papers [2, 3]. LlorensFuster and Sims [5] also proved that a closed, bounded, convex subset of Co with nonempty interior fails the fixed point property for nonexpansive mappings. In the second part of tllis note we prove an analogous statement in the setting of L1 [0, 1]. We also generalize this result to show that every closed, bounded, convex 2000 Mathematics Subject Classification. Primary 47HI0, 47H09, 46E30. The authors wish to thank Professor Kaz Goebel for his helpful suggestions concerning the proof of Theorem 3.2. The second author thanks the Department of Mathematics and Statistics at Miami University for their hospitality during part of the preparation of this paper. He also acknowledges the financial support of Miami University.
© 171
2003 American l\1athematical Society
P.N. DOWLING, C.J. LENNARD, AND B. TURETT
172
subset of L1 [0, 1J that contains a nontrivial order interval must fail the fixed point property. We refer the reader to the text of Goebel and Kirk [4J for any unexplained terminology. 2. The fixed point property in Co We begin this section with the LlorensF\lster and Sims examples. We have slightly modified their examples to simplify the computations. Let r denote the set of sequences 'Y = ("(n)n in the interval (0,1). For all 'Y E r, define K., by
K...,
:=
{x
=
("(ntn)n E Co 112: h 2: t2 2: t3 2: ... 2: O} .
We define the right shift Ton K..., by
T( ("(1 t 1 , 'Y2t2, 'Y3 t3, 'Y4t4, ... ))
:=
("(1, 'Y2t 1, 'Y3 t2, 'Y4t3, ... ).
Note that if x = ("(ntn)n and y = ("(nsn)n are elements of K..." then Ilx  yll sUPn 'Ynlt"  8 n l and IIT(x)  T(y)11 = SUPn 'Yn+1Itn  8 n l· Clearly, if the sequence ("(n)n is decreasing, then T is a nonexpansive mapping on K...,  this is the case considered by LlorensFuster and Sims [5J. However, it is equally obvious that if the sequence ("(n)n is strictly increasing, then T is an expansive mapping on K...,; that is, IIT(x)  T(y)11 > IIx  yll whenever x i= y. We will show that even though some of these sets do not support a nonexpansive right shift, they do support nonexpansive fixed point free mappings. To simplify our computations we will only consider the sets K..., where the sequence 'Y = ("(n)n is in (0,1), is strictly increasing and satisfies 1  'Yn < 4 n for all n E N. EXAMPLE 2.1. Let I be the identity mapping on K..." let T be the right shift defined above, T2 = ToT, T3 = ToT 0 T, and so on. Define R: K..., + K..., by
R
:= ~I
+ b T + 1a T2 + 21
4
T3
+... .
A simple calculation shows that if x = ("(1 t1, 'Y2t2, 'Y3t3, ... ) E K..." then
R(x) = ('Y1(~t1
+ ~),'Y2(~t2 + it1 + i),'Y3(~t3 + it2 + kt1 + ~), ... ). that if R(x) = x, then tn = 1 for all n E N, and thus
It is easily seen x is not an element of K...,; that is, R is fixed point free on K...,. To see that R is nonexpansive on K..." let x = ("(ntn)n and y = ("(nsn)n be elements of K...,. Then, 12 +  4 1 + ... + 2,,1...,1 ) < 1 for each n E N. since 1  4 n < 'Yn < 1, we have 'Yn( ...,,, "",,1 Consequently,
IIR(x)  R(y)11
sup hnl~(tn  sn) n
+ i(tn1
 snd
+ ... + 2~ (t1

< sup hn(~ltn  snl + iltn1  8 n 11 + ... + 2~' It 1 n
< sup {'Yn(_l+ _1_ + ... + +) max 'Yilti  Sil} n 2...,,, 4"",,1 2""1 l::;'i::;n
< sUP'Ynltn n
8n
l
Ilxyll· Thus R is a nonexpansive mapping on K...,.
st)l} Sl
J)}
173
EXAMPLES OF FIXED POINT FREE MAPS
The mapping R, given in example 2.1, is an affine mapping on K"I' Our next example is nonaffine on K"I' EXAMPLE 2.2. For an element x = (xn)n in K"I' we denote by bl,X) the sequence bl,Xl,X2,X3, ... ). We define a mapping S: K"I + Co by S(x):= X, for each x E K"I' where x = (Xl, X2, X2, ... ) is the decreasing rearrangement of the sequence bl' x); that is, X = bl, x)*. Note that
X= (Xl,X2,X3, ... ) =
(11
(~~) ,12 (~:) ,13 (~:) , ... ).
Also Xl 2: X2 2: X3 2: ... 2: 0 and 0 < 11 < 12 < 13 < ... < 1. Therefore ~ > ~ > ~ > ... > O. Since ~ = max("("x) > 1 S(x) does not necessarily belong ~nn~ ~ , to K"I' However, the mapping S is nonexpansive on K"I because the operation of decreasing rearrangement is nonexpansive on Co, so for all x and y in K"I' we have IIS(X)  S(Y)II = IIx 
yll
IIbl,X)*  bl,Y)*11 :::; Ilb1.x)  b1.y)11
=
=
Ilx  YII·
We now introduce a modification U of S that will be nonexpansive and fixed point free on K"I' Define U : K"I + Co by
Since
11j AXj  1j AYj I :::; IXj  Yj I for all j E N, it follows that IIU(x)  U(Y)II :::;
Ilx  yll = IIS(x)  S(Y)II :::; IIx  YII·
Thus U is a nonexpansive mapping on K"I' Furthermore, since 1j AXj =
1j (1 A ~) for all j E Nand 1 2: 1 A ~ 2: 1 A ~ 2:
1 A ~ 2: ... 2: 0, U maps K"I into K"I' To finish, we will show is fixed point free on K"I' Suppose, to get a contradiction, that there exists x E K"I such that x = U(x). Thus, for all j EN, Xj = 1j AXj. A wellknown fact about decreasing rearrangements that we will use is that for for each mEN, all W E
ct,
WI
+ ... + Wm
:::;
wi + ... + w;;'.
Since 1 = (1n)n is strictly increasing with limit 1, while X is decreasing with 1 > Xl 2: 11, there exists a unique kEN such that Xk+1 < 1k+l and Xk 2: 1k. Thus, for all mEN with m > k, we have, k + Xl
+ ... + Xm+l > (Xl + ... + Xk) + (Xl + ... + xm+d (Xl
+ ... + Xk) +
bl A Xl + ... + 1k A Xk + 1k+l A Xk+1 + ... + 1m+1 A Xm+1 = (Xl + ... + Xk) + bl + ... + 1k + Xk+l + ... + xm+d bl + ... + 1k) + (Xl + ... + Xk + Xk+1 + ... + Xm+1) bl + ... + 1k) + (Xl + ... + Xm+l) > bl+"'+1k)+bl+Xl+"'+Xm),
174
P.N. DOWLING, C.J. LENNARD, AND B. TURETT
It follows that for all
Tn
E N with Tn
> k, k
Xm
+l
>
bl+"'+'Yk)k+'Yl='YIL(1'Yj) j=1
> 'Y1 
=
L (1 
'Yj) ~ 'Yl 
j=1
=.
L 4
J
1
= 'Yl 
:3
1
>
:3'
j=1
This contradicts the fact that x E Co and so completes the proof that U is fixed point free on Ky.
3. The fixed point property in £1 [0, 1] One of the most notable works in metric fixed point theory is the construction of Alspach [1] of a nonempty weakly compact convex subset of Ll[O, 1] which fails the fixed point property. We begin this section by recalling some of the details of Alspach's construction. Let C:= {f E Ll[O, 1] : 0::; f(t) ::; 1, for all t E [0, I]}. Now define T: C + C by
Tf(t) := {min{2f (2t), I} max{2f(2t  1)  1, O}
for 0 ::; t ::; ~ for ~ < t ::; 1.
for all f E C. Alspach showed that the mapping T is an isometry on C which has two fixed points; namely 0 and X[O,lj' Alspach also showed that T is an isometric self map of the closed convex subset Co := {f E C : J~ f dm = 1/2} of C, such that T is fixed point free on Co. Here, m denotes Lebesgue measure. We now follow a modification of Alspach's example due to Sine [7]. Define S : C + C by S(f) := X[O,lj  f, for all f E C. The mapping S is clearly an isometry of C onto C. Thus the mapping ST is a nonexpansive mapping on C. Sine proved that ST is fixed point free on C. In [5], LlorensFUster and Sims prove that a closed bounded convex subset of Co with nonempty interior fails the fixed point property. We will use the above construction of Alspach, and modification by Sine, to prove a result analogous to the LlorensFUster and Sims result in the setting of Ll [0, 1]. Specifically we prove the following result. THEOREM 3.1. Let K be a closed, bounded, convex subset of Ll [0, 1] with nonempty interior. Then K fails the fixed point property for non expansive mappings. PROOF. By translating and scaling, we ca.n assume that K contains the unit ball of £1 [0, 1]. Consequently, the set C, constructed above, is a subset of K. Define the mapping R : K + K by
Rf(t) := min{lf(t)l, I}, for 0::;
f
E
t::; 1, for all f
E
K.
It is easily seen that R is a. nonexpansive mapping on K and R(f) E C for all K. Now define U: K + K by
U(f) := ST(R(f)), for all
f
E
K.
The mapping U is nonexpansive since all of the mappings, R, S, and Tare nonexpansive.
EXAMPLES OF FIXED POINT FREE MAPS
175
We now show that U is fixed point free. Suppose that f E K is a fixed point of U, that is, U(f) = f. Since f E K, R(f) E C, and since ST maps C into C, f = U(f) = ST(R(f)) E C. Note that the mapping R restricted to C is the identity on C. Therefore, f = ST(R(f)) = ST(f) and so f is a fixed point of ST in C. This contradicts Sine's result that ST has no fixed point in C [7], and thus the proof is complete. D THEOREM 3.2. Let K be a closed, bOllnded, convex sllbset of Ll[O, 1] that contains an order interval [h,g] := {f E L1[0, 1] : h ::; f ::; 9 a.e.}, for some h,g E Ll[O, 1] with h ::; 9 a.nd h ::f g. Then K fa'ils the fixed point property for nonexpansive mappings,
a a
a
PROOF. By translating by h, we may assume that h = and 9 ~ a.e. with 9 nontrivial. Next, note that there exists a real number c > and a measurable set E with Lebesgue measure rn(E) > 0, such that 9 ~ CXE. By rescaling K by lie, we may assume without loss that c = 1. Now, define the mapping R : K ~ [0, xel ~ K by R(f) := ifi /\ XE. Note that R is nonexpansive and R equals the identity on [0, xel. At this stage, consider E. There exists to in the interval [0,1] such that rn(En [0, to]) = ~ rn(E), Let E1.1 := En [0, to] and E1.2 := En (to, 1]. Clearly E is the disjoint union of E1.1 and E 1,2 and rn(E 1,1) = rn(El,2) = ~ rn(E). Proceed iteratively from here. Similarly to above, there exist pairwise disjoint measurable subsets E2,1, E 2,2, E2,3 and E2,4 of [0, 1] such that El,l = E 2,1 U E2,2, E 1 ,2 = E 2,3 U E 2.4 , and each rn(E2,k) = ~ rn(E). Repeating this construction inductively, we produce a family of measurable subsets (EO,1 := E, En,k : n E N, k E {I, ... , 2n}) of [0,1] such that (XE",k)n.k is a dyadic tree in Ll [0,1]. Moreover, letting the measure v be defined on the measurable subsets of E by v = (1/rn(E))rn, it follows that the Banach space L 1 ( E, v) is isometrically isomorphic to L 1 ( [0, 1], m) = L 1 [0, 1] via the mapping Z defined as follows: Z(XEn, k):= X[kl k), 2l'r'2"'t"
for each XE",k' Then Z is extended to L := the linear span of the functions XE",k in the usual way. Of course, Z is an isometry on L. Finally, since L is dense in Ll(E, v), with dense range in Ll [0,1], Z extends to a linear isometry from L1(E, v) onto Ll[O, 1]. Let W := ST be Sine's variation on Alspach's example, as described above, and note that W maps the order interval C := [0, X[O,I]] into C. Let's use W to define E : [0, xel ~ [0, XE], by E := Z1 W Z. We have that E is a fixed point free Ll [0, 1]isometry on [0, xel. Finally, we define U : K ~ [0, XE] ~ K via U := E R. In a manner analogous to the argument in the proof of Theorem 3.1 above, we see that U is a fixed point free Ll[O, 1]nonexpansive mapping on K. D REMARK 3.3. In [6], MatlI'ey proved that closed, bounded, convex, nonempty subsets of reflexive subspaces of Ll [0, 1] have the fixed point property for nonexpansive mappings. Consequently, Maurey's result, in tandem with Theorem 3.2, shows that reflexive subspaces of U [0,1] cannot contain a nontrivial order interval. In fact, as pointed out by the referee, the argument in the proof of Theorem 3.2 shows
176
P.N. DOWLING, C.J. LENNARD, AND B. TURETT
that infinitedimensional subspaces of £1[0,1] which contain nontrivial order intervals actually contain isometric copies of £1 [0, 1] and thus are nonreflexive. The authors thank the referee for his/her comments. References 1. D. Alspach, A fixed point free nonexpansive mapping, Proc. Amer. Math. Soc., 82 (1981), 423424. 2. P.N. Dowling, C.J. Lennard and B. Thrett, Characterizations of weakly compact sets and new fixed point free maps in co, to appear in Studia Math. 3. P.N. Dowling, C.J. Lennard and B. Thrett, Weak compactness is equivalent to the fixed point property in co, preprint 4. Kazimierz Goebel and W.A. Kirk, Topics in metric fixed point theory, Cambridge University Press, Cambridge, 1990 5. Enrique LlorensFuster and Brailey Sims, The fixed point property in co, Canad. Math. Bull. 41 (1998), no. 2, 413422. 6. B. Maurey, Points fixes des contractions de certains faiblement compacts de Ll, Seminaire d'Analyse Fonctionelle, 19801981, Centre de Mathematiques, Ecole Polytech., Palaiseau, 1981, pp. Exp. No. VIII, 19. 7. R. Sine, Remarks on an example of Alspach, Nonlinear Anal. and Appl., Marcel Dekker, (1981), 237241. DEPARTMENT OF MATHEMATICS AND STATISTICS, MIAMI UNIVERSITY, OXFORD, OH
45056
Email address: dowlinpnGmuohio. edu DEPARTMENT OF MATHEMATICS, UNIVERSITY OF PITTSBURGH, PITTSBURGH, PA
15260
Email address: lennard+lDpi tt. edu DEPARTMENT OF MATHEMATICS AND STATISTICS, OAKLAND UNIVERSITY, ROCHESTER,
48309 Email address: turettlDoakland. edu
MI
Contemporary Mathelnatics Volume 328. 2003
Homotopic composition operators on Hoo (Bn) Pamela Gorkin, Raymond Mortini, and Daniel Suarez We characterize the path components of composition operators on Hoo(B n ), where Bn is the unit ball of en. We give a geometrical equivalence for the compactness of the difference of two of such operators. For n = 1, we give a characterization of the path components of the algebra endomorphisms. ABSTRACT.
1. Introduction
Consider the Hardy space H2 on the unit disk D. Littlewood's subordination principle tells us that for an analytic selfmap ¢ of D and a function f in H2, the function f 0 ¢ is once again in H2. Thus one defines the composition operator C'" on H2 by C",(f) = f 0 ¢. The interplay of operator theory and function theory leads to several interesting results. One of these results is Berkson's theorem on isolation of composition operators (see [1] and [14]): THEOREM 1 (Berkson). Let ¢ be an analytic selfmap of D. If ¢ has mdial limits of modulus one on a set E of positive measure, then for every other analytic selfmap 'l/J of D, the following estimate holds:
IIC",  C",II2:
Jmea;(E),
where C'" and C'" are the corresponding composition opemtors on H2.
Thus, Berkson's theorem tells us that every such operator is isolated in the set of composition operators in the operator norm topology. For example, the identity operator, C z , is at least a distance of ..[f72 from every other composition operator on H2 (as is C"', where ¢ is any inner function). However, not every composition operator is isolated. If ¢ is analytic and ¢ : D ~ sD for some s with 0 < s < 1, then it is easy to check that
Thus, C'" is not isolated. Inner functions induce highly noncompact operators, as well as isolated operators. The operators C'" for which ¢(D) is contained in sD for some s with 0 < s < 1 2000 Mathematics Subject Classification. Primary 47B33; Secondary 47B38. Key words and phrases. composition operator, path components, compact differences.
© 177
2003 American lvlathematical Society
178
PAMELA GORKIN. RAYMOND MORTINI. AND DANIEL SUAREZ
are compact. As Shapiro and Sundberg [14J indicate in their paper, "compact composition operators are dramatically nonisolated." They show that the set of compact composition operators is path connected, and therefore these operators are never isolated. It is interesting to ask which composition operators are, in fact, isolated. Shapiro and Sundberg studied this problem, and showed (among other things) that if ¢ is au analytic selfmap of D that is not an extreme point of the algebra HOC(D), then C> is not isolated; in their words, "isolated composition operators can only be induced by extreme points." This allowed them to exhibit an example of a noncompact nonisolated operator. They also raised several questions at the end of their paper: (1) Characterize the components in Comp(H2), the space of all composition operators on H2. (2) Which composition operators are isolated? (3) Characterize composition operators whose difference is compact. Before stating the final question, we remind the reader that the essential norm of an operator T defined on a Banach space H is the distance to the compact operators; that is, liT lie = inf{IIT  KII : K compact on H}. It is clear that IITII ~ IITlle, and therefore every essentially isolated operator is isolated. In fact, because of the abundance of weakly null sequences in H2, all the results on isolation appearing in Shapiro and Sundberg's paper hold true if we replace the norm with the essential norm (see [14], p. 148). Thus they raised the following question. (4) Is every isolated operator essentially isolated? Other papers of interest on this subject include [10J. Of course, one is not limited to the space H2, and the study of composition operators on various spaces has lead to a large body of literature. In this paper, we are interested in the same problems for composition operators on HOC (Bn), where Bn denotes the open unit ball in en. While the problem 011 H2 seems to be difficult, MacCluer, Ohno and Zhao [l1J were able to obtain partial results about operators on the algebra HOO(D). They showed that for two analytic selfmaps of the disk, C> and C.p are in the same path component in the space of composition operators, Comp(HOC(D)), if and only if IIC",  c.pll < 2. In particular, an operator C> is isolated in Comp(HOC(D)) if and only if IIC",  C",II = 2 for any other analytic selfmap 'IjJ of the disk. The authors show that this result can be rephra..<;ed in terms of the pseudohyperbolic metric, and they posed the question of whether or not every isolated composition operator on HOC is, in fact, essentially isolated. The answer to this la..<;t question was given by Hosokawa, Izuchi and Zheng [8J. Their technique was to develop something called asymptotic interpolating sequences, or a.i.s. for short. Essentially, this definition allowed them to interpolate sequences with a good bound on the norm (see [6J for more information about these sequences). They then used these sequences and Blaschke products to obtain HOC(D) functions that provide good estimates for the essential norm of the difference of two composition operators.
HOMOTOPIC COMPOSITION OPERATORS ON Hoo(Bn)
179
In this paper, we give simpler proofs of the results obtained by Hosokawa, Izuchi and Zheng, and combine them with the proofs of MacCluer, Ohno and Zhao. Because our proofs are significantly simpler and do not refer to asymptotic interpolating sequences or Blaschke products, we are able to obtain the results on the ball in en. While our results do not rely on interpolating sequence results, they do rely on a construction of Gamelin and Garnett relating interpolating sequences to peak sets [5]. The proof we will provide is simple enough to be applicable to other algebras. We conclude the paper with an example of such an application: Using these same techniques, we are able to "lift" these results to obtain a characterization of the path components of endomorphisms of HOO(D). After we completed this paper, we learned that some of the results on composition operators on Hoo(Bn) were also obtained by Carl Toews [15]. Two other papers directly related to the results described here are [3] and [9]. Finally, we mention that recent results on Shapiro and Sundberg's first question in the space H2 can be found in [12].
2. Preliminary results Our goal is to prove results about composition operators on Hoo(Bn) and endomorphisms of HOO(D). In this section, we present proofs of several lemmas that will be important in obtaining estimates on norms and essential norms of operators. Our discussion begins with functions of n variables and Z = (Zl' Z2, .•. , zn), where each Zj is a complex number. As usual, the associated norm of Z is given by
Izi =
(z, z)1/2,
and the unit ball Bn is the set of all Z E en for which Izi < 1. We let Hoo(Bn) denote the space of bounded holomorphic functions on Bn. If n = 1 our situation reduces to the familiar space of functions on the unit disk: HOO(D). We need some background on general uniform algebras, some information specific to Hoo(Bn) and some deeper results for HOO(D). Everything we need is presented in this paper. Let A be a uniform algebra. The maximal ideal space of A, denoted M(A), is the set of complexvalued, linear, multiplicative maps of A that map the identity of A to the value 1 E C. Since evaluations at points of Bn are linear multiplicative functionals on Hoo(Bn), we may think of Bn as a subset of M(Hoo(Bn)). It is well known that M(A) is a compact Hausdorff space when endowed with the weak* topology induced by A * . We will always consider this topology for M(A). For an element a E A, the Gelfand transform of a is a complexvalued map defined on M(A) by a(x) = x(a). This map establishes an isometric isomorphism between A and a closed subalgebra of C(M(A)). It is usual to identify the function with its Gelfand transform, since the meaning is generally clear from the context. For x, y E M(A), the pseudohyperbolic and hyperbolic metrics are defined, respectively, by
p(x, y) = sup{l/(y)1 : I E A, 11/11 ::; 1, and I(x) = O} and y) h( x, y )  Iog 11 + p(x, ( ). p x,y
180
PAMELA GORKIN, RAYMOND MORTINI, AND DANIEL SUAREZ
It is wellknown that P is a [0, 1]valued metric and that h is a [0, +oo]valued metric on M(A). The triangle inequality for h immediately implies that the condition p(x, y) < 1 (i.e. h(x, y) < 00) is an equivalence relation on M(A). If A is a uniform algebra and ¢ : M(A)+M(A) is a continuous map such that a 0 ¢ E A for all a E A, then the map C'" defined by
C",a = a 0 ¢ is an endomorphism of A. We will write C'" E End (A). Conversely, if E E End (A), we may define ¢ : M(A)+M(A) by ¢(x)(a) = x(E(a)), obtaining E = C"'. The Shilov boundary of A is the smallest closed subset of M(A) on which every function in A attains its maximum. We denote the Shilov boundary by 8A. LEMMA 2. Let A be a uniform algebra and let C"', C1/J E End (A). Let V M(A) be a set whose closure contains 8A. Then,
(2.1)
sup xEV
PROOF.
c
2p(¢(x),1/I(x)) J :::; IIC",  C1/J1I :::; 2 sup p(¢(x), 1/I(X)). 1 + 1  p(¢(x), 1/I(x))2 xEV
First we show that if x, y
(2.2 )
1
E
M(A), then
II 2p(x, y) + Jl  p(x,y )2:::; X
II 2 ( ) Y A*:::; p x, y .

The proof uses the techniques of [4, p. 144]. Let I E A with Ilfll < 1. It is clear that 1 = U  f(y))/(1  I(y)f) E A, l(y) = and 11111 :::; 1. By definition of p then lj(x)1 :::; p(x, y), and consequently
°
If(x)  f(y)1 :::;
11 
f(y)f(x)1 p(x, y) :::; 2p(x, y).
Taking the supremum over Ilfll < 1 we get the upper inequality. Now we turn to the lower inequality. For simplicity, we write p = p(x, y). Choose fn E A with I!fnl! < 1, In(x) = and fn(y) > pl/n. Let Pn = fn(Y) and Ln(z) = (tn  z)/(I tnz), where tn = (1 Jl P~)/Pn' Therefore Ln 0 fn E A, IILn 0 Inll :::; 1 and
°
IIx 
yl!A* ~ I(Ln
fn)(x)  (Ln
fn)(Y) I = Itn  L .. (Pn)l·
A simple computation shows that
Itn  Ln(Pn)1 = 1Pn(1  t;) 1+ 2p . ItnPn 1+~ The lemma will follow immediately from (2.2) and the following chain of identities: IIC",C1/J1I
sup sup IU 0 ¢)(x)  U 0 1/I)(x) I 11/11=1 xE8A sup sup IU 0 ¢)(x)  U 0 1/I)(x) I 1I/1I=lxEV
sup sup 11(¢(x))  f(1/I(x))1 11/11=1 sup 11¢(x) 1/I(x)IIA*.
xEV
xEV
o LEMMA
IIC",  C1/J1I
3. Let A be a uniform algebra and let C"', C,,}
< 2 is an equivalence relation.
E
End (A). The condition
HOMOTOPIC COMPOSITION OPERATORS ON Hoo(Bn)
181
PROOF. It is obvious that the relation is reflexive and symmetric. So, suppose that ¢,'l/J and t.p define endomorphisms of A such that IICeI>  Cvlli < 2 and IICob C
=
sup p(¢(x),'l/J(x)) < 1 and
0'2
=
xEM(A)
sup p('l/J(x),t.p(x)) < 1. xEM(A)
Using the hyperbolic metric h on M(A) we obtain
1 + 0'1 h(¢(x), t.p(x)) :::; log  XEM(A) 1  0'1 sup
1 + 0'2 + log  = /3, 1
0'2
and consequently sUPXEM(A) p(¢(x), t.p(x)) :::; (ei3 1)/(ei3 + 1) tion of (2.1) yields the desired result.
< 1. A new applica0
LEMMA 4. Let A be a uniform algebra. If {fn} is a sequence of functions in the unit ball of A tending pointwise to zero on 8A, then {fn} tends to zero weakly in A.
PROOF. As indicated in the introduction, using the Gelfand transform, we
may think of A ~ C(8A). Let x be any element of the dual space of A. By the HahnBanach theorem, x has a continuous norm preserving extension to the space of continuous functions on the Shilov boundary. Therefore, there exists a finite measure ILx on 8A such that x(f) =
r
loA
f dJtx •
But Ilfnll :::; 1 for all n, and fn ...... 0 pointwise on 8A, so we may apply the Lebesgue dominated convergence theorem to conclude that 3;(fn) ...... O. Therefore, the sequence {fn} converges to zero weakly. 0 We will need another estimate, but this will depend on the pseudohyperbolic metric particular to the ball, Bn. For a, z E Bn, let Sa = lal 2 , and the
Jl 
((z' a)) a and Qa = 1 Pa. Relevant a,a computations can be found in [13, p. 25]. On Bn, the pseudohyperbolic metric induced by Hoo(Bn) is given by projections Pa and Qa be given by Pa(z)
(
p a, z
)=
=
la  Pa(z)  saQa(z) I 1 _ (z, a) ,
In what follows, for points a and z in the ball and numbers sand t in the closed interval [0,1]' we let as = a + s(z  a) and asH = a + (s + t)(z  a). LEMMA
(2.3)
5. Let a, z
E
Bn. For s, t
E
[0,1] .satisfying t :::; 1  s we have
tp(a,z) p(a + s(z  a), a + (s + t)(z  a)) :::; 1 _ (1 _ t)p(a, z)·
PROOF. We can assume that a =I z, because otherwise there is nothing to prove. We consider first the case s = O. Since Pa and Qa are linear operators satisfying Pa(a) = a and Qa(a) = 0, the nmnerator of p(a, a + t(z  a)) satisfies
a  Pa(a + t(z  a))  saQa(a + t(z  a)) = ta  tPa(z)  tsaQa(z).
182
PAMELA GORKIN, RAYMOND MORTINI, AND DANIEL SUAREZ
We have assumed that a =I z, and therefore a  Pa(z)  saQa(z) =I O. A simple computation gives
Ia 
p(a,a+t(za)) =
Pa{a + t(z  a))  satQa(z) 1  (a + t(z  a), a)
I
I
t(a  Pa(z)  saQa(z)) + (z, a)  (a, a)  t(z  a, a)
I
1  (z, a)
< 11/p(Z,a) 1(1 t)(z  a,ta)I/la  Pa(z)  saQa(z)ll· But
I(z 
a,a)1 = I(a  Pa(z)  saQa(z),a)1 ::; la  Pa(z)  saQa(z)llal. Therefore t tp(a,z) p(a. at) < = . .  (1/ p(a, z))  (1  t) 1  (1  t)p(a, z)
(2.4)
This proves (2.3) for s = O. For the general case we can assume s =I 1 since otherwise t = 0 and there is nothing to prove. From (2.4) we obtain
<
p(a s, as + t(z  a)) p(as,a s + (t/(I s))(z  as)) t/(1  s) (l/p(a s,z))  (1 (t/(I s)))'
But
p(a + s(z  a), z) p(z + (1  s)(a  z), z) p(z, z + (1  s)(a  z)). So we may apply (2.4) again to conclude that
Is p(as,z)::; (l/p(a,z)) s Combining this with our estimate on p(a s , as+t) above, we see that
t p(a s , as+t) ::; (1/ p(a, z))  (1  t)
o
Simplifying, we obtain the desired conclusion. 3. Composition operators on Hoc (Bn)
In this section we study composition operators on Hoc(Bn); that is, given an analytic selfmap ¢> of the unit ball, we look at maps C'" : Hoc(Bn) ~ Hoc(Bn) defined by C",(f) = f 0 ¢>. These maps are all endomorphisms of the algebra HOC (Bn). For the special case of n = 1 we will say more in the final section of the paper. We are interested here in estimates on the essential norm of the difference of two composition operators. If T is a bounded operator, we denote its essential norm by IITlle. THEOREM
6. Let ¢> and '!/J be holomorphic selfmaps of Bn such that
max{II¢>II, II'!/JII} =
1.
HOMOTOPIC COMPOSITION OPERATORS ON H"""(Bn)
183
Let
e=
max {lim sup p(¢(Z), 'lj!(Z)) , lim sup p(¢(Z), '¢(Z))} . 1'I/J(z)ll
1",(z)ll
Then (3.1) PROOF. By hypothesis there is a sequence of points {Zj} in Bn, such that p(¢(Zj),¢(Zj)) ~ e, and one of them, say {¢(Zj)}, converges to a point ( on the boundary of Bn. Without loss of generality we may assume that p( ¢( Zj ), 'lj!( Zj)) > e1fj. Let kj E Hoo(Bn) be a function of norm one, whose existence is guaranteed by the pseudohyperbolic distance definition, satisfying kj(¢(zj)) > e  1fj and kj('lj!(zj)) = O. Consider the functions
f(z) = (1 + (z, () )/2 and g(z) = (1  (z, () )/2. Then f,g E HOO(B n ), f(() = 1, If(7])1 < 1 for all 7] E aBn satisfying 7] I (, and g(() = O. We will now produce a sequence of functions, {hj}, tending to zero weakly for which Ihj(¢(zj))  hj (1/I(Zj)) I ~ e. We proceed as follows. Let j E N. Since f(¢(zj)) ~ 1, we may choose Zmj so that If(¢(zmj))lj > 1  1 fj. Now since 9 I 0 on Bn, there exists an integer Ij such that
J
Ig(¢(zmj))1 1/ 1j 2: J1 1fj. Consider the functions hj = (gl/1 j )(J3)kmr Then Ilhjll ~ 1, hj (1/I(zmj)) = 0, and Ihj(¢(zmJ)1 > (1  1fj)(e  1/mj). We note that hj ~ 0 on the Shilov boundary and, by Lemma 4, hj ~ 0 weakly. Thus for any compact operator K we have IIC",  C,p
+ KII > 2:
IIC",hj  C,phj + Khjll Ihj(¢(zmj))  hj('lj!(zmj))
+ (Khj)(zmJI ~ e·
= Ihj(¢(zmJ) + (Khj)(zmj)1
This proves the lower inequality in (3.1). For the upper inequality, let I: > 0 and choose 8 with 0 to 1 so that
p(¢(Z), 'lj!(z)) ~
e + I:
on the set {1¢(z)1
< 8 < 1 close enough
> 8} U {1'lj!(z)1 > 8}.
Now choose a = a(I:,8) E (0,1) close enough to one so that p(¢(z),a¢(z)) < I: and p('lj!(Z) , ml'{z)) < I: on the set {1¢(z)1 ~ 8}n{I1/I(z)1 ~ 8}. Since max{lla¢lI, Ila'lj!ll} ~ a < 1, the operator K ~f Co",  Co,p is compact. If Z and ware any two points of B'\ we may view Z and w as elements of the dual space of Hoo(Bn) in the obvious way. Therefore, for any function f in the unit ball of Hoo(Bn) we may apply (2.2) to conclude that If(z)  f(w)1 ~ 2p(z,w). Henceforth, applying (2.2) to the functions f and fo(z) = f(az), we have
I(C",f)(z)  (C,pf)(z)  (Kf)(z)1
~
If(¢(z))  f(a¢(z))1 + If('lj!(z))  f(a'¢(z))1 < 2p(¢(z), a¢(z)) + 2p('lj!(z), a'lj!(z)) < 4c
when z E {I¢I ~ 8} n Hl/)I ~ 8}, while
I(C",f)(z)  (C,pf)(z)  (Kf)(z)1
< If(¢(z))  f('lj!(z)) I + If(a¢(z))  f(a1/l(z))1 < 2p(¢(z), 'lj!(z)) + 2p(¢(z), 'lj!(z)) < 4e + 41:
184 when
PAMELA GORKIN. RAYMOND MORTINI, AND DANIEL SUAREZ Z E
{1cf>1
> 8} U {I'!/JI > 8}. Since the function f is arbitrary, IIG>  G",  KII
4e + 4c, and since c is arbitrary we obtain (3.1).
~
D
COROLLARY 7. Let cf> and '!/J be two holomorphic selfmaps of the unit ball. Then G>  G", is compact if and only if either max {11<;b11, II'!/JII} < 1, or lim sup p(cf>(z), '!/J(z))
=
leI>(z)I~1
lim sup p(cf>(z), '!/J(z))
= O.
1"'(z)I~1
PROOF. It is clear that G> and G", are compact, if max {11cf>11, 11'!/J11l < 1. On the other hand, if max {11cf>11, II'!/JII} = 1 and e is the parameter of Theorem 6, then (3.1) says that Gel>  G", is compact if and only if e = o. D Our next goal is to characterize the path components of composition operators on Hoo(Bn). We write G> '" G", to indicate that there is a normcontinuous homotopy of composition operators joining Gel> with G",. Also, if K denotes the ideal of compact operators, we write G> "'e G", to indicate that there is an essential normcontinuous homotopy of classes {G", + K: cp: Bn 4 B n holomorphic} joining Gel> + K with G", + K. Let cf> be a holomorphic selfmap of Bn. For x E M (HOO (Bn)) we can define
cf>(x) E M(Hoo(Bn)) by the rule <;b(x)(f) ~f x(f 0 cf». Thus we can extend cf> : B n 4Bn to a selfmap of M(Hoo(Bn)), which we also denote by cf>. The continuity of this extension is immediate. We now have everything we need to prove the main theorem of this paper. As indicated in the introduction, this theorem unifies and extends many of the results appearing in [11], as well as [8]. THEOREM 8. Let cf> and '!/J be holomorphic selfmaps of the unit ball in en. Then the following are equivalent.
(a) (b) (c) (d)
G> '" G",. G> "'e G",. IIGeI>  G",II < 2. SUPzEBn p(cf>(z), '!/J(z)) < 1.
PROOF. (a) => (b) is obvious. (c) ¢:} (d). A boundary for HOCJ (Bn) is a closed set F c M (HOCJ (Bn)) such that Ilfll = SUPxEF If(x)1 for all f E HOCJ(Bn). It is clear that the closure B n of B n in M(HOCJ(Bn)) is a boundary for HOCJ(B n ), and since oHOCJ(Bn) is the intersection of all the boundaries [4, p. 10], then oHOCJ(Bn) c F. The equivalence then follows from (2.1). (b) => (c). By hypothesis there is a family {cf>t}, with t E [0, 1], of holomorphic selfmaps of Bn such that cf>o = cf>, cf>1 = '!/J and for every c > 0 there is some 8 > 0 satisfying
< c: if It  sl < 8. Then we can take finitely many points ty = 0 < ... < tm = 1 in [0,1] such that IIG>t.  GeI>tHl lie < 1/2 for every i = 1, . .. , m  1. We claim that IIG>t  Gel>. lie
(3.2)
sup p(cf>t. (z), cf>t'+l (z)) < 1
zEBn
for every i. In fact, if r = max{llcf>d,lIcf>tHlll} < 1, then both functions map B n into the closure of r Bn, and since the pseudohyperbolic diameter of this ball
HOMOTOPIC COMPOSITION OPERATORS ON
Hoo(Bn)
185
is smaller than 1, we are done. If some of the maps have norm 1, then the first inequality of (3.1) tells us that there is some 0 < 8 < 1 close enough to 1 such that sup p(¢t;{Z), ¢tHl (Z)) < 3/4, {I>., 1~t5}U{I>"+11~6} while the set {I¢t, I < 8} n {I¢t'+ll < 8} is mapped by both functions into the ball 8Bn, whose pseudohyperbolic diameter is smaller than 1. Our claim follows. Since the closure of Bn in M(Hoo(Bn)) contains the Shilov boundary, (3.2) and (2.1) imply that IIC>"  C>'<+1 11 < 2 for i = 1, ... , Tn  1. Lemma 3 now says that (c) holds. (d) =} (a). By (d) there exists 0: < 1 such that sup p(¢(z), 'l/J(z)) :; 0:. zEBn
We define a map ¢t = ¢
+ t('l/J 
IIC>,  C>. II
¢) for t E [0,1]. Now, if t < s < 1
< 2 sup p(¢t(z), ¢s(z)) zEBn
< <
2(s  t)p(¢(z), 'l/J(z)) 1  (1  (s  t)) p(¢(z), 'l/J(z)) 20: (s  t) 1 _ 0:'
where the first inequality holds by (2.1), since 8Hoo(Bn) C 13", and the second inequality from (2.3). From this, we see that t 1+ C>t is a continuous mapping. 0 Therefore C'" lies in the same path component as C>. As a corollary, we obtain the following generalization of the work on isolated points in [11]. COROLLARY 9. Let ¢ be a holomorphic selfmap of the ball. Then C> is isolated in the set of composition operators if and only if C> is essentially isolated. PROOF. Since IIC>  C",II ~ IIC>  C",lIe, it is clear that if C> is essentially isolated, then it is isolated. If C> is isolated and 'l/J =I ¢, Theorem 8 implies that SUPzEBn p(¢(z),'¢(z)) = 1. This can only happen if there are points z E Bn such that 1¢(z)I+1 or I'l/J(z) 1+1, and p(¢(z), 'l/J(z))+1. Hence, Theorem 6 says that IIC>  C'" lie ~ 1, and C> is essentially isolated. 0
4. Examples So what are some examples of isolated operators? If ¢ : Bn + Bn has radial limits of Euclidean norm 1 on a set of positive measure, we claim that C> is isolated. If 'l/J =I ¢ there must exist a set of positive measure in 8Bn on which ¢ has radial limits of norm 1 and 'l/J does not equal ¢ (see [13, Ch. 5]). Thus, there exists a sequence {zd c Bn for which ¢(Zk) + ( E 8Bn and 'l/J(Zk) + 'Tf, with 'Tf =I (. Therefore p(¢(Zk), 'l/J(Zk)) + 1 and Lemma 2 tells us that IIC>  C",II = 2. In particular, the automorphisms of B n induce isolated composition operators. It is clear from Theorem 8 that if ¢, 'l/J are holomorphic selfmaps of Bn and O>  C'" is compact, then C> '" C"'. It is not completely clear, though, that the converse fails. In [8] Hosokawa, Izuchi and Zheng constructed an example that shows this for n = 1. By eliminating variables, every example that works for n = 1 can be made to work for general n. Here we construct a simpler example of
186
PAMELA GORKIN, RAYMOND MORTINI, AND DANIEL SUAREZ
holomorphic selfmaps of the ball, 4> and '1/), such that C'" '" C'" but C  C,,) is not compact. Let n = {UJ ED: ~1=!:1 > ~} be a nontangential region in D at the point Z = 1. We want to estimate p(w, (w + 1)/2) for wEn. We recall that p(z, w) = Iz  wi/II  zwl for z, wED. By straightforward calculation,
( ( + 1)/2) 1
p w, w
=
11 lwl 2+ 1 wi Ilw I
(1 lwl)(1 + Iwl) 1 3 Ilw I + ~
~
and
 Iwl21 11+ 1lw
p(w, (w + 1)/2)1
> ~ (1 + 1 Iw12) 1 w
1lwl 2
1+ 11_wI2(1~w)
1+ 1+ Iwl > ~
>
2
 2
when wEn. That is,
1
2
3 ~ p(w, (w + 1)/2) ~ 3
(4.1)
for all wEn. Let c.p : D+n be a onetoone and onto holomorphic function and define 4>, 'I/J : Bn+Bn by
= (c.p(Z1), 0, ... ,0) and that 114>11 = 11'1/)11 = 1. For z
4>(Z1,"" zn) It is clear that
'I/J(Z1, ... , zn) E
= ((c.p(zt) + 1)/2,0, ... ,0).
Bn, a straightforward calculation shows
p(¢(z), 'I/J(z)) = p(c.p(zd, (c.p(zd
+ 1)/2).
Since c.p(zd E n, the inequalities in (4.1) show that p(4)(z), 'I/J(z)) E [1/3,2/3]. Therefore Theorem 8 says that C'" '" C"', while Corollary 7 says that C  C'" is not compact..
5. Endomorphisms of HOO(D) In this section we investigate the path components of elldomorphisms of H OO (D). For x E M(HOO(D)), the Gleason part of x is P(x) = {y E M(HOO(D)) : p(x,y) < I}. Since the condition p(:r, y) < 1 is an equivalence relation, the Gleason parts form a partition of M(HOO(D)). In [7] Hoffman produced a continuous and onto map Lx: D+P(x) such that Lx(O) = x and foLx E HOO(D) for every x E M(HOO(D)) and f E HOO(D). There are two possibilities: either Lx(z) = x for all zED (so P(x) = {x}) or Lx is onetoone. We write G = {x E M(HOO) : Lx is onetoone} and
r
= {x E M(HOO) : Lx = {x}}.
It is wellknown that every endomorphism T of HOO(D) can be factored as T = C",CL", , where 4> is a holomorphic selfmap of D and x E M(HOO(D)). Although it is clear that this factorization is not unique, two different factorizations of the same endomorphism are related in the following way (see [2]): if p(x, y) < 1, then there is a biholomorphic map r of D (depending on x and y) such that Ly(z) = Lx(r(z))
HOMOTOPIC COMPOSITION OPERATORS ON
Hoo(B n )
for every zED. This means that every endomorphism of the form T also be factored as
187
= Cq,CLy can
T = Cq,CLy = Cq,CTCL", = CToq,CLx ' Of course, if x E rand p(x, y) < 1, then x = y and T = C Lr . LEMMA 10. Let x E G and A = Ej=1 )..,jCq,j' where composition operators on H'XJ(D). Then IIACLJI = IIAII.
)..,j
E C and Cq,j are
PROOF. Since IIACLxll ::; IICLxllliAIl ::; IIAII, one direction is easy. For the other direction, if 0 < f < 1, there exists a function f in the ball of HOO(D) such that IIA(f)11 > (1  f)IIAIi. By the definition of the norm, there exists r with o < r < 1 such that n
L )..,jf(cPj(z)) ZETD j=l
sup IA(f)(z)1 = sup
ZETD
> (1  f)2I1AII·
By a result of Hoffman [7, p. 91]' there exist Blaschke products bk such that (bk 0 Lx)(z) ~ z uniformly on compact subsets of D. But rD is a precompact subset of D, and therefore cPj (r D) is precompact for each j. That is, there is 0 < 0: < 1 such that Uj=l cP j(rD) C o:D. Fix {J with 0: < {J < 1. Since f is analytic, there is 8> 0 such that for z, wE (JD with Iz  wi < 8 we have If(z)  f(w)1 < f. Clearly we can also require 8 < (J  0:. Therefore we may choose k sufficiently large so that I(b k 0 Lx)(cPj(z))  cPj(z)1 < 8 for all z E rD. Thus, for k that large, z E rD and f as above, (b k o Lx)(cPj(z)) E (3D and consequently If(bk(Lx(cPj(z)))  f(cPj(z))1 < f. Therefore there exists a constant At depending only on nand )..,l"",)..,n such that II ACLx II
~
sup I(ACLx(f
bk))(z)1
zErD
n
sup
IL
zErD j=l
)..,j(f 0 bk 0 Lx)(cPj(z))1
n
>
sup
IL
zErD
> Letting
f ~
)..,jf(cPj(z))I Mf
j=l
(1  f)2I1AII Mf.
0 yields the desired result.
o
THEOREM 11. Let T 1, T2 E End(HOO(D)). Then the following ar'e equivalent.
(a) T1 rv T2 in End(HOO(D)). (b) IIT1  T211 < 2. (c) There exist x E M(HOO(D)) and holomorphic selfmaps cP. 'ljJ of D such that T1 = Cq,C Lx • T2 = CtfJCLx and IICq,  CtfJlI < 2. PROOF. Suppose that (a) holds. Then there is a homotopy
G: [0, 1] ~End(HOO(D)) with G(O) = T1 and G(I) = T 2. We can find finitely many points 0 = it < ... < tn = 1 such that IIG(tj)  G(tj+d II < 2 for j = 1, ... , n  1. Lemma 3 then says that IIG(O)  G(I)1I < 2. Suppose that (b) holds and write T1 = CLxoq, and T2 = CLyo"" where x,y E M(HOO(D)) and
PAMELA GORKIN, RAYMOND MORTINI, AND DANIEL SUAREZ
188
p(Lx(¢(O)), Ly(
 CLyo'!'lI = 2. Thus (b) implies that p(x,y) < 1. If x E r, then Tl = T2 = C Lx ' If x E G and we write T2 = CLxo.p, where 1/J is a holomorphic selfmap of D. Lemma 10 says that 2> IITI  T211 = IICt/>  C,pll, so (c) holds. If (c) holds Theorem 8 says that there is a homotopy of composition operators F(t), with t E [0,1] such that F(O) = Ct/> and F(l) = C,p. By Lemma 10, G(t) ~f F(t)CLx is a homotopy of endomorphisms connecting Tl with T 2, which proves (a). 0 Acknowledgement. The last author thanks Bucknell University for its hospitality and peaceful environment during the preparation of this paper. References [1] E. Berkson, Composition opemtors isolated in the uniform opemtor topology, Proc. Amer. Math. Soc. 81 (1981),230232. [2] P. Budde, Support sets and Gleason parts, Michigan Math. J. 37 (1990), 367383. [3] P. Galindo and M. Lindstrom, Factorization of homomorphisms through HOO(D), preprint. [4] T. Gamelin, Uniform algebras, PrenticeHall, Inc., Englewood Cliffs, N. J., 1969. [5] T. Gamelin and J. Garnett, Distinguished homomorphisms and fiber algebras Amer. J. Math. 92 (1970), 455474. [6] P. Gorkin and R. Mortini, Asymptotically interpolating sequences in uniform algebms, J. London Math. Soc., to appear. [7] K. Hoffman, Bounded analytic functions and Gleason parts, Ann. of Math. (2) 86 (1967), 74111 . [8] T. Hosokawa, K. Izuchi, and D. Zheng, Isolated points and essential components of composition opemtors on HOC, Proc. Amer. Math. Soc. 130 (2002), no. 6, 17651773. [9] U. Klein, Kompakte multiplikative Opemtoren auf uniformen Algebren, Mitt. Math. Sem. Giessen 232 (1997), 1120. [10] B. MacCluer, Components in the space of composition opemtors, Integr. Equ. Oper. Theory 12 (1989), 725738. [11] B. MacCluer, S. Ohno, and R. Zhao, Topological structure of the space of composition oper'ators on HOC, Integr. Equ. Oper. Theory 40 (2001), 481494. [12] J. Moorhouse and C. Toews, Differences of composition opemtors, preprint. [13] W. Rudin, Function theory in the unit ball of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], 241. SpringerVerlag, New YorkBerlin, 1980. [14] J. Shapiro and C. Sundberg, Isolation amongst the composition opemtors, Pacific J. Math., 145 (1990), 117151. [15] C. Toews, Topological components of the set of composition opemtors on HOC (B N ), preprint.en,
DEPARTMENT OF MATHEMATICS, BUCKNELL UNIVERSITY, LEWISBURG, PENNSYLVANIA
17837
Email address: pgorkinlDbucknell. edu DEPARTEMENT DE MATHEl\fATIQUES, UNIVERSITE DE METZ, ILE DU SAULCY,
F57045 METZ,
FRANCE
Email address:mortinilDponcelet.univmetz.fr DEPARTA~IENTO DE MATEMATICA, UNIVERSIDAD DE BUENOS AIRES, PAB. I, CIUDAD UNIVER
(1428) NUNEZ, CAPITAL FEDERAL, ARGENTINA Email address:dsuarezlDdm.uba.ar
SITARIA,
Contemporary Mathematics Volume 328, 2003
Characterization of conditional expectation in terms of positive projections J.J. Grobler and M. De Kock ABSTRACT. A description of positive, order continuous projections in ideals of measurable functions is given in terms of conditional expectationtype operators. The dual of such an operator can also be represented as a conditional expectation operator. We use this result to characterize conditional expectation in terms of a positive, order continuous projection that preserves one, and such that an extension of the dual also preserves one.
1. Introduction
Dodds, Huijsmans and De Pagter (see [2]) give a complete description of positive projections in ideals of measurable functions in terms of conditional expectationtype operators. Let E be an ideal of measurable functions such that Loo(O, E, Jl) C E C L1 (0, E, Jl). They characterize a positive, order continuous projection T : E + E with the property that T and the operator dual T' preserves one, in terms of conditional expectation. We extend this result to an ideal of measurable functions L with the property that Loo(O, E, Jl) C L, but we omit the assumption L C Ll (0, E, Jl). In this case we prove that a positive, order continuous projection S : L + L, which is onepreserving and with the property that an extension of its dual is also onepreserving, can be characterized in terms of the conditional expectation. In order to prove this extension, we need the fact that if an operator can be characterized in terms of conditional expectation, then its dual can also be characterized in terms of conditional expectation.
2. Preliminaries Let (0, E, Jl) be a afinite measure space. The vector lattice of all Jla.e. finite Emeasurable functions on 0, with the usual identification of Jla.e. equal functions, is denoted by Lo(O, E, Jl). M+(O, E, Jl) denotes the set of all (equivalence classes of) real, positive, Jlmeasurable functions into [0,00]. We define x Vy := sup{x, y}. A linear subspace G of a vector lattice E is called a sublattice if x Vy belongs to G for all x, y E G. A linear subspace A in which Ixl :::; IYI, with yEA implies x E A, is called an ideal. A subset AcE is order bounded if A is contained in an order interval, i.e. A is bounded from above and below. We denote the set of order bounded linear operators © lRO
2003 American Mathematical Society
190
J.J. GROBLER AND M. DE KOCK
from the vector lattice E into itself by Cb(E). Cb(E) is a Dedekind complete vector lattice (see [3), p3). All vector lattices considered will be Dedekind complete. A net (xQ)QEf in E is called order convergent if there exists a net (YQ)QEr satisfying (YQ) 1 0 and Ix  x", I ~ y", for all Q E r, where r is an index set. We write X > x in order. If Eo is a subaalgebra of E, then we denote the restriction of p, to Eo by IL again. Et denotes the collection of subsets of Eo of positive measure. The characte1"istic function of a set A E E is denoted by lA. We write 1 rather than In. For any measurable function Jon 0, the support of f is denoted by supp(J) = {w EO: f(w) =1= o}. Our attention will, to a great extent, be focused on ideals of measurable functions on (0, E, I.L), i.e., on ideals L in the vector lattice Lo(O, E, p,). The set Z E E is called an Lzero set if every .f E L vanishes p,a.e. on Z. There exists (modulo ILllUll sets) a maximal Lzero set ZI in E and the set 0 1 = 0 \ ZI is called the carrier of the ideal L. There exists a sequence An i 0 1 in E such that 1.L(An) < 00 and IAn E L for all n E N, (see [5], p143). Clearly, the carrier of L is equal to n, if and only if L is order dense in Lo(n, E, p,). Let L c Lo(O, E, p,) be an order dense ideal with order continuous dual L~. We identi(y L~ with an ideal L' of functions in Lo(n,E,p,), and we will assume that L' is again an order dense ideal (which is always the case if L is a Banach function space; (see [5), Theorem 112.1). Equivalent to this assumption is that L ~ separates f gd,.L the points of L. The duality relation between Land L' is given by (J, g) = for f ELand gEL' (see [5), Section 86). Let 8 E Cb(L) with L an ideal of functions in Lo(O, E, p,). We define its order continuous adjoint 8' : L' > L' by (g, 8' J) = (8g, J) for all f E L' and gEL (see [5], Section 97). Then 8' E Cb(L'). If there is no reason for confusion, we will denote (n, E, p,) by (E) only. Q
In
DEFINITION 1. Let E and F be vector lattices and let T : E > F be a linear operator. Then (i) T is positive (denoted by T ~ 0) whenever Tx ~ 0 for all x ~ 0; T is called strictly positive (denoted by T »0) if Tx > 0 for all x > O. (ii) T is order continuous whenever Txc< + 0 in order for every net (x satisfying X + 0 in order. (iii) T is order bounded if it maps order bounded subsets into order bounded subsets. Q )
Q
For a Banach function space (E, II . liE) defined on some finite measure space ~ E ~ Ll (E, I.L), we define the following (see [2), p69).
(n, E, p,) for which Loo(E, p,)
DEFINITION 2. (i) The linear map T : E + E is called averaging if for all f E Loo(E) and all gEE we have that T(JTg) = Tf· Tg. (ii) T: E + E is called contractive if II T II ~ 1. DEFINITION 3. Let (n, E, p,) be a probability space (i.e. 1.L(n) = 1) and let Eo be a subaalgebra of E. For fELl (E), we denote by lFP(J I Eo) the ILa.e. unique Eomeasurable function with the property that
i
lFf'(J I Eo)dl.L
=
i
fdp,
for all A E Eo. The function lFP(J I Eo) is called the conditional expectation of f with respect to Eo. If there is no reason for confusion, we will denote the p,a.e. Eomeasurable function lElL('IE o) by lE('IE o) only. The existence of lE(J I Eo) is a consequence of
CHARACTERIZATION OF CONDITIONAL EXPECTATION.
191
the RadonNikodym theorem. The conditional expectationlE('IEo) can be extended from a mapping from Ll (E) into itself, to a mapping from M+(E) into itself. If f E M+(E,), then 1E(f I Eo) E .l\J+(E) is defined by 1E(f I Eo) = suplE(fn I Eo), where 0 :::; fn E Ll (E) (71. = 1,2, ... ) satisfy 0 :::; fn i f J,La.e. The conditional expectation operator has the following properties. For a proof of properties (i) to (vi) we refer to [4], p7; for property (vii) we refer to [3], p7. 1. (i) lE(o:f + /1g I Eo) = 0:1E(f I Eo) + /11E(g I Eo) for all f,g E M+(E) and for all 0:::; 0:,/1 E R (ii) 0 :::; f :::; 9 in M+(E) implies that 0 :::; 1E(f I Eo) :::; lE(g I Eo) and if 1E(lfll Eo) = 0, then it follows that f = O. By virtue of positivity we have 11E(f I Eo)1 :::; 1E(lfll Eo). (iii) 0:::; fn i f IJ,a.e. implies that 0:::; lE(fn I Eo) i 1E(f I Eo) J,La.e. (iv) lE(gf I Eo) = glE(f I Eo) for all f E M+(E) and all 9 E M+(Eo). (v) If 9 E M+(Eo) and f E M+(E), then fA gdJ,L = J~ fdJ,L for all A E Eo if and only if 9 = 1E(f I Eo) IJ,a.e. (vi) If Eo c Ao are subaalgebras of E, then 1E(f I Eo) = 1E(1E(f I Ao) I Eo) for all 0:::; f E M+(E). (vii) If f E M+(E) is such that 1E(f I Eo) E Lo(E), then we also have that f E Lo(E). PROPOSITION
DEFINITION
4. The domain domlE('IE o) of 1E('IEo) is defined by
domlE('IE o) := {f E Lo(E) : 1E(lfll Eo) E Lo(Eo)}. Clearly, domlE(·IE o) is an ideal in Lo(E) which contains L 1 (E). For f E dom 1E(·1 Eo), we define: 1E(f I Eo) := 1E(f+ I Eo) 1E(r I Eo). This defines a positive linear operator 1E('IEo) : domlE('IE o)
+
Lo(Eo) C Lo(E).
Let (n, E, l.l) be a probability space and let L carrier n. Set
c
M(L) = {m E Lo(E) : 1E(lmfll Eo) E L
Lo(n, E, J,L) be an ideal with
V f E L}.
Since L C Lo(E), we have that mf E domlE('IE o) for all m E M(L) and f E L. For m E M(L) we define Smf : L + L by
Smf := lE(mf I Eo)
V f E L.
Sm is order bounded and ISml :::; Simi' Sm is also order continuous. The following proposition will be applied in the sequel. A proof can be found in [3], (p8). PROPOSITION
2. Let (n, E, J,L) be a probability space and Eo C E a suba
algebra. (i) If f E domlE('IE o) and 9 E Lo(Eo), then it follows that gf E domlE('IE o) and lE(gf I Eo) = glE(f I Eo). (ii) If f E Lo(E), then f E domlE('IE o) if and only if there exists a sequence {A n }:'=1 in Eo such that An in and .
{
JAn
IfldJ,L <
00
V
71.
= 1,2, ....
J.J. GROBLER AND M. DE KOCK
192
Moreover, if f E domlE(·IE o), then, for all A E Eo with
i
lEU I Eo)dft =
i
fA Ifldft < 00,
fdft·
The following lemma will be applied in t.he sequel. LEMMA
3. For a linear subspace N of Loo(E) the following statements are
equivalent. (i) There exists a subaalgebra Eo such that N = Loo(Eo). (ii) N is a subalgebra of Loo(E) containing the constants such that fn N, Ifni::; u E Loo(E) (11. = 1,2, ... ) and fn  t f a.e. imply that fEN.
E
The proofs of the following propositions and corollaries rely mainly on t.he proofs by Dodds, Huijsmans and De Pagter (see [2]). Let L c Lo(E) be an ideal of measurable functions such that Loo C L. We then have the following. PROPOSITION
4. Let S : L
t
L be an or'der continuous, positive linear opemtor
for which (i) Sf E Loo(E) whenever f E Loo(E),
(ii) SUSg) = Sf· Sg for' all f E Loo(E) and all gEL. Then there exists a subaalgebra Eo of E and there exists a 0 ::; m E M(L) such that Sf = lE(mf I Eo) for all f E L. We use the following proposition in the proof of the main result. PROPOSITION
5. For a linear operator S : L
t
L, the following statements are
equivalent.
(i) S is positive and or'der continuous, S2 = S, SI = 1 and the range R(S) of S is a sublattice.
(ii) There exist a subaalgebra Eo of E and a function 0 ::; m lE(mIEo) = 1 such that Sf = lE(mf I Eo) for all f E L.
E
M(L) with
Because the range of a strictly positive projection is a sublattice, we obtain the following result. COROLLARY
6. For a linear operator S : L
t
L the following statements are
equivalent.
(i) S is a strictly positive, order continuous project'ion with SI = 1. (ii) There exists a subaalgebra Eo of E and a strictly positive function m M(L) with lE(ml Eo) = 1 such that Sf = lE(mfl Eo) for all f E L.
E
In the following proposition we consider the case where the operator no more preserves onc, but where the image of the indicator function is strictly positive. We derive a similar result as in Corollary 6 for S strictly positive. PROPOSITION 7. Let S : L ments are equivalent.
t
L be a linear operator, then the following state
(i) S is a positive or'der continuous projection onto a sublattice such that SI is strictly posit'ive. (ii) There exist a s'nbaalgebra Eo ofE, 0 ::; m E Lo(E) and a strictly positive function k E Lo(E) with lE(mk I Eo) = 1, such that Sf = klE(mf I Eo) for all f E L.
CHARACTERIZATION OF CONDITIONAL EXPECTATION.
193
We have the following basic characterization of conditional expectation on L1 (E). A proof can be found in [2], p71. PROPOSITION 8. (Douglas R.G. and Seever) If T is a continuous linear map on L 1(E), then the following statements are equivalent.
(i) There exists a subaalgebra Eo of E such that for all f E L1 (E) we have that Tf = lEU I Eo). (ii) T is a contractive projection which preserves 1. 3. Main characterization of conditional expectation. We prove that if an operator can be characterized in terms of the conditional expectation, then its dual can also be characterized in terms of the conditional expectation. As before, we let L c Lo(E) be an ideal of measurable functions which contains Loo(E). LEMMA 9. If S : L + L is a linear operator such that Sf = lEU IEo) for all f E L, then S' : L' + L' satisfies S' 09 = lE(g I Eo) for all Emeasurable 09 E L'.
ProoF. Let f ELand gEL'. Then (j, S'g)
(Sf,g) l
SfgdJ.l
l l
lE iL U I Eo)gdJ.l
!l
n
lEiL(glEiLU I Eo) I Eo)dJ.l
llEiLU I Eo) ·1E1L(g I Eo)dJ.l l
f S'lE iL (g I Eo )dJ.l
Thus, we have proved that (3.1)
(j,S'g)
=
(j,S'IE(gIE o )).
For any Eomeasurable g, we have that (3.2) It follows from (3.2) that for Eomeasurable 9 we have that S'g = 9 and from (3.1) 0 and (3.2), for arbitrary gEL' that S'g = S'lE(g I Eo) = lE(g I Eo). Now we are able to prove the main result, where we characterize conditional expectation in terms of a positive, order continuous projection and an extension of its dual. PROPOSITION
10. If S is a linear map, then the following statements are equi
valent.
(i) Ther'e exists a subaalgebra Eo of E such that Sf = lEU I Eo) for' all f E L. (ii) S: L + L is a positive order continuous projection such that SI = 1 and S' has an extention S' : L' + Loo + L' + Loo satisfying S'I = 1.
194
J.J. GROBLER AND M. DE KOCK
Proof. (i) =} (ii) It follows from Proposition 5 that S is a positive, order continuous projection such that SI = 1. Since the conditional expectation operator is defined on L1 (E) into L1 (E), it follows that
S : L n L 1(E)
4
Denote the restriction of S to L n L1 (E) by
8' : L' + Loo(E) and 8' is an extension of S' : L' have (3.3)
4
4
L n L 1(E).
8.
Then
L' + Loo(E)
L'. Because for
f
E
L' and gEL n L1 (E), we
(g, 8' J) = (8g, J) = (Sg, J) = (g, s' J).
Since L n L1 (E) is dense in L, (3.3) holds for all gEL, and so 8' f = S' f for all f E L'. For all gEL' + Loo(E), it follows from Lemma 9 that 8'g = lE(g 1 Eo), so 8'1 = 1, by the properties of conditional expectation. (ii) =} (i) We first note that since Loo (E) c L we have L' C L1 (E) and also that L' is dense in L 1 (E). An argument of Ando (see [1], (p401)) shows that S' is contractive for the L1 (E)norm. In fact, if gEL', it follows from the assumption SI = 1 that,
In IS'
gld{t
In In IgIIS(1
S' 9 sgn S' gd{t
<
sgnS'g)ld{t
< InlgIS(lsgnS'gl)dJL <
In In
IglSld{t Igld{t.
Since L' is dense in L1 (E) for the L1 (E)norm, we can extend S' to a contraction on LdE). Since S is a projection, the same holds for S'. Thus, by Proposition 8, there exists a subaalgebra Eo of E such that S' 9 = lE(g 1 Eo) for every 9 E Ll (E) and therefore also for all gEL'. It follows from Lemma 9 that S" f = lEU 1Eo) for all f E L". By restricting S" to L, we therefore have that Sf = lEU 1 Eo) for all f E L. 0
References [1] ANDO, T., 1966, Contmctive projections in Lpspaces, Pacific J. Math., 17,391405. [2] DODDS, P.G., HUIJSMANS, C.B., and DE PAGTER, B., 1990, Chamcterizations of Conditional Expectation typeopemtors, Pacific J. Math., 141,5576. [3] GROBLER, J.J. and DE PAGTER, B., 1999, Opemtors representable as Multiplication Conditional Expectation opemtors, To appear in J. of Operator Theory. [4] NEVEU, J., 1975, Discretepammeter martingales, North Holland/American Elsevier, Amsterdam Oxford New York. [5] ZAANEN, A.C., 1982, Riesz spaces II, North Holland, Amsterdam, New York. SCHOOL FOR BUSINESS MATHEMATICS, POTCHEFSTROOM UNIVERSITY FOR CHE, POTCHEFSTROOM 2520, SOUTH AFRICA" MATHEMATICS DEPARTMENT, KENT STATE UNIVERSITY, KENT, OH 44240 Email address:srsjjgClpuknet.puk.ac.za • mdekockClmath. kent. edu
Contemporary Mathematics Volume 328, 2003
The Krull nature of locally C* algebras Marina Haralampidou ABSTRACT. Any complete locally mconvex algebra, whose normed factors in its ArensMichael decomposition are Krull algebras is also Krull. In particular, any locally COalgebra is a Krull algebra. Considering perfect projective systems, we give another proof of the fact that any Frechet locally COalgebra is a Krull algebra. Furthermore, a proper complete locally mconvex H* algebra with continuous involution and a normal unit is a locally C* algebra, hence Krull. The class of Krull (topological) algebras is closed with respect to cartesian products, topological algebra isomorphic images, and perfect projective limits.
1. Introduction and preliminaries
Every closed (left) ideal of a COalgebra E is the intersection of the (closed) maximal regular (left) ideals containing it (see, for instance, [2: p. 56, Theorem 2.9.5]. Thus, E is a Krull algebra in the sense of Definition 1.1. A natural question arises here whether, in general, any locally COalgebra is Krull. In that direction, using the ArensMichael decomposition, we get that a complete locally mconvex algebra (E,(Po)oEA) is a Krull algebra, if each factor Eo. = E/ker(po.)' Q E A is a Krull algebra (Proposition 2.1). As a consequence, we get that any locally COalgebra is Krull (Corollary 2.2). Besides, a proper complete locally mconvex H*algebra (E, (Po.)o.EA) with continuous involution and a unit element e, so that po.(e) = 1 for every Q E A, is a Krull algebra (Corollary 2.6). Based on the notion of a perfect projective system (Definition 2.7), we provide another proof of the fact that any Fn3chet locally C* algebra is Krull (Theorem 2.10). By the term topological algebm we mean an algebra, which is a topological linear space such that the ring multiplication is separately continuous (see [11: p. 4, Definition 1.1] and/or [12: p. 6]). A topological algebra E is called a Q'algebm, if every maximal regular left or right ideal in E is closed (see [5: p. 148, Definition
1.1]). A locally mconvex algebm is a topological algebra E whose topology is defined by a family (Po.)o.EA of submultiplicative seminorms, i.e. Po.(xy) :::; Po.(x)Po.(y) for 1991 Mathematics Subject Classification. Primary 46H05, 46H10, 46H20. Key words and phrases. Krull algebra, Q'algebra, ArensMichael decomposition, locally C*algebra, perfect projective system of topological algebras, perfect projective limit algebra, Frechet locally COalgebra, proper algebra, locally mconvex H*algebra. © 2003 American Mathematical Society 195
MARINA HARALAMPIDOU
196
all x, y E E, 0: E A (see for instance [11] and/or [12]). Such a topological algebra is denoted by (E, (PoJ"'EA)' A complete metrizable locally mconvex algebra E is called a Prechet locally mconvex algebra. In this case, the topology of E is defined by a countable family (Pn)nEN of submultiplicative seminorms. A C*seminorm is a seminorm P on an involutive algebra E, satisfying the C*condition, namely, p(x*x) = p(x)2 for every x E E [13: p. 1, Definition 1]. Such a seminorm is submultiplicative and *preserving [ibid. p. 2, Theorem 2]. A locally preC* algebra is an involutive locally (m) convex algebra (E, (P",)",EA), such that each p"" 0: E A is a C* seminorm, while a complete algebra, as before, is called a locally C* algebra [8: p. 198, Definition 2.2]. A Frechet locally C* algebra is an involutive Frechet locally (m) convex algebra (E, (Pn)nEN) where each Pn is a C* seminorm. A locally mconvex H* algebra is an algebra E equipped with a family (P"')"'EA of Ambrose seminorms in the sense that P"" 0: E A arises from a positive pseudoinner product <, >"" such that the induced topology makes E into a locally mconvex topological algebra. Moreover, the following conditions are satisfied: For any x E E, there is an x* E E, such that
< xy,z >",=< y,x*z >", < yx, z >",=< y, zx* >", for any y, z E E and 0: E A. x* is not necessarily unique. In case, E is proper (viz. Ex = (0), implies x = 0), then x* is unique and * : E + E : x f+ x* is an involution (see [4: p. 451, Definition 1.1 and p. 452, Theorem 1.3]). Throughout of this work the considered algebras are over the field of complexes. To fix notation we recall the following. Let (E, (P"')"'EA) be a complete locally mconvex algebra and
(1.1)
P'" : E
+
E/ker(p",) == E", : x
f+
p"'(x) :=x + ker(p",)
the respective quotient maps. Then Ilx",ll", := p"'(x), x E E, 0: E A defines on E", an algebra norm, so that E", is a normed algebra and the morphisms P"" 0: E A are continuous. E"" 0: E A denotes the completion of E", (with respect to II . II",). A is endowed with a partial order by putting 0: :::; /3 if and only if p"'(x) :::; P(3(x) for every x E E. Thus, ker(p(3) <;;;; ker(p",) and hence the continuous (onto) morphism
(1.2)
j",(3 : E(3
+
E", : x(3
f+
j",(3(x(3) := x""
0::::;
/3
is defined. Moreover, j ",(3 is extended to a continuous morphism !",(3 : E(3
+
E""
Thus, (E"" j",(3), (E",'/",(3), 0:, /3 E A with (resp. Banach) algebras, so that
(1.3)
E
2:!
0: :::;
0::::;
/3.
/3 are projective systems of normed
 
lim E", 2:! lim E", (ArensMichael decomposition)
within topological algebra isomorphisms (cf., for instance, [11: p. 88, Theorem 3.1 and p. 90, Definition 3.1] and/or [12: p. 20, Theorem 5.1]). Concerning the following notion see [7]. DEFINITION 1.1. A topological algebra is called a Krull algebra, if every proper closed left (resp. right) ideal is contained in a closed maximal regular left (resp. right) ideal.
THE KRULL NATURE OF LOCALLY C'ALGEBRAS
197
For the statements (i) and (iii) in the next proposition see [7: Lemma 3.8). PROPOSITION 1.2. Let E, F be topological algebms and ¢ : E 7 F a continuous epimorphism. Then the following hold true: (i) If E is a Krull algebm and ¢ closed, then F is a Krull algebm. (ii) If E is a Krull algebm and F a Q'algebm, then F is a Krull algebm. (iii) If F is a Krull algebm and ¢ closed with ker(¢) ~ I for every proper closed left or right ideal in E, then E is a Krull algebm. (iv) If F is a Krull algebm and ¢ a closed injection, then E is a Krull algebm. PROOF. (ii) For a proper closed left ideal J in F, ¢l(J) is a proper closed left ideal in E with ker(¢) ~ ¢l(J) and J = ¢(¢l(J)) (see also [3: p. 316, Proposition B.5.4)). Thus, ¢l(J) ~ M for some closed maximal regular left ideal Min E. Hence J ~ ¢(M), so that ¢(M) is a maximal regular left ideal in F (ibid.), closed by Q'. Similarly, for proper closed right ideals. (iv) Immediate from (iii). 0 COROLLARY 1.3. A topological algebm is a Krull algebm if and only if a topological algebm isomorphic image of it is so. PROPOSITION 104. Let (E"')"'EA be a family of topological algebms and F = II"'EA E", the respective cartesian product topological algebm. Then F is a Krull algebm, if each E"" Q E A is a Krull algebm. The converse is true in case the factors are Q' algebms. PROOF. Consider the canonical continuous epimorphisms (projections)
(1.4)
11"", :
F
7
E", : x = (X"')"'EA
1+
1I"",(x)
:=
x""
Q
E A.
Let I be a proper closed left ideal in F. Since the multiplication is separately continuous, it follows that the closure 11"",(1) of the left ideal 11"", (I), is a closed left ideal in E",. Moreover, 11"",(1) i E", for some Q E A. Otherwise, II"'EA 11"",(1) = F. It is easily seen that n"'EA 11";;1 (E",) = ILEA E",. Besides, I = I = n"'EA 11";;1(11"",(1)). Hence I = II"'EA E", = F, a contradiction. Now, since E", is a Krull algebra, 11"",(1) ~ M, for some closed maximal regular left ideal M. Hence I ~ 1I";;1(M) with 1I";;1(M) a closed maximal regular left ideal in F. Similarly, for proper closed right ideals. The above argument shows that F is a Krull algebra. 0 For the rest of the assertion apply (ii) of Proposition 1.2. 2. The Krull property for locally C*algebras
We provide first the following result akin to that of Proposition 104. PROPOSITION 2.1. Let (E, (P"')"'EA) be a complete locally Tnconvex algebm, such that the norrned algebms E"" Q E A in its ArensMichael decomposition, are Krull algebms. Then E is a Krull algebm, as well. On the other hand, if E is a Krull algebm, then a factor E", is a Krull algebm if it is also a Q' algebm. PROOF. E ~ lim E", within a topological algebra isomorphism, say ¢ (see f(1.3)). Consider the continuous epimorphic image p",(I) (see (1.1)) of a proper closed left ideal I in E. Claim that the closed left ideal p",(I) is proper in E", for
MARINA HARALAMPIDOU
198
some Q E A. Suppose the contrary. Then based on M. Exarchakos, concerning the first equality of the next rels, we get ¢(E)
= ll!!!Ea = ll!!!Pa(I) = ll!!!(fa(¢(I))) = ¢(I),
here fa denotes the restriction to ll!!! Eo of the projection map 7ra : TIaEA Eo + Eo, Q E A (cf. also [11: p. 87, Lemma 3.2 and p. 89, (3.24)]; we note that ¢(I) is a closed left ideal in lim Eo). Thus, E = I, which is a contradiction. So, since tEa, Q E A is a Krull algebra, it follows that Pa(I) ~ M for some closed maximal regular left ideal M, and hence I ~ p~l(Pa(I)) ~ p~l(M). Besides, p~l(M) is a maximal regular left ideal in E, closed by the continuity of Po. An analogous result holds for proper closed right ideals. The last part of the assertion follows from (ii) of Proposition 1.2. 0
By [1: p. 32, Theorem 2.4], the factor normed algebras, in the ArensMichael decomposition of a locally C*algebra, are C*algebras and hence Krull (see, for instance [2: p. 56, Theorem 2.9.5]). Thus, Proposition 2.1 implies the next. COROLLARY 2.2. Every locally C*algebm is a Krull algebm. By Proposition 1.4 and Corollary 2.2, we get the next. COROLLARY 2.3. The cartesian product of locally C* algebras is a Krull (locally C* ) algebm. In view of Corollary 2.2, Theorem 4.7 in [6: p. 3732] is improved as follows: THEOREM 2.4. A locally C* algebm is dual if and only if it is complemented. Let (E, (Pa)aEA) be a proper complete locally mconvex H* algebra with continuous involution. Then E can be made into a locally preC* algebra, via a family (qa)aEA of C*seminorms given by (2.1)
so that, (2.2)
qa(X) ~ Po(x) for every x E E,
Q
E A.
(Namely, the respective topology on E is weaker than the given one). Moreover,
(2.3)
Po(xy) ~ qa(x)Pa(Y) for every x, y E E,
Q
E A.
(See [9: p. 265, Proposition 2.3]). In that framework we get the next two results. PROPOSITION 2.5. Let (E, (Po)oEA) be a proper complete locally mconvex H*algebm with continuous involution and a unit element e. Then the following are equivalent: 1) po(e) = 1 for every Q E A (: normal unit). 2) (E, (Pa)aEA) is a locally C*algebm. PROOF. 1) ==> 2): Let qa, Q E A be the seminorms given by (2.1). By (2.3), p",(x) ~ qo(x) for every x E E, Q E A. Hence (see also (2.2)) Po = qa for every Q E A. Namely, (E, (Pa)aEA) is a locally C*algebra. 2) ==> 1): C*property implies Pa(e)(lpo(e)) = 0 for every Q E A. If po(e) = 0 for some Q E A, then Ileoli o = 0, where eo = e + ker(Pa) is the respective unit
THE KRULL NATURE OF LOCALLY C'ALGEBRAS
199
element in the factor algebra EOl == EOl (see aslo [1: p. 32, Theorem 2.4] and [11: p. 91, Theorem 4.1]). Thus eOl = 0, which is a contradiction. Thus, pOl(e) =I 0 for every Q E A, hence POl (e) = 1 for every Q E A. 0 As a consequence of Corollary 2.2 and Proposition 2.5 we have the next. COROLLARY 2.6. Every proper complete locally mconvex H* algebm with continuous involution and a normal unit is a Krull algebm. Our next aim is to provide another proof to the fact that a F'rechet locally C*algebra is Krull (see Corollary 2.2). To do this, we use the notion involved in the next. DEFINITION 2.7. A projective system {(EOl , fOl,8)}OlEA of topological algebras is called perfect, if the restrictions to the projective limit algebra
= ~EOl = {(x Ol ) E
II
EOl: fOl,8(X,8) = XOl , if Q:S; (3 in A} OlEA of the canonical projections 7r0l : I10lEA EOl + E Ol , Q E A, namely, the (continuous algebra) morphisms
(2.4)
(2.5)
E
fOl = 7rOl IE =limE", : E
+
EOl ,
Q
E A,
t
are onto maps. The resulted projective limit algebra E = lim EOl is called perfect (topological) algebm. ~
LEMMA 2.8. Every Frechet locally mconvex algebm (E, (Pn)nEN) gives a perfect projective system of normed algebms. PROOF. For any n
:s; m in N, the connecting maps
(2.6)
with
fnm(X + ker(Pm)) = x + ker(Pn) are onto algebra morphisms (see, for instance, [11: p. 86, (3.6) and (3.7)]). So, since {(En, fnm)}nEN is a denumerable projective system of normed algebras, it follows that fn, n E N (see (2.5)) are onto, as well (see [10: p. 229, Theorem 8]). 0 The proof in the next result is an adaptation of that in Proposition 2.1. PROPOSITION 2.9. Any perfect projective limit of Krull algebms is a Krull algebm. PROOF. Let {(EOl,fOl,B)}OlEA be a perfect projective system of Krull algebras. Consider the projective limit algebra E = limEOl (see (2.4)), which is a closed subalgebra of the cartesian product topological algebra I10l EA EOl (see, for instance, [11: p. 84, Lemma 2.1]). For a proper closed left ideal I in E, fOl(I) is a (closed) left ideal in E Ol , Q E A. If f Ol (I) = EOl for every Q E A, then ~
1= limfOl(I) ~
= limfOl(I) = limEOl = E, ~
~
(see also [ibid. p. 87, Lemma 3.2]), which is a contradiction. Thus, fOl(I) =I EOl for some Q E A. Since E Ol , Q E A is a Krull algebra, there exists a closed maximal regular left ideal, say M, with fOl(I) ~ M and hence I ~ f;;l(1Ol(1)) ~ f;;l(M),
200
MARINA HARALAMPIDOU
where J;;l(M) is a closed maximal regular left ideal in E and this terminates the proof for closed left ideals. Similarly, for closed right ideals. D THEOREM
2.10. Any Prichet locally C*algebm is a Krull algebm.
PROOF. Let (E, (Pn)nEN) be an algebra as in the statement. By [1: p. 32, Theorem 2.4], the respective normed algebras En, n E N in the ArensMichael decomposition of E, are C* algebras and hence Krull (see, for instance, [2: p. 56, Theorem 2.4.5]. In particular, {(En' Jnm)}nEN is a perfect system of normed algebras (see Lemma 2.8 and relation (2.6)). Proposition 2.9 assures that the projective limit algebra lim En is a Krull algebra and hence E is a Krull algebra, as it fol+lows from Corollary 1.3 and the fact that E ~ lim En within a topological algebra +isomorphism (see (1.3)). D References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
C. Apostol, b*algebms and their representation, J. London Math. Soc. 3(1971), 3038. MR 44:2040. J. Dixmier, C*Algebms, NorthHolland, Amsterdam, 1977. MR 56:16388. R.S. Doran and V.A. Belfi, Chamcterizations of COAlgebras. The Gel'fandNa'tmark Theorems, MarcelDekker, 1986. MR 87k:46115. M. Haralampidou, On locally convex H*algebms, Math. Japon. 38(1993), 451460. MR 94h:46088. M. Haralampidou, Annihilator topological algebras, Portug. Math. 51(1994), 147162. MR 95f:46076. M. Haralampidou, On complementing topological algebms, J. Math. Sci. 96(1999), 37223734. MR 2000j:46085. M. Haralampidou, On the Krull property in topological algebms (to appear). A. Inoue, Locally C* algebras, Mem. Faculty Sci. Kyushu Univ. (SerA) 25(1971), 197235. MR 46:4219. A. EI Kinani, On locally preC*algebm structures in locally mconvex H*algebms, Thrk. J. Math. 26(2002), 263271. G. Kothe, Topological Vector Spaces, I, SpringerVerlag, Berlin, 1969. MR 40:1750. A. Mallios, Topological Algebms. Selected Topics, NorthHolland, Amsterdam, 1986. MR 87m:46099. E.A. Michael, Locally multiplicativelyconvex topological algebms, Mem. Amer. Math. Soc. 11(1952). (Reprinted 1968). MR 14,482a. Z. Sebestyen, Every C*seminorm is automatically 8ubmultiplicative, Period. Math. Hung. 10(1979), 18. MR 80c:46065.
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ATHENS, PANEPISTIMIOPOLlS, ATHENS 15784, GREECE Email address:mharalamOcc.uoa.gr
Contemporary Mathematics Volume 328, 2003
Characterizations and automatic linearity for ring homomorphisms on algebras of functions Osamu Hatori, Takashi Ishii, Takeshi Miura, and SinEi Takahasi ABSTRACT. Automatic linearity results for certain ring homomorphisms between two algebras, in particular, semisimple commutative Banach algebras with units are proved. For this purpose a representation by using the induced continuous mapping between the maximal ideal spaces and ring homomorphisms on the field of complex numbers is given. Ring homomorphisms on certain noncomplete metrizable algebras into the algebras of analytic functions are also considered. A characterization of the kernel of complexvalued ring homomorphism on a commutative algebra is given. As a corollary of the results a complete description of ring homomorphisms on the disk algebra into itself is given in terms of prime ideals.
Introduction A ring homomorphism between two algebras is a mapping which preserves addition and multiplication. If we assume that the mapping is linear, then it is an ordinary homomorphism. In the case where the two algebras are just the field C of complex numbers, the assumption cannot be avoided; there are ring homomorphisms of C into C which are not linear nor conjugate linear (cf. [9]). The history of ring homomorphisms on C probably dates back to the investigation of Segre [19] in the nineteenth century and that of Lebesgue [12]. A similar remark applies to finitedimensional Banach algebras. But this is not the case for several infinitedimensional ones; for instance, Arnold [1] proved that a ring isomorphism between the two Banach algebras of all bounded operators on two infinitedimensional Banach spaces is linear or conjugate linear (cf. [5]). Kaplansky [8] proved that if p is a ring isomorphism from one semisimple Banach algebra A onto another, then A is a direct sum Al EB A2 EB A3 with A3 finitedimensional, p linear on All and p conjugate linear on A 2 . It follows that a ring isomorphism from a semisimple commutative Banach algebra onto another with infinite and connected maximal ideal space is linear or conjugate linear. 2000 Mathematics Subject Classification. Primary 46JlO, 46E25; Secondary 46J40. The first, the second, and the fourth author were partialy supported by the GrantsinAid for Scientific Research, The Ministry of Education, Science, Sports and Culture, Japan.
© 201
2003 AJnerican Mathematical Society
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O. HATORI, T. ISHII, T. MIURA, AND S.E. TAKAHASI
It is interesting to study ring homomorphisms on Banach algebras which are not necessarily injective or surjective. We may expect that a number of ring homomorphisms on infinitedimensional Banach algebras are automatically linear or conjugate linear. By a routine work we see that a a ring homomorphism is reallinear if it is continuous. On the other hand, we can also arrive at automatic linearity for several ring homomorphisms by results in [14, 15, 20, 21, 13]; they studied and characterized *ring homomorphisms between commutative Banach algebras with involutions and ring homomorphisms on regular commutative Banach algebras with additional assumptions. The heart of this paper is automatic linearity results for certain ring homomorphisms of a much more general nature. Throughout the paper A and B denote semisimple commutative Banach algebras with units eA and eB respectively. The maximal ideal space for A is denoted by MA. In this paper, we denote the Gelfand transform of a E A also by a. For a ring homomorphism of C into C, we simply say a ring homomorphism on C. Let T be a ring homomorphism on C and x E MA. Then the complexvalued mapping p on A defined by
p(a) = T(a(x)),
aEA
is a typical example of a ring homomorphism. Semrl [20, Example 5.4] showed that there exists a complexvalued ring homomorphism other than this type. In section 2 we show that if a ring homomorphism of A into B satisfies a certain condition, say (m), then it is represented by a modified version of the above. Many ring homomorphisms satisfy this weak and rather natural condition (m): *ring homomorphisms on involutive algebras; p{A)(y) = C for every y E M B ; p(A) contains a subalgebra of B. Thus our result generalizes the previous ones in [14, 20, 21, 13]. In section 3, by using results in section 2, we deduce some automatic linearity results for ring homomorphisms: p with (m) is reallinear on a closed ideal of finite co dimension in A; if p(CeA) = CeB and p(A) contains an element with an infinite spectrum, then p is linear or conjugate linear. It is a natural question: under the two hypotheses (1) p(CeA) c CeB and (2) p(A) contains an element with an infinite spectrum, does it follow that p is linear or conjugate linear? We give an affirmative answer under stronger hypotheses: (1) and (2)' p(A) contains an element whose spectrum contains a nonempty open subset. Problems in the same vein are also considered not only for Banach algebras but also for algebras of analytic functions. Bers [3] proved that if U and V are plane domains and H(U), H(V) are the rings of analytic functions on U, V respectively, then any ring isomorphism of H(U) onto H(V) is induced by a conformal (or anticonformal) equivalence of V with U, thus the ring isomorphism is linear (or conjugate linear). Nakai [17] and Rudin [18] have shown this also holds for open Riemann surfaces. Ring homomorphisms which are not necessarily injective or surjective are also considered by many mathematicians (cf. [7, 10]). Among them, Becker and Zame [2] have proved automatic continuity and linearity for ring homomorphisms from certain complete metrizable topological algebras into the algebra of analytic functions on connected, reduced analytic spaces. In section 4 we also consider ring homomorphisms into the algebras of analytic functions. In particular, we consider the case of a ring homomorphism p from the algebra Rs of rational functions on C with poles off a subset SeC into an algebra of analytic funtions. Here Rs is a metrizable topological algebra, but it cannot be
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a Banach algebra by the Baire category theorem. We show that, for certain subsets S, p is automatically linear or conjugate linear, if the range of p contains a 11011constant function. We also give an example of S such that a ring hommomorphism that is neither linear nor conjugate linear, and whose range contains nonconstant functions, is possible. In the final section we study ring homomorphisms into C: ring homomorphisms whose ranges contain only constant functions. We characterize the kernels of ring homomorphisms from a unital commutative algebra into C, which is compared with the one to one correspondence between maximal ideals and complex (linear) homomorphisms on commutative Banach algebras. As a corollary we show that there exists an injective ring homomorphism from an algebra which consists of analytic functions into C. We also give a complete description of the ring homomorphisms on the disk algebra in terms of prime ideals. We say that a ring homomorphism 7 on C is trivial if 7 = 0 or 7(Z) = Z (resp. z) for every Z E C. Other ring homomorphisms on C are said to be nontrivial. We note some properties of nontrivial ring homomorphisms on C, which are used later in this paper. For a proof of the existence of nontrivial ring homomorphisms, historical comments, and further properties, see [9]. It is easy to see that every nonzero ring homomorphism 7 on C fixes rational real numbers and 7(i) = i or i. If 7 is nontrivial, then 7 does not preserve complex conjugation. (This is a standard fact. Here is a proof. Suppose 7 does preserve complex conjugation: 7(Z) = 7(Z) for every Z E C. Then 7(JR) C JR, that is, 7 is a ring homomorphism on the set of all real numbers R If x> 0, then 7(X) = (7( JX))2 > O. It follows that 7 is order preserving on R Since 7(r) = r for every rational real number r, we have 7(X) = x for every real number x. Thus 7(Z) = Z (resp. 7(Z) = z) for every Z E C if 7(i) = i (resp. 7(i) = i), which is a contradiction.) It is easy to see that T is nontrivial if and only if 7 is discontinuous at every (resp. one) point in C. Thus, if T is nontrivial, then it is unbounded on every neighborhood of zero. It follows that there exists a sequence {w n } of complex numbers which converges to 0 such that IT(Wn)1 tends to infinity as n > 00 if 7 is nontrivial. If the ring homomorphism on C is onto, then it is said to be a ring automorphism on C. Note that there is a nonzero ring homomorphism on C which is not a ring automorphism. We also note that there is a nontrivial ring automorphism on C (cf. [9, 11]).
1. Partial representation If ¢ is a nonzero complex homomorphism on A, then there exists a unique x E MA such that ¢(a) = a(x) for every a E A. By this fact a wellknown representation of a (linear) homomorphism VJ from A into B follows: There exists a continuous mapping defined on {y E MB : VJ(a)(y) :I 0 for some a E A} into MA such that
VJ(a)(y)
= a((y)),
a E A,
y E {y E MB : VJ(a)(y)
:I 0 for
some a E A}.
On the other hand, if is a continuous mapping of MB into AfA and Ty is a ring homomorphism on C for every y E AfB , then
p(a)(y) = Ty(a((y)),
a E A,
y E MB
defines a ring homomorphism from A into the algebra of all complexvalued functions on M B . Thus it defines a ring homomorphism from A into B under the condition that T. (a ( (. )) is in B for every a E A, and this is the case when M B
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is finite. A problem is the converse: Is every ring homomorphism represented as above? A negative answer is known even in the case where B = C by the example due to Semd [20, Example 5.4]. Nevertheless, we show that a partial representation is still possible in this section. DEFINITION 1.1. Let p be a ring homomorphism of A into Band y a point in M B . The induced ring homomorphism py of A into C is defined by
py(a) = p(a)(y), Let Ie : C
+
A be defined by IdA)
= AeA
aE
A.
for every A E C. We denote Ty
= pyole.
For every y E M B , the induced mapping Ty is a ring homomorphism on C. DEFINITION 1.2. Let p be a ring homomorphism of A into B. We denote: Mo = {y E MB : Ty = a}; !vlt = {y E MB : Ty(Z) = Z for every Z E C}; ALl = {y E MB : Ty(Z) = Z for every Z E C}; Md,l = {y E MB : Ty is nontrivial and Ty(i) = i}; Md,l = {y E !vIB : Ty is nontrivial and Ty(i) = i}. LEMMA 1.3. Let p be a ring homomorphism of A into B. Then M o, Ml U Md,l and M1 UMd,l are clopen (closed and open) subsets of MB. The subsets M1 and Ml are closed in M B . PROOF. By the definitions it is easy to see that Mo = {y E MB : p(ieA)(y) = a}, M1 U Md,l = {y E MB : p(ieA)(y) = i}, and ALl U Md,l = {y E MB : p(ieA)(y) = i}, so they are clopen since p(ieA) is continuous on M B . Next we show that Jl,ft is a closed subset of MB. Let y E Md,l' Since Ty is nontrivial, there exists a complex number A such that Ty(A) =I A. Put
Then G is an open neighborhood of y. We also see that G n M1 = 0. It follows that ]\,{1 is a closed subset of MB since M1 U M d,l is clopen. In the same way, we see that M1 is a closed subset of M B . 0 Suppose that p is a ring homomorphism of A into B. If y E M 1 , then it is easy to see that Py is a nonzero complex homomorphism on A. Thus there exists a unique cp(y) in MA with
p(a)(y) = a(CP(y)),
a E
A.
In a way similar to the above we arrive at a partial representation as follows:
p(a)(y) =
a, { a(CP(y)), a(CP(y)),
yEMo, yE M 1 , y E M_ 1 .
If y E Md,l U Md.1, then the situation is complicated, in particular, ring homomorphisms with large Md,l U Md,l are possible (cf. [20, Examples 5.3 and 5.4]).
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2. Ring homomorphisms which satisfy the condition (m) In general the kernel of a nonzero ring homomorphism of A into C is a prime ideal and need not be a maximal ideal. (See section 5 in this paper.) In this section we consider ring homomorphisms P of A into B which satisfy the condition that the kernel of the induced ring homomorphism Py for each y E MB defined by
Py(f) = p(f)(y),
f EA
is a maximal ideal. DEFINITION 2.1. Let P be a ring homomorphism of A into B. We say that P satisfies the condition (m) if Py is zero or ker Py is a maximal ideal of A for every yEMB. By the following Lemma 2.2, if py(A) = C for every y E M B , in particular, if p(A) :J CeB, then (m) is satisfied. A *ring homomorphism also satisfies the condition (m). (See the proof of Corollary 2.5.) LEMMA 2.2. Let Po be a nonzero ring homomorphism of A into C. Then the following are equivalent. (1) The kernel ker Po of Po is a maximal ideal of A. (2) The equation po(A) = PO(CeA) holds. (3) There exist a nonzero ring homomorphism 7 on C and an x E MA such that the equation po(a) = 7(a(x)) holds for every a E A. In this case 7 = Po 0 Ie. Such a 7 and x are unique. (4) The mnge Po(A) is a subfield ofC which contains a nonzero complex number. PROOF. First we show that (1) implies (2). Suppose that ker Po is a maximal ideal. Then there exists a nonzero complex homomorphism
p(a)(y) = {7y(a(
yEMo.
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In particular, p(a)(y) = a(q>(y)) for every y E Ml and p(a)(y) = a(q>(y)) for every y EM_I' The set q>(Md,l U Md,d is a (possibly empty) finite subset of MA and q>l(X) n (Md.l U Md,d is an open subset of MB for each x E q>(Md,l U Md,d. PROOF. Let y E MB \ Mo. Then by the condition (m) ker PY is a maximal ideal of A, so by Lemma 2.2 there exists a unique q>(y) E MA such that
py(a) = PY o Ic(a(q>(y))) holds for every a E A. By the definition Ty = py 0 Ie and since py(a) = p(a)(y) we see that p(a)(y) = Ty(a(q>(y))) holds for every a E A. If y E M l , then Ty(A) = A for every A E C, so p(a)(y) = a(q>(y)). If y EM_I, then Ty(A) =). for every A E C, so p(a)(y) = a(q>(y)). If y E M o, then p(a)(y) = O. Put Md = Md,l UMd,l. We show that q>(Md) is a finite subset of MA. Suppose not. Then there is a countable subset {Xn}~=l of q>(Md). For each n choose a point Yn E Md with q>(Yn) = Xn · Since TYI is unbounded near zero, there exists an al E A such that lIalli < 2 1 and hI (al(xd)1 > 2. By induction we can find, for every n, an E A such that an(xd = ... = an(xnd = 0, lIanll < 2 n , and
ITYn(an(xn))1 > 2n + ITYn(al(Xn) + ... + anl(xn))l· (Choose b2 E A with b2 (xd = 0 and b2(X2) = 1. Since TY2 is unbounded near zero, there is a nonzero complex number 02 such that 1021 < Ilb211 l 2 2 and ITY2(0)1 > 22+ITy2 (al(x2))I· Then put a2 = 02b2. We have a2(xl) = 0, IIa211 < 2 2, and ITy2 (a n (x2))1 > 22+ITy2 (al(x2))1. Suppose that al, .. ' ,anl E A are choosenso that the conditions are satisfied. Choose bn E A with bn(xd = ... = bn(xnd = 0 and bn(xn) = 1. Since TYn is unbounded near zero, there is a nonzero complex number On such that 10nl < Ilbn ll 1 2 n and
ITYn (on)1 > 2n + ITYn (al(xn) + ... + anl(xn))l· Then a2 = onbn is a desired function for n.) Then E::'=l an converges in A, say to a. Then a(xn) = al(x n ) + ... + an(xn) since the Banach norm on A dominates the uniform norm on AlA. On the other hand
so that p(a) is unbounded, which is a contradiction proving that q>(Md) is a finite set. Let q>(Md) = {Xl, ... , xn} and Yj = q>l(Xj) n Md for each j = 1,2, ... , n. Choose an a E A such that a(xl) = 1, a(x2) = ... = a(xn) = O. Then p(a)(y) = 1 if y E Y l while p(a)(y) = 0 if y E Md \ Y l . Because p(a) is continuous, Y1 is clopen in Md; but Md is open in M B , so Y l is open in MB. In the same way we see that Yj is an open subset of MB for each j = 2,3, ... n. Finally we prove that q> is continuous. Since Yj is open and q>(Yj) = Xj, we only need to prove that q> is continuous at each point in Ml U M_ l . Let y E Ml and {y>.hEA be a net which converges to y. Without loss of generality we may assume that {y>.} c Ml U Md,l since Ml U Md,l is clopen. Suppose that {q>(y>.)} does not converge to q>(y), that is, there is an open neighborhood G of q>(y) such that for every A E A there exists a A' 2:: A with q>(YN) rf. G. There exist a finite number of points aI, ... , am in A and a positive real number e such that
{x
E
MA : laj(x)  aj(q>(y))1 < e,j = 1,2, ... , m}
C
G.
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Since {p(aj)(y,x)} converges to p(aj)(Y) for each j, there exists a Ao E A such that
JTy>. (aj(iP(y,x)))  aj(iP(y))J < e holds for every A ~ AO and j = 1,2, ... , m. Suppose that A ~ Ao, then there exists a A' ~ A such that iP(y,x/) f/. G, so that Jajl(iP(y,x/))  ajl(iP(y))J ~ e for some j'. lt follows that YN E Md,l. We also see that iP(y,x/) E {Xl, ... ,Xn } \ {iP(y)}. There exists an a E A such that a(iP(y)) = 1 and a = 0 on {Xl, ... ,Xn } \ {iP(y)}. We conclude that for every A with A ~ Ao there exists a A' ~ A such that p(a)(y,x/) = 0 and p(a)(y) = 1, which is a contradiction since p(a) is continuous on M B . Thus we have that {iP(y,x)} converges to iP(y), so iP is continuous at y. In the same way we see that iP is continuous at each point in M_ I . We have proved that iP is continuous on MB \Mo. 0 Note that the set Md,l U Md,l need not be a finite set or even a closed subset of AlB (cf. [20, Example 5.3]). In [21] the authors proved the following corollary in the case where A is regular and satisfies a certain additional condition. Now we can remove these conditions. COROLLARY 2.4. Let p be a ring homomorphism from A into B. Suppose that py(A) = C for every y E M B . Then there exists a continuous mapping iP of MB into MA and a nontrivial ring automorphism Ty on C for every y E Md,l U Md,l s1Lch that Y E MI , a(iP(Y))' { p(a)(y) = a(iP(y)), Y EM_I, Ty(a(iP(y))), Y E Md,l U Md,l.
Moreover iP(Md,1 U Md,d is a finite subset of MA. PROOF. By Lemma 2.2 we see that ker py is a maximal ideal, so the condition (m) is satisfied. The conclusion follows by Theorem 2.3. In particular, Ty = Py 0 Ie is onto, thus it is a nontrivial automorphism on C for y E Md,l U Md,l. 0
Theorem 2.1 in [13] for the case of unital and semisimple commutative Banach algebras is also deduced from Theorem 2.3 COROLLARY 2.5. Suppose that A is involutive and B is symmetrically involutive. Let p be a *ring homomorphism. Then MB = Mo U MI U MI and there exists a continuous function iP from MB \ Mo into AlA such that
a(iP(Y))' p(a)(y) = { 0, :a('=iP..,...( y77")) ,
yE
MI ,
yE Mo,
y EM_I·
PROOF. Since p is a *ring homomorphism, it it easy to see that Ty(Z) = Ty(Z) for every Z E C and for every y E MB \ Mo. It follows that Ty is 0 or linear or conjugate linear. Thus the conclusion follows. 0
3. Automatic linearity One of the reasons for ring homomorphisms between infinitedimensional Banach algebras to be linear or conjugate linear is that the range contains an element with large spectrum. In this section we show evidence of this.
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O. HATORI, T. ISHII, T. MIURA, AND S.E. TAKA HAS I
COROLLARY 3.1. Let p be a ring homomorphism of A onto B. If MB contains no isolated point, then p is reallinear. If MB is infinite and connected, then p is linear or conjugate linear. PROOF. Since p is a surjection, py(A) = C for every y E M B , so the condition (m) is satisfied by Lemma 2.2. We also have that the induced mapping is injective. Thus Md,l U Md,l is a (possibly empty) finite set. Because Md,l U Md,l is open (by Lemma 1.3), each point of Md,l U Md,l is isolated in M B . If MB contains no isolated point, then Md,l U Md,l = 0. Thus p is reallinear. If lvIB is infinite and connected, then MB contains no isolated point, so Md,l U Md,l = 0. It follows by Lemma 1.3 that MB = lvh or MB = M_ l . Thus P is linear or conjugate linear. 0 COROLLARY 3.2. Let p be a ring homomorphism of A into B. Suppose that p satisfies the condition (m). Then there exists a (possibly empty) finite subset {Xl, ... , xn} of M A such that p is reallinear on the finitecodimensional closed ideal {a E A: a(xj) = O,j = 1,2, ... ,n} of A. PROOF. Put {Xl,""X n } = (Md,l UMd,d. (The set is finite by Theorem 2.3.) Then for every a E {a E A: a(xj) = O,j = 1,2, ... ,n} p(a)(y)
=
{Ty(a((Y))), 0,
y E Ml U M_l' Y E Mo U Md,l U Md,l.
Since Ty is reallinear for every y E Ml U M_l' the conclusion follows.
o
COROLLARY 3.3. Let p be a ring homomorphism from A into B such that p(CeA) = CeB. Then we have that Mo = 0, and there exists a continuous mapping from MB into MA such that one of the following three occurs. (1) P is linear: p(a)(y)
= a((y)),
a E A,
y E MB .
a E A,
y E MB .
(2) p is conjugate linear: p(a)(y)
= a((y)),
(3) There exists a nontrivial ring automorphism T on C such that p(a)(y)
= T(a((y))),
a E A,
y E M B.
In particular, if there exists an a E A such that the spectrum of p( a) is an infinite set, then p is linear or conjugate linear.
PROOF. For every y E MB, we have py(CeA) = C, so py(A) = py(CeA) = c. Thus ker Py is a maximal ideal of A by Lemma 2.2, so that the condition (m) is satisfied. Since p(ieA) E CeB, MB = Ml U Md,l or MB = M_l U Md,l. Suppose that Md,l U Md,l = 0. Then (1) or (2) occurs. Suppose that there exists some Yd E Md,l and some Yl E M l · Then there is a complex number>' with Tyd (>') =I= >., so that p(>.eA) is not a constant function, which contradicts our hypothesis. Thus Md,l =I= 0 implies that MB = Md,l. It is also easy to see that Ty is identical for every y E Md,l since p(CeA) consists of constant functions. Thus (3) follows. In the same way we see that (3) follows if Md,l =I= 0. Suppose that there exists an a E A such that p(a)(MB) is infinite, then (3) does not occur since ( M B) is a finite set in this case. It follows that p is linear or conjugate linear. 0
AUTOMATIC LINEARITY FOR RING HOMOMORPHISMS
209
Corresponding results for ring homomorphisms on rings of analytic functions are proved by Kra [10, Theorem I]. Suppose that p is a ring homomorphism from A into B which satisfies two conditions: p(CeA) C CeB; there exists an a E A such that p(a)(MB) is infinite. Does it follow that p is linear or conjugate linear? Although the authors do not know the answer, we can provide a positive answer under a stronger condition. THEOREM 3.4. Let p be a ring homomorphism from A into B. Suppose that the following two conditions are satisfied: (i) p(CeA) C CeB; (ii) there exists an a E A such that p(a)(MB) contains a nonempty open subset ofC. Then p is linear or conjugate linear. PROOF. Since p(CeA) C CeB we may suppose that pole is a nonzero ring homomorphism on C. We have two possibilities: po Ie(i) = i; po Ie(i) = i. We show that, in the first case, po Ie(z) = z for every complex number z, so it will follow that p is linear on A. (In the same way we see that p is conjugate linear if pole( i) = i.) Suppose that pole( i) = i. We show that pole is continuous on C. For this it is enough to show that pole is continuous at O. Suppose not. Then there is a sequence {w n } of nonzero complex numbers which converges to 0 such that {poIe(w n )} does not converge to O. Without loss of generality we may assume that Ip 0 Ie(w n ) I ~ 00 as n ~ 00. Let a be in A such that p(a)(MB) contains a nonempty open subset G of the complex plane. Let s be a complex number in G such that the real part and the imaginary part of s are both rational numbers. Put Zn = S + 1/ po Ie(w n ). Then there is a positive integer mo such that Zm E G for every m ~ mo since Ip 0 Ie(w n )I ~ 00 as n ~ 00, so ZmeB  p(a) ~ B 1 . Thus we have (8 + l/wm)eA  a ~ AI. Then 8 + l/wm is in the spectrum of a for every m ~ mo, which is a contradiction since Is + l/wnl ~ 00 as n ~ 00. It follows that pole is continuous at 0, thus on C, so pole( w) = w for every complex number w since p(ieA) = i. Then we see that p is linear on A. D Note that either of the two conditions (i) and (ii) in the above theorem itself does not suffice for p to be linear or conjugate linear. Let T be a nontrivial ring automorphism on C. Suppose that x E MA and tp from A into C is defined by tp(a) = a(x) for every a E A. Put P = TO tp. Then p is a ring fomomorphism with (i) since p(CeA) = C, but P is neither linear nor conjugate linear; p is not even reallinear. Let D be the closed unit disk in the complex plane. Let D + 3 = {z E C: Iz  31::; I} and X = D U (D + 3). Define p(f)(z) = {f(Z)' f(z  3),
zED zED+3
for every f E C(D). Then p is a ring homomorphism from C(D) into C(X) with the condition (ii). But p is neither linear nor conjugate linear. Even more is true. There is a ring homomorphism with the condition (ii) which is not reallinear. Recall that the disk algebra A(D) is the algebra of all complexvalued continuous functions on D which are analytic on the interior D of D. Suppose that
K = {O} U {l/n : n is a positive integer}, X
= {2} U D
and Y
= K U {z E C : Iz  31::; I}.
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Let
A = {f
E C(X) :
flD
E
A(D)}
and B = C(Y), where CO denotes the algebra of all complexvalued continuous functions on '. Let ¢ be the ring homomorphism from C into C(K) defined in [20, Example 5.3]. For f E A, put
f(2), y=O, p(f)(y) = { ¢(f(2))(1/n), y = lin, f(y  3), Iy  31 :::; 1. Then p is a ring homomorphism and satisfies the condition (ii). But (i) is not satisfied and p is not reallinear on A.
4. Ring homomorphisms into algebras of analytic functions Suppose that A is a completely metrizable topological algebra with an identity and r(X, Ox) is the algebra of global sections of a connected reduced complexanalytic space (X, Ox). Becker and Zame [2] proved among other things that if p is a ring homomorphism from A into r(X, Ox) such that the range of p contains a nonconstant section, then p is linear or conjugate linear. This is not the case for ring homomorphisms on noncomplete algebras. (Suppose that P is the algebra of polynomials on C and H(CC) is the algebra of entire functions. Let T be a nontrivial ring homomorphism on C and define p on P by p(Eanz n ) = ET(an)Zn for every polynomial E anz n . Then p is a ring homomorphism, but it is neither linear nor conjugate linear. By Theorem 5.1 there also exists an injective ring homomorphism from Pinto C since {O} is a prime ideal in P.) Nevertheless we show automatic linearity results for ring homomorphisms on certain noncomplete metrizable algebras. THEOREM 4.1. Suppose that A is a complex (commutative or noncommutative) algebra with unit e. Suppose that Y is a nonempty set and B is a complex algebra of complexvalued functions on Y which contains the constant functions. Suppose that for every nonconstant function b E B the range of b contains a nonempty open subset of c. Let p be a ring homomorphism from A into B. If there exists an element a in A such that the resolvent set of a contains a nonempty open subset G of C and p( a) is nonconstant, then p is linear or conjugate linear. PROOF. It is easy to see that p( e) = 0 or 1 since the range of a nonconstant function in B contains a nonempty open set. If p( e) = 0, then p is 0 on A and so p is linear. Suppose that p(e) = 1. In the same way as above, we see that p(ie) = i or i. We show that p is linear if p(ie) = i. (If p(ie) = i, then 15 defined by 15(a) = p(a) will be linear by what we will show, so p will be conjugate linear.) We will show that p(,Xe) = ,X for every complex number 'x. First we show that p(Ce) C cc. Suppose not. Then there is a complex number ,x such that p(,Xe) is a nonconstant function. Note that Re'x or Im'x is irrational since p(xe) = x for every rational real number x by a simple calculation. Since p('xe)(Y) contains a nonempty open set, there exists r E p('xe)(Y) with Rer and Imr both rational. Then p(,Xe)  r is not an invertible element in B. Therefore (,x  r)e is not invertible in A, that is, ,x = r, which is a contradiction. Thus we have proved that p(Ce) C C, or p induces a ring homomorphism on cc. We denote the induced ring homomorphism also by p.
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211
Suppose that p is nontrivial on C. Since p(a) is not a constant, the interior of p( a) (Y) contains a complex number s whose real and imaginary parts are both rational, by our assumption for B. Since p is nontrivial, there exists a sequence {w n } of nonzero complex number such that Ip(wn)1 tends to infinity as n + 00 and s+ l/w n E G for every n. Since p(i) = i we see that p(s+ l/wn ) = s+ 1/ p(w n ) and we may assume that
s + l/p(w n ) E p(a)(Y) for every n. Thus p(a)  (s + 1/ p(wn )) is not invertible in B. It follows that a  (s + l/w n )e is not invertible in A, which is a contradiction, proving that p is trivial. Since p(i) = i, we have that p(A) = A for every complex number A. We conclude that p is linear on A. 0 The spectrum of each element in a Banach algebra is compact, so the conditions for A in Theorem 4.1 are satisfied by every Banach algebra with unit. Since the range of nonconstant analytic function is a nonempty open subset of C, algebras of global sections on connected, reduced complexanalytic spaces satisfy the condition for B in Theorem 4.1. Thus we have the following, which is a version of a more general result of Becker and Zame [2, Theorem 3.1]. But our proof is considerably simpler. COROLLARY 4.2. Let Ao be a Banach algebra with unit. Suppose that p is a ring homomorphism from Ao into r(X, Ox), the algebra of global sections on a connected, reduced complexanalytic space (X , Ox). If p( Ao) contains a nonconstant section, then p is linear or conjugate linear.
Let S be a subset of C. We denote by Rs the algebra of all rational functions on C with poles off S. Although Rs is a wlital algebra, it cannot be a Banach algebra by the Baire category theorem. If S = C, then Rs = P, and so there is a ring homomorphism p on Rc into r( X , Ox) for (X, Ox) = C such that p(Rc) contains a nonconstant function, while p is neither linear nor conjugate linear. In the case where C \ S contains an interior point, the situation is different; in this case we prove an automatic linearity result. COROLLARY 4.3. Let S be a subset of C whose complement contains interior points. Suppose that (X, Ox) is a connected, reduced complexanalytic space and r(X, Ox) is the algebra of global sections. Suppose that p is a ring homomorphism from Rs into r (X , Ox). If the range of p contains a nonconstant section, then p is linear or conjugate linear. PROOF. In the same way as in the proof of Theorem 4.1 we see that p(C) C C. Suppose that z denotes the identity function: z(w) = w for every complex number w. Then we have that p(z) is nonconstant. (Suppose not. Then p(f) is a constant section for every f E Rs.) On the other hand z  A is invertible for every A E C \ S, which contains a nonempty open set. Thus the conditions in Theorem 4.1 are satisfied. It follows by Theorem 4.1 that p is linear or conjugate linear. 0
Note that every ring homomorphism p of R0 into r(X, Ox) is constantvalued for the empty set 0. (We see that p(C) C C as before. Suppose that p(f) is not a constant section for some nonconstant rational function f. Then there is a complex number r in p(f)(X) with rational real and imaginary parts. It follows that f  r is not invertible in R0, which is a contradiction.)
212
O. HATORI, T. ISHII, T. MIURA. AND S.E. TAKAHASI
Let A be one of the algebras P, njj or the disk algebra A(D), where jj denotes the closed unit disk in C, and H(D) the algebra of analytic functions on the open unit disk. Although both P and no are dense in the disk algebra, automatic linearity results for ring homomorphisms on these algebras are different from each other. Suppose that p is a ring homomorphism from A into H(D) such that the range of p contains a nonconstant function. If A = njj (resp. A(D), then p is linear or conjugate linear by Corollary 4.3 (resp. Corollary 4.2). But that is not the case for A = P. The ring homomorphisms defined by p(2:a n z n ) = 2:1'(a n )zn for polynomials 2: anz n are neither linear nor conjugate linear for nontrivial ring homomorphisms l' on C.
5. Complexvalued ring homomorphisms In this section we consider ring homomorphisms into the complex number field C. Suppose that A is a complex algebra and p is a nonzero ring homomorphism from A into C. Then the kernel ker p of p is a prime ideal. Recall that a proper ideal I of A is said to be a prime ideal of A if fg E I implies that f E I or gEl. By using wellknown results of algebra, we see the converse is also valid; for every prime ideal such that the cardinal number of the quotient algebra of the algebra by the ideal is equal to that of the continuum, there exists a ring homomorphism into C whose kernel coincides with the ideal. Let K be an extension field of a field k. (Here and after a field means a commutative field.) We recall a subset S of K is said to be algebraically independent over k if the set of all finite products of elements in S is linearly independent over k. A subset T of K which is algebraically independent over k and is maximal with respect to the inclusion ordering is said to be a transcendence base of Kover k. By definition, for every transcendence base T of Kover k, K is algebraic over the quotient field k(T) of the polynomial ring of T over k. There exists a transcendence base of Kover k (cf. [11, Theorem X.l.I]). Using the same argument as in [9] we can prove the following (cf. [11, 20]). (This might be a standard fact. But we present here with a proof for the convenient of the readers.) THEOREM 5.l. Let A be a commutative complex algebra with unit e. Suppose that I is a prime ideal of A such that the cardinal number of AI I is that of the continuum c. Then there exists a ring homomorphism p from A into C such that kerp=I. PROOF. The quotient algebra AI I has no nonzero divisor of zero, for I is a prime ideal. We denote by K the field of fractions over AI I. Let Q be the field of complex numbers whose real and imaginary parts are both rational. Let TK be a transcendence base for Kover Q and T a transcendence base for Cover Q. Then the cardinal number of TK (resp. T) is c since that of AI I (resp. q is c. There exists au injection a defined from TK onto T. Since TK is algebraically independent, there is a unique extension from Q(TK ) onto Q(T), which is also denoted by a, and a is a ring homomorphism. Since C is algebraically closed and K is an algebraic extension of Q(TK), there exists an extension of a which defines a ring homomorphism of K into C by Theorem VII.2.8 in [11]. We also denote it bya. Let h be the natural homomorphism of A onto AI I. Put p = a 0 h. Then p is the desired ring homomorphism. D
AUTOMATIC LINEARITY FOR RING HOMOMORPHISMS
213
Note that the corresponding ring homomorphism p is not unique. Let T be any nonzero ring homomorphism on C. Then TOp is a ring homomorphism on A with ker p = ker TOp. As a corollary of Theorem 5.1 we display a pathological feature of ring homomorphisms on algebras of analytic functions into C; even injection can be possible. COROLLARY 5.2. Let A be a unital algebra which consists of holomorphic functions on a domain in Then there exists an injective ring homomorphism of A into Co
cn.
PROOF. Since the ideal containing only zero is a prime ideal and the cardinality of A is the same as that of the continuum, there exists a ring homomorphism p of A into C whose kernel consists only of zero, by Theorem 5.1. Then p is an injective ring homomorphism. 0 Note that the injective ring homomorphism in Corollary 5.2 can never be surjective if A contains nonconstant functions since A is not a field. Note also that every ring homomorphism from a unital commutative C* algebra into C cannot be injective if the dimension of the algebra is greater than one since {O} is not a prime ideal in this case. Together with the results in the previous sections we give a complete description of ring homomorphisms on the disk algebra A(D). COROLLARY 5.3. Let p be a nonzero ring homomorphism on the disk algebra into itself. Then ker p is a prime ideal. If the range of p contains a nonconstant function, then p is linear or conjugate linear; there exists 'P E A(D) with 'P(D) c fJ such that zED, f E A(D) p(f)(z) = f 0 'P(z), or
p(f)(z) = f
z E fJ,
'P(z),
f E A(fJ).
On the other hand, suppose that'P E A(D) with 'P(D) a(f)(z) = f
'P(z),
c
D. Then
ZED,
f
E A(fJ)
zED,
f
E A(D)
defines a linear ring homomorphism and a(f)(z) = f
'P(z),
defines a conjugate linear ring homomorphism.
PROOF. A(D) has no nonzero divisors of zero, so the kernel of any ring homomorphism from complex algebra with unit element into A(D) must be a prime (algebra) ideal. If p(A(D) contains a nonconstant function, then by Theorem 4.1 we see that p is linear or conjugate linear. Suppose that p is linear. Then it is well known and easy to prove, since the maximal ideal space of A(D) is the closed unit disk D, that there exists 'P E A(fJ) with 'P(D) c D such that p(f)(z) =
f
'P(z)
holds for every f E A(D) and zED. Suppose that p is conjugate linear. Let h : A(fJ) + A(fJ) be defined as h(f)(z) = f(2),
f
E A(D),
zED.
214
O. HATORI, T. ISHII, T. MIURA, AND S.E. TAKAHASI
Then hop is a linear ring homomorphism on the disk algebra. It follows that there exists r.p E A(D) with r.p(D) c D such that
h 0 p(f)(z)
=
1 0 r.p(z) ,
zED,
I
E
A(D).
Thus we see that
p(f)(z) = 1 0 r.p(z), holds for every I E A(D) and zED. Conversely, suppose that r.p E A(D) with r.p(D) c D. Then it is easy to see that u(f)(z)
=
1 0 r.p(z),
zED,
IE A(D)
zED,
IE A(D)
defines a linear ring homomorphism and
u(f)(z) = 1 0 r.p(z),
defines a conjugate linear ring homomorphism.
o
Let n be a positive integer and An(D) the subalgebra of those I in A(D) whose nth derivative I(n) on D is continuously extended up to D. An(D) is a unital commutative Banach algebra with the norm IIIlIn = L~=o III(k)lIoo/k! for I E An(D), where II . 1100 is the supremum norm on D. Then Corollary 5.3 is also valid for An(D). Prime ideals in A(D) and An(D) are studied in [16]. (See also [4] for the case of A(D).) Mortini proved that every nonzero prime ideal is contained in a unique maximal ideal. He in fact showed that a nonzero and nonmaximal prime ideal in An(D) (resp. A(D)) is dense in exactly one of the ideals {J E An(D) : 1(>") = 1'(>..) = ... = I(j)(>..) = O} for some 0 ~ j ~ n (resp. {I E A(D) : 1(>") = O}), >.. E aD. We also see by a theorem of Dietrich [4] that the cardinal number of the set of all prime ideals of A(D) which is contained in a maximal ideal {J E A(D) : 1(>") = O}, >.. E aD is 2', the cardinal number of the set of all the subsets of the continuum. Thus we see that there are 2' ring homomorphisms on the disk algebra. Acknowlegement. The authers would like to thank Professor KenIchiroh Kawasaki for his valuable comments. They also would like to thank the referees for their careful reading of the paper and their valuable comments.
References [1] B. H. Arnold, Rings of opemtors on vector spaces, Ann. of Math. 45(1944), 2449 [2] J. A. Becker and W. R. Zame, Homomorphisms into analytic rings, Amer. Jour. Math. 101(1979), 11031122 [3] L. Bers, On rings of analytic junctions, Bull. Amer. Math. Soc. 54(1948), 311315 [4] W. E. Dietrich, Jr., Prime ideals in uniform algebms, Proc. Amer. Math. Soc. 42(1974), 171174 [5] M. Eidelheit, On isomorphisms of rings of linear opemtors, Studia Math. 9(1940),97105 [6] J. B. Garnett, Bounded analytic functions, Academic Press, New York, 1981 [7] H. Iss'sa, On the meromorphic junction field of a Stein variety, Ann. Math. 83(1966), 3446 [8] 1. Kaplansky, Ring isomorphisms of Banach algebms, Canadian J. Math. 6(1954),374381 [9] H. Kestelman, Automorphisms of the field of complex numbers, Proc. London Math. Soc. 53(1951), 112 [10] 1. Kra, On the ring of holomorphic functions on an open Riemann surface, Trans. Amer. Math. Soc. 132(1968),231244 [11] S. Lang, Algebm (second edition), AddisonWesley, California, 1984. [12] M. H. Lebesgue, Sur les tmnsformations ponctuelles, tmnsformaant les plans en plans, qu'on peut definir par des procedes analytiques, Atti della R. Acc. delle Scienze di Torino 42(1907), 532539
AUTOMATIC LINEARITY FOR RING HOMOMORPHISMS
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[13J T. Miura, Star ring homomorphisms between commutative Banach algebrns, Proc. Amer. Math. Soc. 129(2001), 20052010 [14J L. Molnar, The rnnge of a ring homomorphism from a commutative C' algebrn, Proc. Amer. Math. Soc. 124(1996), 17891794 [15J L. Molnar, Automatic surjectivity of ring homomorphisms on H* algebrns and algebrnic differences among some group algebrns of compact groups, Proc. Amer. Math. Soc. 128(2000), 125134 [16J R. Mortini, Prime ideals in the algebrn An(D), Complex Variables Theory AppJ. 6(1986), 337345 [17J M. Nakai, On rings of analytic and meromorphic functions, Proc. Japan Acad. 39(1963), 7984 [18J W. Rudin, An algebrnic charncterization of conformal equivalence, Bull. Amer. Math. Soc. 61(1955), 543 [19J S. de Corrado Segre, Un nuovo campo di ricerche geometriche, Atti della R. Acc. delle Scienze di Torino 25(1889), 276301 [20J P. Semrl, Nonlinear perturbations of homomorphisms on C(X), Quart. J. Math. Oxford (2) 50(1999),87109 [21J S.E. Takahasi and O. Hatori, A structure of ring homomorphisms on commutative Banach algebrns, Proc. Amer. Math. Soc. 127(1999),22832288 DEPARTMENT OF MATHEMATICAL SCIENCE, GRADUATE SCHOOL OF SCIENCE AND TECHNOL
9502181 JAPAN Email address:hatorilDmath.se.niigatau.ae.jp
OGY, NIIGATA UNIVERSITY, NIIGATA
NIIGATA CHUO HIGH SCHOOL, NIIGATA
9518126 JAPAN
DEPARTMENT OF BASIC TECHNOLOGY, ApPLIED MATHEMATICS AND PHISICS, YAMAGATA UNI
9928510 JAPAN Email address:miura«lyz.yamagatau.ae.jp
VERSITY, YONEZAWA
DEPARTMENT OF BASIC TECHNOLOGY, ApPLIED MATHEMATICS AND PHISICS, YAMAGATA UNI
9928510 JAPAN Email address:sineiOemperor.yz.yamagatau.ae.jp
VERSITY, YONEZAWA
Contemporary :r...1athematics Volume 328. 2003
Carleson Embeddings for Weighted Bergman Spaces Hans Jarchow and Urs Kollbrunner ABSTRACT. We are going to discuss Carleson measures for the standard weighted Bergman spaces A~ (1 < a < 00, 0 < p < (0). These are finite, positive Borel measures J.L on the unit disk in IC such that, given 0 < q < 00, A~ embeds, as a set, continuously into Lq(J.L). Such measures have been closely investigated by V.L. Oleinikov and B.S. Pavlov [15], W.W. Hastings [6] and D.H. Luecking [11], [12]. We complement their results, in particular by characterizing compactness, order boundedness and related (absolutely) summing properties of the canonical embedding A~ <t Lq(J.L).
1. Introduction
The Carleson measures under investigation are finite, positive Borel measures Jl. on the open unit disk l[J in the complex plane such that, given 1
<
<
00
and 0 < p, q < 00, the (classical) weighted Bergman space A~ is a subset of Lq(J.l) and the embedding is bounded. Such measures have been characterized, even in a more general setting, by V.L. Oleinikov and B.S. Pavlov [15J, W.W. Hastings [6J and D.H. Luecking [11], [12J. Our first main topic is to complement their results by characterizing when the canonical embedding I : A~ ~ Lq(J.l) is compact. We will see that this is always the case if p > q. Our second main topic is to characterize when I is order bounded, that is, the unit ball of A~ is a subset of some order interval in Lq(J.l). As a consequence, we obtain necessary and sufficient conditions for I to have specific (absolutely) summing properties. Our results extend corresponding ones for composition operators which have been obtained e.g. in [3]' [4], [20]' [21J. In fact, they can also be viewed as results on composition operators which may have rather unusual range spaces. They apply, for example, to pointwise multipliers. Many of the results to be presented remain valid for measures J.l on D for which f f> f induces just a bounded linear map A~ ~ Lq(J.l) (not necessarily injective). This is rather straightforward; precise formulations, however, require somewhat 1991 Mathematics Subject Classification. Primary 46 E 15,47 B 38, 47 B 10; Secondary 46 B 25, 30 H 05, 32 H 10. Key words and phrases. Weighted Bergman spaces, Carleson measures, composition operators, compactness, order boundedness, absolutely summing operators. The results of this paper are part of the dissertation of the second named author written at the University of Ziirich under the supervision of the first.
© 217
2003 Alnerican Mathematical Society
HANS JARCHOW AND URS KOLLBRUNNER
218
clumsy notation. Also, there is an immediate extension to complex Borel measures on 1lJ whose variation is (a, p, q)  Carleson. We are indebted to the referee for providing Example 8 and for bringing to our attention the paper [15] by V.L. Oleinikov and B.S. Pavlov. 2. Weighted Bergman spaces Throughout the paper, we will use standard results and notation from (quasi) Banach space theory. We will work on the open unit disk 1lJ = {z E C: Izl < I} in the complex plane. The space 1i(1lJ) of all analytic functions 1lJ > C is a F'rechet space with respect to the topology of uniform convergence on compact subsets of 1lJ. Let da be normalized area measure on 1lJ. For each a > 1,
dao(z) :=
(a + 1) (1 lzI 2 )O da(z)
is a probability measure on 1lJ. For each 0 < p < Bergman space is defined to be A~ := A~
00,
the corresponding weighted
1i(1lJ) n P(ao ).
is closed in LP (a 0); it is a Banach space if p 2:: 1 and a p  Banach space if Its (p ) norm will be denoted by II . Ilo,p. A~ is a Hilbert space and has a reproducing kernel:
o < p < 1.
Ko(z, w) = K(z, w)o+2; here K(z,w) = (1  ZW)l is the reproducing kernel for the Hardy space H2. For reasons like this, the scale of Hardy spaces is often considered as the scale of weighted Bergman spaces which corresponds to a = 1. Some of the results below actually remain true for this case, and some can even be extended to analytic Besov spaces B~ (f E B~ {::} f' E A~+p). Nevertheless, in this paper we will only deal with the case 1 < a < 00. 3. Carleson measures All measures on 1lJ will be finite, positive Borel measures. Let 1 < a < 00 and 00 be given. We say that a measure /L on 1lJ is an (a, p, q)  Carleson measure if A~ c Lq(/L) and the embedding A~ '+ Lq(/L) is continuous: there is a constant C> 0 such that IIfIILq(~) ::; C ·lIfIIA~ "If E A~. Given an (a, p, q)  Carleson measure, the canonical embedding I : A~ > Lq(/L) will be referred to as a Carleson embedding. As mentioned in the introduction, a number of the results to follow remain true if we just require that f 1+ f induces a bounded linear map A~ > Lq(/L). Also, complex measures whose variation is (a, p, q)  Carleson can be incorporated. Moreover, there are extensions to analytic functions of several variables. However, we are not going to discuss such generalizations in this paper. We say that /L is a compact (a,p, q)  Carleson measure if the embedding A~ '+ Lq(/L) exists and is compact. For example, an a.e. positive function h E Lq(a/3) defines the bounded multiplier Mh : A~ > Lq(a/3) : f 1+ fh iff the measure h q da/3 is (a,p, q)  Carleson. Moreover, discrete (a, p, q)  Carleson measures on 1lJ can be defined using appropriate versions of 'sampling sequences', etc.
o < p, q <
CARLESON EMBEDDINGS FOR WEIGHTED BERGMAN SPACES
219
An important example is obtained by looking at the composition operator C", : J 0 'P induced by a nonconstant analytic function 'P : 10 > 10. Clearly, C",: A~ > A~ exists iff a o o'P 1 is (o:,p,q)Carleson. More generally, an arbitrary measure J.l on 10 is (0:, p, q)  Carleson if and only if, for every analytic map 'P : 10 > 10, CI{J maps A~ boundedly into Aq(J.l) := 1i(1O) n Lq(J.l). In fact, the condition applied to the identity of 10 shows that J.l is (o:,p,q)Carleson. On the other hand, if J.l is (0:, p, q)  Carleson and 'P : 10 > 10 is analytic, then CI{J : A~ > Aq(J.l) is welldefined and bounded. For nonconstant functions 'P, the condition is further equivalent to J.l 0 'P 1 being (0:, p, q)  Carleson. This allows an interpretation of Carleson embeddings, and in particular of multipliers as above, as composition operators. However, in such a general setting the range space of a composition operator might be unpleasent, and desirable properties may not be available. For example, Aq(J.l) embeds continuously into 1i(1O) if and only if Aq(J.l) is a closed subspace of Lq(J.l) and all point evaluations Aq(J.l) > C : J f+ J(z), z E 10, are continuous. (0:, p, q)  Carleson measures have been characterized, even in a more general setting, by V.L. Oleinikov and B.S. Pavlov [15], W.W. Hastings [6] and D.H. Luecking
J
f+
[11],[12]. The hyperbolic metric on 10 is given by
. lJd(J "I 1 _ J(J
e(z, w) := l~f
where the infimum extends over all smooth curves 'Y in 10 joining z and w. For w E 10 and r > 0, let Br (w ) = {z E 10 : e( z, w) < r} be the corresponding hyperbolic disk. Actually, the particular choice of r > 0 doesn't really matter in our context. Let us agree to write A~ '+ Lq(J.l) if A~ is a subset of U(J.l) and the embedding is continuous. Similar for other function spaces. The Carleson measures under consideration can be characterized in terms of the function
THEOREM
3.1. Let 1 < 0: <
00
and 0 < p, q <
00
be given.
(a) IJp::; q, then A~
'+
Lq(J.l) iJ and only iJ Ho,p,q(w) is bounded on 10.
(b) IJp> q, then A~
'+
Lq(J.l) if and only if Ho,p,q
E
L/!!q(A).
Here dA(z) = da(z)(IJzJ2)2 is the Mobius invariant measure on 10. In [12] Luecking presents an interesting proof of (b) which is based on the inequalities of Khinchin and Kahane for Rademacher functions (see e.g. [2]). It is wellknown (compare K. Zhu [22]) that there is a constant C = C(o:, r) > 0 such that 1 C . ao(Br(w)) ::; (IJwJ2)o+2 ::; C· ao(Br(w)) 'Vw E 10 . Therefore we may also say that Theorem 3.1 refers to properties of the function w f+ J.l(Br (W))I/ q(1 _JWJ2)(o+2)/p .
220
HANS JARCHOW AND URS KOLLBRUNNER
It also follows that Ha,p,q E L~(A) if and only if Jt(B r (·))/C7 a (B r (·)) is in LP/(pq) (C7 a ). This will be used in the proof of Theorem 4.3 below. If p ~ q; then the relevant parameter in Theorem 3.1 is q (0: + 2)/p, whereas for p > q and fixed 0:, dependence is on p / q. As a first immediate consequence we may state: COROLLARY
if and only if A~
3.2. For any 1 < 0: < "> Ltq(Jt).
00,0
< p,q < 00 and t > 0,
A~ ">
Lq(Jt)
In turn, this leads to: 3.3. Let 1 < 0:,0:' < 00 and 0 < p, pi, q, q' < 00 be given. (a) Ifp ~ q, pi ~ q' and q. (0: + 2)/p = q'. (0: ' + 2)/p', then A~ "> Lq(Jt) iff A~, "> Lq' (Jt).
COROLLARY
(b) Ifp> q and p/q = pi /q', then A~
">
Lq(Jt) iff A~
">
Lq' (Jt).
For a large range of parameters, this allows a reduction to Hilbert spaces as follows: 3.4. Suppose that 1 < 0:,0:' < 00 and 0 < p, q < 00. (a) Ifp ~ q and 0:' + 2 = q. (0: + 2)/p, then A~ "> Lq(Jt) iff A~,
COROLLARY
(b) If p > q and pi /2 = 2/q' = p/q, then A~ A~ "> L 2 (Jt). A special known case occurs when we take Jt Horowitz [7]).
">
Lq(Jt) iff A~
3.5. Suppose that 1 < 0:, (3 < 00 andO < p, q < (a) If p ~ q, then A~ "> A~ iff (0: + 2)/p ~ ((3 + 2)/q. ">
A~ iff (0: + l)/p
">
L 2 (Jt).
Lq' (Jt) iff
= C7fJ for some (3 > 1 (see C.
COROLLARY
(b) If p > q, then A~
">
00.
< ((3 + l)/q.
There are several ways to modify the domain space of a composition operator. In a systematic fashion, we may proceed as follows; cf. [4]. Each of the kernel functions K a ( Z, .) is bounded (z E lV), and ._ ( 1  IzI2 ) (a+2)/p
(1 _
ka,p,z(w),has (p) norm one in representation
A~
(0 < p <
00).
ZW)2
The functions
f
E A~ which admit a
00
f(w) =
L
an
Vw E lV ,
ka,p,zn(w)
n=l
where the scalars linear space, say
an
satisfy
En lanl <
00
and the
Zn's
are taken from lV, form a
This is a Banach space with norm
IlfIIA~)
00
:= inf
{L lanl:
(*) holds} .
n=l
In fact, the map e1(lV)
+
A~) : (az)zEllJ
14
EZEllJ
azka,p,z is a metric surjection.
CARLESON EMBEDDINGS FOR WEIGHTED BERGMAN SPACES
221
It is immediate that
• if 0
00
are such that (a+2)/p
= (/3+2)/q,
Atomic decomposition is available for weighted Bergman spaces (e.g. [10], [1], [22]), hence
• A~
S:!
A~l) (with equivalent norms).
In particular, if a> p2, then A~), alias Ab with /3+2 = (a+2)/p, is isomorphic to fl. Moreover: • If p 2: 1 then A~) '+ A~ (boundedly and densely).
A~) = Ab (boundedly and densely). In fact, it was shown by J.H. Shapiro [19] that in the latter case Ab is the Banach • If 0
< p < 1, then A~
'+
envelope of A~, that is, the Banach space which is obtained by completing A~ with respect to the biggest norm which is smaller than the given pnorm. PROPOSITION 3.6. Let 1 < a < 00 and 0 < p ::; q < 00, q 2: 1. The following are equivalent: (i) J.L is an (a,p, q)  Carleson measure. (ii) A~)
'+
Lq(J.L) (boundedly).
(iii) SUPzEV IIko,p,zlbu.t) < 00. PROOF. (i) =? (ii) is obvious if p 2: 1 and follows from the above result of J.H. Shapiro [19] if p < 1 ::; q: A~) is the Banach envelope of A~, and so the convex hull of BAP is dense in B A(p), (ii) =? (iii) is trivial, and (iii) =? (i) is immediate from the followi~g estimate in ;hich C is a constant depending only on 0 < r < 1: J.L(B r (W))l/ q 1 )l/q Br{w) dJ.L(z) (1  IwI 2){o+2)/p = (1 lwI 2){o+2)/p .  (1 
<

12){o+2)/p
I

W
C· (
.
(1 (1
Br{w)
1 d ( )) l/q (1 lwI2)2q{o+2)/p J.L Z
r1(1(1  wz)2{o+2)/p I
IwI 2){o+2)/p qd
lv
Z) l/q
J.L()
= C ·lIko,p,wIILq{~).
o
Now apply Theorem 3.1(a).
Actually, in the last step, no restriction on p and q is needed: Ho,p,q(w) is bounded whenever A~) '+ Lq(J.L). There is another interesting consequence of Theorem 3.1, Corollary 3.2, and Proposition 3.6: COROLLARY 3.7. If 1 ::; q < p < 00 and (p/q)  2 < a < L~(A) implies Ho,p,q(w) E LOO(A).
00,
then Ho,p,q(w) E
PROOF. Define a' > 1 by a' + 2 = q(a + 2)/p. Then, with £T = £T(A),
Ho,p,q E Lpq/{pq) <=> A~
'+
Lq(J.L)
=? A~/q) = A~, '+
<=> H O ',l,l E L oo <=> Ho,p,q E L oo
.
L 1 (J.L)
o
HANS JARCHOW AND URS KOLLBRUNNER
222
EXAMPLE 3.8. (a) Let a, p, q be as in Corollary 3.7, let (an) be a sequence in foo \ fP/(pq) , and let (1]n) be a sequence in llJ such that (!(1]n,1]k) :::: r· 8nk for all n, k, and such that llJ = Un Br(1]n). Consider the measure JL = En bn 811n on llJ where bn = lanl' (1I1]nI 2 )q("'+2)/p and note that (b n ) E fl. A calculation reveals that H""p,q is in LOO(A) but not in uq/(pq) (A). We conclude that the converse in Corollary 3.7 doesn't hold. (b) In Proposition 3.6, (ii) {::} (iii) is true for arbitrary 0 < p, q < 00, and (i) ~ (ii) holds trivially whenever q :::: 1. But (iii) ~ (i) fails for 1 ~ q < p and a + 2 > p. In fact, if JL is as in (a), then A~ 'f+ Lq(JL) since H""p,q tfLpq/(pq)(A), but Hy,l,q = H""p,q E LOO(A) if we put "I = (a + 2)/p  2. From A~) = A~l) = A~ we conclude that A~ '> Lq(JL). 4. Compactness We shall frequently make use of the following classical result: THEOREM 4.1 (Pitt's Theorem). If 0 < p < q < 00 then every operator f q + fP is compact. This was obtained in 1936 by H.R. Pitt [16] for p :::: 1. For an extension see H.P. Rosenthal [17]. The result as stated was proved recently by E. Oja [14]. By atomic decomposition, A~ and fP are isomorphic (see [10], [1]). Combining this with Pitt's theorem we see that in particular the embedding in Corollary 3.5.(b) is compact. It will follow from the next theorem that the embedding in 3.5.(a) is compact iff (a + 2)/p < (f3 + 2)/q. Our characterization of compactness of Carleson measures splits into two parts. We consider the case p ~ q first. Here the characterization is as expected: THEOREM 4.2. Let 1 < a < 00 and 0 < p ~ q < statements are equivalent: (i) JL is a compact (a,p, q)  Carles on measure. (ii) A~)
'>
with 1 ~ q. The following
Lq(JL) compactly.
(iii) z+ limlllk""p,zIILq(ll) (iv)
00
= O.
lim Ho.,p,q(w) = O.
Iwl+l
PROOF. (i) ~ (ii): If p :::: 1 then nothing is to prove since A~) '> A~. If p < 1 ~ q then, by [19], A~) is the Banach envelope of A~, and the convex hull of B A~ is dense in B A!:)' Hence relative compactness of B A~ in the Banach space Lq(JL) entails relative compactness of B A<.!) in Lq(JL).  Here we have used Bx to denote the unit ball of a (quasi) Banach space X. (ii) ~ (iii): Suppose that (iii) doesn't hold. Then there exist an c > 0 and a sequence (zn) in llJ such that limn+ oo IZnl = 1 and Ilko.,p,zn IIL(Il) > c for all n. By (ii), we may assume that (k""p,zn)n converges to some f E Lq(JL). But lim n + oo IZn I = 1 implies f = 0 since clearly (k""p,zn) tends to zero pointwise: contradiction. (iii) ~ (iv): The estimate proved in (iii) ~ (i) of Proposition 3.6 provides us with a constant C = C(r) > 0 such that JL(Br(W))l/q ~ C, (1  IwI 2 )(",+2)/p . IIk""p,w IIL(Il) for all W E llJ.
CARLESON EMBEDDINGS FOR WEIGHTED BERGMAN SPACES
223
(iv) =} (i): We apply Lemma 4.3.6 of K. Zhu [22]: there exists an integer N such that for sufficiently small r there is a sequence (TJn) in llJ having the following properties:
(1) llJ = U~=l Br(TJn), (2) B r / 4 (TJm) n B r / 4 (TJn) = 0 whenever m # n, (3) Every Z E llJ is contained in at most N of the sets B 2r (TJn)' Note that limn+oo ITJnl = 1 follows from (2). Let now (In) be a sequence in BA~' By Montel's theorem, some subsequence (Ink) converges uniformly on compact sets to some I E 1i(llJ). By Fatou's lemma, even I E BA~' Put gk := I  Ink' Vk E N. By our hypothesis there exists, for any given c > 0, an integer ne such that J.L(Br(TJn)) :::; c' (1  ITJnI 2)q(a+2)/p if n ~ ne' Therefore, with constants depending only on the indicated parameters,
n~, Lr(T/n) Igk(ZW dJ.L(z)
=
n~, L,.(T/,.) (lgk(ZW)q/PdJ.L(z)
< C(r)· n~E Lr(T/n) COa(B:r(TJn))
L2,0(T/n)
r < C(r)· C(r, 0)' "~ (1 _IJ.L(B 2(17n)) )q(a+2)/p 1
n~nE
< C(r)·C(r,o)·c·
TJn
L ({ n~nE J
IgdwWdaa(w)) q/p dJ.L(z)
(1
p
Igk(W)1 daa(w)
)q/P
B 2r (T/n)
Igk (wWda a (w)r/ p
B2r(T/n)
< C(r)· C(r, 0) . c· (
L1
Igk(wWdaa(w) riP
(L
Igk (wWda a (w) ) q/p :::;
n~n.
< C(1')' C(r, 0) . Nq/p . c·
B2r(T/n)
c· c ;
here C = C(r,o,N,p,q). If we choose now ke EN such that Ln
Iv
A slight modification of the argument used to prove (i) =} (ii) shows that compactness of A~ <+ Lq(J.L) implies compactness of A~, <+ Lq(J.L) if 1 < 0,0' < 00, 0< p < p' :::; q and (0 + 2)/p = (0' + 2)/p'. It can be shown that (i) {::} (iii) {::} (iv) is true even for arbitrary 0 < q < 00. In the case p > q, we can prove: THEOREM 4.3. Suppose that 1 < 0 < 00 and 0 < q < p < 00. Regardless 01 the (0, p, q)  Carleson measure J.L, the embedding A~ <+ Lq (J.L) is always compact. For composition operators C
+
Ah this is due to W. Smith and L. Yang
[21]; see also [4].
PROOF. Put s = p/(p  q) and recall that J.L(Br())/aa(Br (·)) E £B(aa) whenever J.L is (0, p, q)  Carleson.
HANS JARCHOW AND URS KOLLBRUNNER
224
Let Un) be a bounded sequence in A~. For some subsequence, f = limk+oo fnk exists in 'H(lU). As before, f E A~, hence we may assume that Un) is a null sequence in 'H(lU) and that IIfnllo,p :::; 1 for all n. There are constants Cr , Cr > 0 such that, for each n,
Inf1lJ Ifnlq dp.
< Cr' = Cr'
f
l1lJ a o
(;r (W)) lBr(w) f Ifn(zWdao(z) dp.(w)
flao(B~(w)) I Br (w)(z) Ifn(zW dp.(w) dao(z)
(4.1) = Cr'
f Ifn(zW lBr(z)a f (B1 ( )) dp.(w) dao(z) rW  f q p.(Br(z))
l1lJ
O
< Cr' Cr" l1lJ1fn(z) I ao(Br(z)) dao(z) .
. f
By hypothesIs,
p.(Br(zW
l1lJ ao(Br(Z))8dao(z) <
00.
Therefore, given c > 0, there is an r E E (0,1) such that (4.2)
11lJ\r 1lJ
_
p.(Br(zW ( ) ( / )8 (B ( ))8 da o z < c 2 .
rZ uniformly on rEV, so that E
ao
Un) tends to zero limn+oo Ir,v Ifn (z)IPda o (z) = cordingly, we may choose n E EN such that, for n 2: n E ,
o.
Ac
(4.3)
(4.2)
<
Combine (4.1), (4.4) and (4.5) to find a constant for n 2: ne:'
c

2
Cr > 0 such that I1lJ Ifnlq dp. :::; Cr·c 0
Essential parts of Theorem 4.3, if not the entire theorem, can be proved by using other methods. We sketch three possibilities: • Suppose that 1 < Q < 00, 1 :::; q :::; 2 < P < 00 and v is any measure. Every operator u : A~ + Lq(v) is compact. In fact, by Kwapien's theorem (see [8], or [2], 12.19), the operator u admits a factorization u : A~ ~ 2 .2:. Lq(v). By atomic decomposition and Pitt's theorem, w is compact, and so is u.
e
CARLESON EMBED DINGS FOR WEIGHTED BERGMAN SPACES
225
• Rosenthal's extension [17J of Pitt's theorem admits the same conclusion for q ~ 1 and p > max{2, q}. • Suppose that 1 < a < 00, 1 < p < 00, and pis (a,p, 1)  Carleson. Then I : A~ '+ £1 (p) is compact.
Since A~ is reflexive it suffices to verify that I is completely continuous. Accordingly, let (fn)n be a weakly null sequence in A~. Then fn(z) t 0 for each z E llJ. Being weakly null in L1(p), (fn) is uniformly integrable. Now limn ..... oo IIfnl11 = 0, by a theorem of Vitali (see W. Rudin [18], p.133). Standard results from interpolation theory on the preservation of compactness by interpolated operators lead from either of these special results to (at least parts of) Theorem 4.3. 5. Order bounded and absolutely summing operators
Our Banach lattices will be complex Banach lattices; see e.g. P. MeyerNieberg
[13J for the construction of such an object from a real Banach lattice. Let X be a Banach space and Y a closed subspace of a Banach lattice L. An operator u : X t Y is called order bounded if there is a non  negative h E L such that lufl :::; h for f in Bx, the unit ball of X. Thus we require u to map Bx into the order interval {g E L : Igl :::; h} of L. Note that L is part of the definition! Every 1.L E C(X, Y) is order bounded when Y is considered as a subspace of C(K) for some compact Hausdorff space K. Let I be an order interval in the Banach lattice L. Its span, Z, is a Banach lattice with respect to L's order and (a multiple of) I's gauge functional as its norm. Z is an abstract M  space with unit and so, by a wellknown theorem of S. Kakutani, isometrically isomorphic (as a Banach lattice) to C(K) for some compact Hausdorff space K; see again [13J. It follows that every order bounded operator u : X t Y c L factorizes X ~
~ C(K)
L L where K is as before and j is the canonical embedding. In this paper, L will be a space LP(p) which results in close ties with absolutely summing operators. Recall that a Banach space operator u : X t Y is (q,p)summing (p:::; q), written u E IIq.p(X, Y) , if there is a constant C such that, for every choice of n E N and x!, . . . ,X n EX, Z
In other words, u is (q,p)summing iff every weak ePsequence, i.e. every sequence (xn) in X which satisfies 2:::=11(x*,x n )iP < 00 for all x* E X*, is taken to a strong eq  sequence, i.e. 2:::=1 Iluxnllq < 00 holds. (p, p)  summing operators are called p  summing; the corresponding notation is IIp(X, Y) = IIp,p(X, Y) . We refer to [2J for details on these concepts and in particular for the following facts: • If Hand K are Hilbert spaces and q ~ 2, then IIq,2(H, K) is the corresponding Schatten q  class. Moreover, for any 1 :::; p < 00, IIp(H, K) is the class of Hilbert  Schmidt operators. • If 1 :::; P :::; 2, then every operator from C(K) to LP(lI) is 2  summing.
HANS JARCHOW AND URS KOLLBRUNNER
226
• If p > 2, then every operator C(K) r  summing for every r > p.
7
U(v) is (p, 2) summing, and
Moreover: • If u : X
Lp(v) is order bounded then u is p  summing. Here v is any measure. In the last statement, the converse fails. But: 7
• If u* p  summing then u is order bounded.
More precisely, we have the following result due to D.J.H. Garling [5]: • Let 1 ::; p < 00. A Banach space operator u : X 7 Y has a p  summing adjoint if and only if, for every measure v and operator v : Y 7 LP(v), the composition v 0 u : X 7 LP(v) is order bounded.
In particular: • An operator u : L 2 (VI) Schmidt.
7
L 2 (V2) is order bounded iff it is Hilbert
We are going to characterize order boundedness of Carleson embeddings
A~ ~
Lq(J.L). To this end we introduce, for s > 0, the Banach space
Xs
:= {f: lU
7
C: f measurable, sup(1lzI 2 )Slf(z)1 < oo} . zE1U
and its closed subspace
Xs := Xs n H(lU) . It is easy to see that
A~ ~
X(o.+2)/p and that the index (o.+2)/p is best possible.
THEOREM 5.1. Let 1 < 0. < 00, 0 < p < 00 and 1 ::; q < s := (0. + 2)/p, the following statements are equivalent: (i)
A~ ~
00.
Then, with
U(J.L) order boundedly.
(ii) A~) ~ U(J.L) order boundedly. (iii) (1lzI 2 )S E U(J.L). (iv)
XS ~ Lq(J.L)
boundedly.
(v) XS ~ Lq(J.L) order boundedly. (vi) XS ~ Lq(J.L) boundedly.
(vii) XS
~
Lq(J.L) order boundedly.
PROOF. (i)::::} (ii) is obtained as before, by considering separately the cases p ;:::: 1 and p < 1. In order to prove (ii)::::} (iii) it suffices to look at the functions ka,p,q' (iii)::::} (iv) and (iv)::::} (vi) as well as (iv) <=?{v) and (vi) ¢:} (vii) are easily verified. Finally, for (vii)::::} (i), just observe that A~ ~ Xs. 0 Various statements related to boundedness of Carleson embeddings do have 'order bounded counterparts'. The first example is:
00
COROLLARY 5.2. Suppose that 1 < 0.,0.' < 00,0 < p,p' < are such that q. (0. + 2)/p = q'. (0.' + 2)/p'. Then
A~ ~ U (J.L) order boundedly
¢:}
00
and 1::; q,q' <
A~, ~ U' (J.L) order boundedly.
Again, in many cases, reduction to Hilbert spaces is possible.
CARLESON EMBEDDINGS FOR WEIGHTED BERGMAN SPACES
227
COROLLARY 5.3. If 1 < a, ci < 00, 0 < p < 00 and 1 :::; q < 00 are such that a' +2 = q. (o+2)/p, then A~ '+ Lq(f.L) order boundedly if and only if A~, '+ L2(f.L)
exists as a Hilbert  Schmidt operator. A special case occurs when f.L is a measure conclude:
U'(3.
From (iii) of Theorem 5.1 we
5.4. If 1 < a,(3 < 00, 0 < p < 00 and 1 < q < order boundedly if and only if (a + 2)/p < ((3 + l)/q .
COROLLARY
A~
'+
A~
00,
then
In such case, we can even factorize: A~ '+ Xs '+ A~ where s = (a + 2)/p.  There is of course no problem in verifying this corollary directly. If (a + 2)/p < ((3 + l)/q, then A~ '+ A~ is order bounded and so qsumming; here 1 < a, (3 < 00 and 1 :::; p, q < 00. In many cases, the converse (which doesn't hold for general Banach space operators) is true for Carleson embeddings. To see why, we need to look at special sets. With each z E U, we associate the 'interval' in au:
I(z) :=
{I:I ei9 :
11"(1
Izl) :::; () :::; 11"(1  IZI)} ,
and the 'squares' in U:
R(z) := {w E U:
Izl < Iwl < 1, (w/lwl)
E I(z)}
and
Q(z) := {w E R(z) :
Iwl < (1 + Izl)/2} .
It can be shown that the sets Q(Tln) form a partition of U whenever (TIn) is a [sufficiently fine] sequence of points in U having the properties (1)  (3) listed in part (iii) =} (iv) of the proof of Theorem 4.2. PROPOSITION 5.5. Let f.L be a (a, p, q)  Carles on measure with 1 < a < 00 and 1 < p, q < 00. If p* :::; q < 00, then A~ '+ L q(f.L) is order bounded iff it is q  summing iff it is (q, p*)  summing.
This is due to T. Domenig [3] for composition operators acting between weighted Bergman spaces. For the sake of completeness, we sketch a proof of the proposition which follows closely Domenig's arguments. PROOF. If I : A~ '+ Lq(f.L) is order bounded, then it is qsumming and so (q,p*)summing. Suppose now that I is (q, p*)  summing. By standard  but lengthy  calculations it can be shown that the functions
HANS JARCHOW AND URS KOLLBRUNNER
228
form a weak ff sequence in A~. Hence we get from our hypothesis that, with a suitable constant C, 00
>
L llvn(zW d/L(z) 2 L 1 n
n
IlJ
Ivn(zW d/L(z)
Q(T/n)
L [ n
> C·
1Q (T/n)
lzI 2 )(<>+2)/p E Lq(/L),
2 )l/p· )q(a+2)
11 1Jn z
L1 n
Thus (1
((1 11JnI
l
(1 lzI 2)q(<>+2)/Pd/L(z) .
Q(T/,,)
and so ] is order bounded by Theorem 5.1.
More is available. Consider the Rademacher functions
Tn : [0, 1]
~
1R : t
f+
sign sin (2nrrt) , n E N
(or any sequence of independent symmetric Bernoulli variables). Given 0 < p < 00, Khinchin's inequality assures the existence of positive constants Ap and Bp such that, for any finite collection of scalars al,"" an: Ap'
(
L n
)
lakl 2
k=l
1/2 ::; ([1 10 IL akTk(t) I dt n
P
) l/p
::; Bp'
k=l
(
L n
)
lakl 2
1/2 .
k=l
Pursuing the fate of this inequality within the framework of Banach spaces leads to the theory of type and cotype of Banach spaces, and to the following related class of operators (compare [2], Chs. 11 12). A Banach space operator u : X > Y is almost summing,
u E IIas(X, Y) , if there is a constant C such that, for any choice of finitely many vectors from X,
(10[III {; rk(t)UXk 112 dt )1/2 ::; C x.~~x. ( n
n
{;
Xl, ... ,
Xn
)1/2
I(x*, xk)1 2
It is known that each of the operator ideals IIp is properly contained in IIas. Moreover, if 1 ::; p < 00 and r = max{p,2} then IIash X) c II r ,2(', X) whenever X is an £P space, or the Schatten pclass Sp(H) for some Hilbert space H. In addition, it was shown by S. Kwapien [9] that • if H is a Hilbert space and u is in IIas(H, Y) then the adjoint 11,* : Y* > H
is 1  summing. See [2], p.255 for details. We have the following application to Carleson embeddings. The argument is the same as for composition operators between weighted Bergman spaces [3]. PROPOSITION 5.6. Let /L be an (a,p, q)  Carles on measure where 1 ::; q < 00., and 2 ::; p < 00. The embedding] : A~ '+ Lq(/L) is almost summing if and only if it is order bounded. PROOF. Define "I > 1 by ("I + 2)/2 = (a + 2)/p. Since p 2 2, A~ :::::} ]: A~ '+ Lq(/L) is almost summing :::::} ]* is Isumming (Kwapien) :::::} ] is order bounded (Garling)
'+
A~.
CARLESON EMBEDDINGS FOR WEIGHTED BERGMAN SPACES
229
Combining the preceding two propositions yields: COROLLARY 5.7. Let 1 < ct < 00 and 1 ::; p, q < 00 be such that p 2: min {q* ,2} and let 11, be an (ct, p, q)  Carleson measure. The embedding I : A~ "> Lq(/.l) is q  summing iff it is order bounded. PROOF. Only sufficiency requires proof. If p 2: q*, then Proposition 5.5 settles the case. And if p 2: 2, then I, being q  summing, is almost summing, and so order bounded by Proposition 5.6. 0
References [IJ R.R. Coifman, R. Rochberg, G. Weiss, Facto'rization theorems for Hardy spaces in seveml variables. Ann. of Math. (2) 103. (1976),611635. [2J J. Diestel, H. Jarchow, A. Tonge: Absolutely Summing Opemtors. Cambridge University Press 1995. [3J T. Domenig, CompoS'ition opemtors on weighted Beryman sp(,ces and Hardy spaces. Dissertation University of Zurich 1997. [4J T. Domenig, H. Jarchow, R. Riedl, The domain space of an analytic composition opemtor. Journ. Austral. Math. Soc. 66 (1999), 5665. [5J D.J.H. Garling, Lattice bounding, Radonifying and summing mappings. Math. Proc. Camb. Phil. Soc. 77 (1975), 327333. [6J W.W. Hastings, A Carleson measure theQrem for Beryman spaces. Proc. Amer. Math. Soc. 52 (1975), 237241. [7J C. Horowitz Zeros of functions in the Bergman spaces. Duke Math. Journ. 41 (1974), 693710. [8J S. Kwapien, On a theorem of L. Schwartz and its applications to absolutely summing opemtors. Studia Math. 38 (1970), 193201. [9J S. Kwapien, A remark on p  summing opemtors in fr  spaces. Studia Math. 34 (1970), 277278. [lOJ J. Lindenstrauss, A. Pelczynski, Contributions to the theory of classical Banach spaces. Journ. Funct. Anal. 8 (1971), 225249. [l1J D.H. Luecking, Multipliers of Bergman spaces into Lebesgue spaces. Proc. Edinb. Math. Soc. 29 (1986), 125131. [12J D.H. Luecking, Embedding theorems for spaces of analytic functions via Khinchine's inequality. Mich. Math. Journ. 40 (1993), 333358. [13J P. MeyerNieberg, Banach Lattices. SpringerVerlag 1991. [14J E. Oja, Pitt Theorem for nonlocally convex spaces f p • Preprint. [15J V.L. Oleinik, B.S. Pavlov, Embedding theorems for weighted classes of harmonic functions. Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov 22 (1971),94102. Transl. in Journ. Soviet Math. 2 (1974), 135142. [16J H.R.Pitt, A note on bilinear forms. Journ. London Math. Soc. 11, 171174 (1936). [17J H.P. Rosenthal, On quasicomplemented subspaces of Banach spaces with an appendix on compactness of opemtors from LP(p,) to Lr(/I). Journ. Funct. Anal. 4 (1969), 176214. [18J W. Rudin, Real and Complex Analysis. 3 rd ed., McGrawHill 1987. [19J J.H. Shapiro, Mackey topologies, reproducing kernels, and diagonal maps on the Hardy and Bergman spaces. Duke Math. Journ. 43 (1976), 187202. [20J W. Smith, Composition opemtors between Bergman and Hardy spaces. Trans. Amer. Math. Soc. 348 (1996) 23312348. [21J W. Smith, L. Yang, Composition opemtors that impro1Je integmbility on weighted Beryman spaces. Proc. Amer. Math. Soc. 126 (1998) 411420. [22J K. Zhu, Opemtor Theory in Function Space.~. Marcel Dekker, New York 1990.
HANS JARCHOW AND URS KOLLBRUNNER
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INSTITUT FUR MATHEMATIK, UNIVERSITAT ZURICH, WINTERTHURERSTRASSE
190, CH 8057
ZURICH, SWITZERLAND
Email address:jarchowlDmath.unizh.ch INSTITUT FUR MATHEMATIK, UNIVERSITAT ZURICH, WINTERTHURERSTRASSE ZURICH, SWITZERLAND
Email address:kollbrunlDmath.unizh.ch
190, CH 8057
Contemporary Mathematics Volume 328, 2003
Weak* extreme points of injective tensor product spaces Krzysztof Jarosz and T. S. S. R. K. Rao ABSTRACT. We investigate weak* extreme points of the injective tensor product spaces of the form A ®. E, where A is a closed subspace of C (X) and E is a Banach space. We show that if x E X is a weak peak point of A then f (x) is a weak*extreme point for any weak*extreme point f in the unit ball of A ®. E C C (X, E). Consequently, when A is a function algebra, f (x) is a weak*extreme point for all x in the Choquet boundary of A; the conclusion does not hold on the Silov boundary.
1. Introduction
For a Banach space E we denote by E1 the closed unit ball in E and by BeE1 the set of extreme points of E 1 . In 1961 Phelps [16] observed that for the space C(X) of all continuous functions on a compact Hausdorff space X every point f in Be (C (X))1 remains extreme when C (X) is canonically embedded into its second dual C (X)**. The question whether the same is true for any Banach space was answered in the negative by Y. Katznelson who showed that the disc algebra fails that property. A point x E OeE1 is called weak* extreme if it remains extreme in BeEi*; we denote by B;E1 the set of all such points in E 1. The importance of this class for geometry of Banach spaces was enunciated by Rosenthal when he proved that E has the RadonNikodym property if and only if under any renorming the unit ball of E has a weak* extreme point [19]. While not all extreme points are weak* extreme the later category is among the largest considered in the literature. For example we have: strongly exposed S;; denting S;; strongly extreme S;; weak* extreme. We recall that x E E1 is not a strongly extreme point if there is a sequence Xn in E such that Ilx ± xnll t 1 while IIxnll ~ 0 (see [3] for all the definitions). We denote by O;E1 the set of strongly extreme points of E 1. It was proved in [14] that e E O;E1 if and only if e E o;Ei* (see [9], [13], or [17] for related results). Examples of weak* extreme points that are no longer weak* extreme in the unit ball of the bidual were given only recently in [6]. In this paper we study the weak* extreme points of the unit ball of the injective tensor product space A®,E, where A is a closed subspace of C(X). Since C(X)®,E Both authors were supported in part by a grant #0096616 from DST/INT/US(NSFRP041)/2000.
© 231
2003 American Mathematical Society
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KRZYSZTOF JAROSZ AND T. S. S. R. K. RAO
can be identified with the space C(X, E) of Evalued continuous functions on X, equipped with the supremum norm, elements of A ®e E can be seen as functions on X. We are interested in the relations between 1 E a; (A ®£ E) and 1 (x) E a;E1 , for all (some?) x E X. Since any Banach space can be embedded as a subspace A of a C (X) space no complete characterization should be expected in such very general setting. For example if A is a finite dimensional Hilbert space naturally embedded in C (X), with X = Ai, and dim E = 1, then any norm one element 1 of A ®£ E is obviously weak* extreme however the set of points x where 1 (x) is extreme is very small consisting of scalar multiples of a single vector in Ai. Hence in this note we will be primarily interested in the case when A is a sufficiently regular subspace of C (X) and/or x is a sufficiently regular point of X. It wa..<; proved in [5] that
1 E a:C(X, Eh
(1.1)
{::::::::}
[I (x)
E a:EI , for all x EX].
It follows from the arguments given during the proof of Proposition 2 in [6] that for a function 1 E (A ®€ E)l we have
[I (x)
E a;El for all x E X with c5 x E aeAi]
=> 1 E a; (A ®e E)I'
where we denote by c5x the functional on A of evaluation at the point x. In this paper we obtain a partial converse of the above result (Theorem 1). Our proof also shows that if 1 E a; (A ®€ E)l then 1 (x) E a;El for any weak peak point x (see Def. 1), extending one of the implications of (1.1). It follows that when A is a function algebra then any weak* extreme point of Al is of absolute value one on the Choquet boundary ChA (and hence on its closure, the Shilov boundary) and consequently is a strongly extreme point [17]. Since we have concrete descriptions of the set of extreme points of several standard function algebras (see e.g. [12], page 139 for the Disc algebra) one can give easy examples of extreme points that are not weak* extreme. Recently several authors have studied the extremal structure of the unit ball of function algebras ([1], [15], [18]). It follows from their results that the unit ball has no strongly exposed or denting points. Our description that strongly extreme and weak* extreme points coincide for function algebras and are precisely the functions that are of absolute value one on the Shilov boundary completes that circle of ideas. We also give an example to show that the weak* extreme points of (A ®, E)l in general need not map the Shilov boundary into aeE1 • Considering the more general case of the space of compact operators K (E, F) (we recall that under assumptions of approximation property on E or F*, K (E, F) can be identified with E* ®£ F) we exhibit weak*extreme points T E K (fP)1 for 1 < p =I= 2 < 00 for which T* does not map unit vectors to unit vectors. Our notation and terminology is standard and can be found in [3], [4], or [11]. We always consider a Banach space as canonically embedded in its bidual. By E(n) we denote the nth dual of E. By a function algebra we mean a closed subalgebra of a C (X) space separating the points of X and containing the constant functions; we denote the Choquet boundary of A by ChAo
2. The result
a;
As noticed earlier, for A c C (X) and a point 1 E (A ®£ Eh we may not have 1 (x) E a;EI for all x E X even in a finite dimensional case. Hence we need to define a sufficiently regular subset of X in relation to A.
WEAK' EXTREME POINTS
233
DEFINITION 1. A point Xo E X is called a weak peak point of A C C (X) if for each neighborhood U of Xo and £ > 0 there is a E A with 1 = a (xo) = Iiall and la (xo)1 < £ for x E X\U; we denote by opA the set of all such points in X.
There are a number of alternative ways to describe the set opA. If Xo E X is a weak peak point of A C C (X), /l is a regular Borel measure on X annihilating A, and al' is a net in A convergent almost uniformly to 0 on X\ {xo} and such that a.., (xo) = 1 then /l({xo}) = liml'Jxal' = O. Hence if a* E A* and VI,V2 are measures on X representing a* we have VI ({Xo}) = V2 ({xo}), consequently V
~ v ( {xo}) is a well defined functional on A * .
On the other hand if X {xo} E A ** then /l ( {xo}) = 0, for any annihilating measure /l, and Xo is a weak peak point. To justify the last claim notice that Al is weak*dense in Ai* so X{xo} is in the weak*closure of the set K = {f E Al : f (xo) = I}. Let U be an open neighborhood of Xo and Ax\U be the space of all restrictions of the functions from A to X\U. We define the norm on Ax\U as sup on X\U. Lct K X\U be the set of restrictions ofthe functions from K and cl (Kx\U) be the norm closure of Kx\U C Ax\U. If 0 ~ clKx\U then there is G E (Ax\U)* , represented by a mcasure 1] on X\U and separating Kx\U from 0: Re G (h) > a > 0
=
X{xo}
(/l)
I
for all hE clKx\U·
The measure 1] extends G to a functional on A so K is functionally separated from 0 in A contrary to our previous observation. Hence 0 E clKx\u so there is a function in K that is smaller then £ outside U which means that Xo is a weak peak point. The concept of weak peak points is well known in the context of function algebras where opA coincides with the Choquet boundary ([8]' p. 58). For more general spaces of the form Ao 'f1 {foa E C (X) : a E A} I where A C C (X) is a function algebra and fo a nonvanishing continuous function on X we have ChA ~ opAo. Spaces of these type appear naturally in the study of singly generated modules and Morita equivalence bimodules in the operator theory [2J. THEOREM 1. Let E be a Banach space, X a compact Hausdorff space, and A a closed subspace of C (X). If f E A ®, E is a weak* (strongly) extreme point of the unit ball then f (x) is a weak* (strongly) extreme point of the unit ball of E for any x E opA. In particular if f E (C(X,E))I is a weak*(strongly) extreme point then f(x) is a weak* (strongly) extreme point of EI for all x EX.
We first need to show that for a weak peak point Xo E X there exists a function in A not only peaking at Xo but that is also almost real and almost positive. LEMMA
1. Assume X is
(L
compact Hausdorff space, A is a closed subspace of
C (X), and Xo is a weak peak point of A. Then for each neighborhood U of Xo and £
> 0 there is g
E A such that
Ilgll = 1 = g (xo) , (2.1)
Ig (x)1
IIRe+ g where Re+ z
= max{O,Rez}.
< £, for all x
gil < £,
E X\U, and
KRZYSZTOF JAROSZ AND T. S. S.
234
R. K.
RAO
PROOF. Put U1 = U and let gl E A be such that IIg111 = 1 = gl (xo) and Ig1 (x)1 < e for x
i
U1.
Put U2 = {x E U1 : Ig1 (x)  11 < e} and let g2 E A be such that IIg211
= 1 = g2 (xo)
and Ig2 (x)1 < e for x
i
U2·
Put U3 = {x E U2 : Ig2 (x)  11 < e}. Proceeding this way we choose a sequence {gn}n>l _ in A. Fix a natural number k such that k> 1e: and put 1 k
g=
k Lgj· j=l
We clearly have Ilgll = 1 = 9 (xo) and Ig (x)1 < e for x i U. Let x E U, then either x belongs to all of the sets Uj , j k, in which case Ig (x)  11 < e, or there is a natural number p < k such that x E Up \ Up+!. In the later case we have
:s
Ig(X)_P~II=~
tgj(Pl) ;=1
<
.!. (
(lg1 (x)  11
pl
1
< k e + k Hence IIRe+ 9 
gil
+ ... + Igp1 (x) 
11)
+ Igp (x)1 + (lgp+dx)1 + ... + Igk (x)!)
 k
)
kp
+ k e < e. o
< e.
We are now ready to finish the proof of the Theorem. PROOF. Suppose! (xo) is not a weak*extreme point. Then by [9] there is a 1 + ~ and e* (en) ~ O. sequence en in E1 and e* E Ei such that II! (xo) ± enll By the Lemma there is a sequence gn in A such that
:s
Ilgnll = 1 = gn (xo) , (2.2)
ign (x)1
<.!., n
IIRe+ gn  gnll
< .!., n
if II! (x)  ! (xo) II
~ .!., n
and
Hence II! (x) ± 9 (x) e II < max { sUPllf(x)f(xo)II~,* {II! (x)11 + Ign (x)llIenll}, } n n sUPllf(x) f(xo)lI< {II! (x) ± gn (x) en II}
'*
:s max {I +.!.,.!. + II! (xo) ± Re+ gn (x) en II + .!.} n n n 3
<  1+. n Therefore II! ± gnenll > 1 but (8 (xo) ® e*)(gnen) = gn (xo) e* (en) ~ O. This contradiction shows that! (xo) is a weak*extreme point. The same line of arguments shows that! (x) is strongly extreme for any strongly extreme! E (A ®, E)l' 0 Since for a function algebra A the Choquet boundary C hA coincides with 8p A ([8], p. 58) and the Shilov boundary 8A is equal to the closure of ChA we have:
235
WEAK' EXTREME POINTS
1. Let A be a function algebra, E a Banach space and f a weak* extreme point. Then
COROLLARY
Eh
E
(A ®e
f(x) E 8;E1 , for x E ChA, and Ilf (x)11 = 1, for x E 8A. REMARK 1. Theorem 1 is not valid for the spaces WC(X, E) of Evalued continuous functions with E quipped with the weak topology. Even a strongly extreme point of WC(X, Eh need not assume extremal values at all points of X [13].
We next give an example of a function algebra A and a 3dimensional space showing that a weak* extreme point f E (A ®e Eh need not take extremal values on the entire Shilov boundary. Since E is finite dimensional this function f maps the Choquet boundary into the set of strongly extreme points but f is not a strongly extreme point. E
EXAMPLE
1. Put
Q = {(z,w,O) E (:3: Izl2 + Iwl 2 ::; I} U {(O,w,u) E (:3: max{lwl, lui}::; I}, and B = convQ. Let 11·11 be the norm on (:3 such that B is its unit ball. Note that (z,w,O) is an extreme point ofB ifflwl =I 1 and Iz12+lw12 = 1. PutE = ((:3,11·11), X'!!:. {O} x {I} x lD>u {(sint,cost,O): 0::; t::; 11"},
fo : X
+
df
El, fo (x) = x, and A = {h E C (X) : h (0, 1,·) E A (lD>)},
where A (lD» is the disc algebra. We have ChA = {O} x {I} x 8lD> U {(sin t, cos t, 0) : 0 < t < 11"} . The function fo is in A ®e E and takes extremal values on the Choquet boundary of A so it is a weak* extreme points of (A ®e E)l. However fo (0, ±1, 0) = (0, ±1, 0) are not extreme points of Ei while (0, ±1, 0) are in the Shilov boundary of A. Since E is finite dimensional clearly the function fo maps the Choquet boundary of A into the set of strongly extreme points of E 1 • We next show that f is not a strongly extreme point. Let gn E A be such that Ilgnll
gn (sint, cost, 0)
= 1 = gn
(sin
~,cos ~,o) ,
= 0, for ~ < t::; 1, and n
gn (0, 1, z) = 0, for z E lD>. Put fn = (O,O,gn) E A ®e E. We have (fo ± fn)(a, b, c) Hence IIf ± fnll
+
(0,1, c) for = { (a, b, ±gn) for
(a,b,c) E {O} x {I} x 8lD> (a, b, c) E {(sin t, cos t, 0) : 0 ::;
t ::; 11"} .
1 but Ilfnll ~ 0 so f is not a strongly extreme point.
In the next Proposition we consider a more general setting of compact operators. For a Banach space E we denote by C(E) the space of all linear bounded maps on E, by K(E) the set of all compact linear maps, and by S(E) the set of
KRZYSZTOF JAROSZ AND T. S. S. R. K. RAO
236
unit vectors in E. Since K (E, C (X)) can be identified with C (X, E*) our result on weak* extreme points taking weak* extremal values can be interpreted as follows T E a;K(E,C(X))l
==}
T* (a;C(X);) c a;Er.
Thus more generally one can ask whether T* (a; Ft) c a; Ei for any TEa; K(E, Fh. We give a class of counter examples with the help of the following proposition. PROPOSITION 1. Let E be an infinite dimensional Banach space such that K(E) is an Mideal in C(E). 1fT E K(Eh then T*(aeEi) ct. S(E*).
We recall that a closed subspace M of a Banach space E is an AIideal if there is a projection P E C (E*) such that ker P = Ml. and liP (e*)II+lle*  P (e*)11 = Ile*ll, for all e* E E* (see [11] for an excellent introduction to 1\/ideals). PROOF. Since qE) is an AIideal it follows from Corollary V1.4.5 in [11] that E* has the RadonNikodym property and hence the IP (see [10]). Also since qE) is a proper Mideal it fails the IP. It therefore follows from Theorem 4.1 in [10] that there exists a net {x~} c e Ei such that x~ > Xu in the weak* topology with Ilxoll < 1. Suppose T*(aeEi) c S(E*). Since T* is a compact operator by going through a subnet if necessary we may assume that T*(x~) > T*(xo) in the norm. Thus 1 = IIT*(xo)11 < 1 and the contradiction gives the desired conclusion. 0
a
EXAMPLE 2. Banach spaces E for which K(E) is an AI ideal in C(E) have been well extensively studied. Chapter VI of [11] provides seveml examples including E = p , 1 < p < 00, as well as properties of these spaces. It was observed in [6] that for p # 2 there are weak*extreme points in the space K(ePh. It follows from the last proposition that the adjoint of these weak" extreme points do not even map extreme points to unit vectors.
e
A strongly extreme point remains extreme in all the dual spaces of arbitrary even order. A weak* extreme point remains extreme in the second dual but may not be extreme in the fourth dual. Hence the property of remaining extreme in all the duals of even order is placed between the strong and the weak* type of extreme points. It would be interesting to describe that property in terms of the original Banach space alone. A procedure for generating extreme points which have this property but are not strongly extreme was described in [6]. PROPOSITION 2. Let X be a compact Hausdorff space, A a closed subspace of C(X), and E a Banach space. Suppose Xo E X is a weak peak point and f E A®. E is an extreme point in the unit ball of all the duals of even order. Then f (xo) is an extreme point of the unit ball of all the duals of E of even order. PROOF. Since the space A ®. E** can be canonically embedded in (A ®. E)** [7] we have, for any natural number n
A ®. E(2n) C (A ®( E(2n2))** C (A ®. E)(2n).
If f E A ®. E is an extreme point of (A ®. E)(2n+2) then it is a weak*extreme point of (A ®. E)(2n), as it also belongs to A ®. E(2n) it is a weak*extreme point of A ®. E(2n). Hence by our theorem f (xo) is an extreme point of E~2n). 0 The next proposition characterizes strongly extreme points in terms of ultrapowers.
WEAK' EXTR.EME POINTS
PROPOSITION
of the unit ball
El
237
3. An element e of a Banach space E is a strongly extreme point if and only if (e).:F is an extreme point of (E.:Fh
PROOF. If e ¢. O;El then there is a sequence {en}n~l eEl with lie ± enll + 1 and infnEN Ilenll > o. Thus II (e).:F ± (en).:F11 = 1 and II (en).:F11 =I 0 so (e).:F is not an extreme point. If (e).:F ¢. oe(E.:Fh then there is 0 =I (en).:F E (E.:Fh with 1 = II (e).:F ± (en).:F11 = lim.:F lie ± enll· Thus for every € > 0 the set {n E N: lie ± enll ~ 1 + €} is none empty as an element of F. Hence there exists a sequence {k n } such that lie ± ek n II + 1 but Ilek" II A 0 so e is not a strongly extreme point. 0
References [1] P. Beneker and J. Wiegerinck, Strongly exposed points in 'Uniform algebras, Proc. Amer. Math. Soc. 127 (1999) 15671570. [2] D. Blecher and K. Jarosz, Isomorphisms of function modules, and generalized approximation in modulus, Trans. Amer. Math. Soc. 354 (2002), 36633701 [3] R. D. Bourgin, Geometric aspects of convex sets with the RadonNikodym property, LNM 993, Springer, Berlin 1983. [4] A. Browder, [ntroduction to Function Algebras, W. A. Benjamin, New York 1969. [5] P. N. Dowling, Z. Hu and M. A. Smith, Extremal structure of the unit ball of C(K, X), Contemp. Math., 144 (1993) 8185. [6] S. Dutta and T. S. S. R. K. Rao, On weak*extreme points in Banach spaces, preprint 2001. [7] G. Emmanuele, Remarks on weak compactness of operators defined on injective tensor products, Proc. Amer. Math. Soc., 116 (1992) 473476. [8] T. Gamelin, Un'iform Algebras, Chelsea Pub. Comp., 1984. [9] B. V. Godun, BorLuh Lin and S. L. Troyanski, On the strongly extreme points of convex bodies in separable Banach spaces, Proc. Amer. Math. Soc., 114 (1992) 673675. [10] P. Harmand and T. S. S. R. K. Rao, An intersection property of balls and relations with Mideals, Math. Z. 197 (1988) 277290. [11] P. Harmand, D. Werner and W. Werner, Mideals in Banach spaces and Banach algebras, Springer LNM No 1547, Berlin 1993. [12] K. Hoffman, Banach spaces of analytic functions, Dover 1988. [13] Z. Hu and M. A. Smith, On the extremal structure of the unit balls of Banach spaces of weakly continuous functions and their duals, Trans. Amer. Math. Soc. 349 (1997) 19011918. [14] K. Kunen and H. P. Rosenthal, Martingale proofs of some geometric results in Banach space theory, Pacific J. Math. 100 (1982) 153175. [15] O. Nygaard and D. Werner, Slices in the unit ball of a uniform algebra, Arch. Math. (Basel) 76 (2001) 441444. [16] R. R. Phelps, Extreme points of polar convex sets, Proc. Amer. Math. Soc. 12 (1961) 291296. [17] T. S. S. R. K. Rao, Denting and strongly extreme points in the unit ball of spaces of operators, Proc. Indian Acad. Sci. (Math. Sci.) 109 (1999) 7585. [18] T. S. S. R. K. Rao, Points of weaknorm continuity in the unit ball of Banach spaces, J. Math. Anal. Appl., 265 (2002) 128134. [19] H. Rosenthal, On the nonnorm attaining functionals and the equivalence of the weak' KMP with the RNP, Longhorn Notes, 198586. DEPARTMENT OF MATHEMATICS AND STATISTICS. SOUTHERN ILLINOIS UNIVERSITY, EDWARDSVILLE.
IL 620261653, USA Email address: kjaroszlDsiue. edu URL: http://www.siue.edu/kjarosz/
R. V. COLLEGE POST, BANGALORE 560059. INDIA Email address:tsslDisibang.ac . in
INDIAN STATISTICAL INSTITUTE,
Contemporary Mathematics Volume 328. 2003
Determining Sets and Fixed Points for Holomorphic Endomorphisms KangTae Kim and Steven G. Krantz The authors study the fixed point sets of a holomorphic endomorphism of a domain in complex space. Sufficient (and necessary) conditions are givenon the number and configuration of the fixed pointsfor the endomorphism to be forced to be the identity. The proofs depend on certain key ideas from differential geometry, particularly the notions of cut locus and Hadamard ABSTRACT.
length space.
1. Introduction
This article concerns the study of the concept of determining set for a collection of holomorphic mappings. We first give the definition. DEFINITION 1.1. Let M be a complex manifold, and let Aut (M) be the collection of biholomorphic mappings of M into itself. We call a subset Z c M a determining set for Aut (M) (or, equivalently, an Aut (M)determining set), if any map f E Aut (M) satisfying f(p) = p for every p E Z is in fact the identity map of
M. We observe first that this article is related to the authors' collaboration with Burna Fridman and Daowei Ma (see [FKKM]), which was originally inspired by the following remarkable theorem in complex dimension one. THEOREM 1.2. Let n be a domain in the complex plane C and let f : n + n be a biholomorphic (conformal) mapping. If there are three distinct points Pl,P2,P3 in n such that f(pj) = Pj, for j = 1,2,3, then f is the identity map. The higherdimensional analog of this theorem given in [FKKM] is as follows: THEOREM 1.3. (FridmanKimKrantzMa [FKKM]) Let M be a connected, complex manifold of dimension n admitting a complete invariant Hermitian metric. 2000 Mathematics Subject Classification. 32H02, 32H50, 32H99. Key words and phrases. fixed point set, holomorphic mapping, cut locus, Hadamard length
space. K. T. Kim supported in part by grant ROl199900005 from The Korean Science and Engineering Foundation. Steven G. Krantz supported in part by grant DMS9988854 from the National Science Foundation.
© 239
2003 American Mathematical Society
240
KANGTAE KIl\1 AND STEVEN G. KRANTZ
Then a determining set consisting of n + 1 points exists for the automorphisms of .M. Furthermore, the choice of such a determining set is generic. Throughout this paper we shall discuss both endomorphisms and automorphisms. If M is a complex manifold then an endomorphism of .I'IJ is any holomorphic mapping
DETERMINING SETS FOR HOLOMORPHIC ENDOMORPHISMS
241
2. The Case of Convex Domains Let us consider a bounded strongly convex domain n in en with a smooth (Ck, k 2 6) boundary. By the wellknown work of Lempert ([LEM]), for each pair of distinct points p, q E n, there exists a unique holomorphic map 'P : ~  n such that (1) 'P(O) = p and 'P(~) = q for some ~ E ~, and (2) 'P*dn = dLl, where d denotes the Kobayashi distance. We call such a map 'P a complex geodesic joining p and q. We now consider the holomorphic endomorphisms of n fixing two given points. LEMMA 2.1. Let n be a bounded, strongly convex domain in en with Ck smooth boundary for some k 2 6. Let p, q E n be two distinct points and let 'P denote a complex geodesic joining p and q. If a holomorphic mapping f : n  n satisfies the condition that f(p) = p and f(q) = q, then it holds that f('P(()) = 'P(() for every (E ~. PROOF. Let f and p, q be as in the hypothesis. Let 'P : ~  n be a complex geodesic joining p and q, with 'P(O) = P and 'P(~) = q. Then let "( : [0, f]  ~ be the unit speed geodesic in ~ with "((0) = 0 and "((e) =~, where £ = dLl(O,~). Let o :S t :S f and let r = "((t). Then we see that
dn(p, q)
< < <
dn(f(p), f(q)) dn (f(p), f('P(r))) + dn(f('P(r)), f(q)) dn('P("((O)), 'P("((t))) + dLl('P("((t)), cp("((£))) dLl(,,((O), "((t)) + dLl("((t), "((f)) dLl(O,~)
dn(p, q). Because of the distancedecreasing property of the Kobayashi metric and the fixed point conditions, we see from the above that
where dn(p, q) = £. Notice that every Kobayashi distance ball is strictly convex, as our domain n is a strongly convex domain with smooth boundary (see [LEM]). Hence the above observations together with the fact that
dn(p,cp(r)) = t, dn('P(r)) = f  t imply that f('P(r)) = 'P(r). Consequently, the map f fixes every point in the set 'P 0 "(( [0, f]). Hence the two maps f 0 'P and 'P of ~ into n coincide along a curve in the unit disc ~. Therefore f 0 'P(() = 'P(() for every ( E ~, as claimed. 0
In other words, we have shown that any holomorphic endomorphism of a bounded strongly convex domain in en fixing two distinct points must fix every point that belongs to the complex geodesic passing through the two fixed points. We immediately ohtain the following general result on the determining sets for holomorphic endomorphisms of a bounded convex domain.
KANGTAE KIM AND STEVEN G. KRANTZ
242
LEMMA 2.2. Let 0 be a bounded, strongly convex domain in en with C k (k ~ 6) smooth boundary. Let PO,Pl,'" ,Pn be points in 0 chosen in such a way that the complex geodesics passing through Po and Pj (j = 1, ... n) have tangent vectors at Po that are linearly independent over IC. Then any holomorphic mapping f: 0 > 0 fixing po, ... ,Pn must fix every point ofO. PROOF. Notice that the current hypothesis together with the preceding lemma implies that dfpo is the identity map. Therefore a theorem of H. Cartan implies that f is in fact the identity mapping. 0 We remark that the choice for Po, ... ,Pn is generic. To formulate this notion more precisely, we consider the cartesian product rrnHo of (n + 1) copies of O. In fact it is shown in [FKKM] that there exists an open dense subset U of rrn+10 such that any element of U gives (n + 1) points that satisfy the sufficiency condition of the preceding lemma. We summarize the result more elegantly in the following statement. THEOREM 2.3. For a bounded, strongly convex domain in en, there exists a collection of n + 1 points such that any holomorphic endomorphism of the domain fixing them must fix every point in the domain. Moreover, the choice of such n + 1 points is generic. REMARK 2.4. We point out that the result of this section concerns the class of bounded, strongly convex domains, which is a rather special collection of objects. However, in compensation, we emphasize that we have treated general holomorphic endomorphisms, rather than just biholomorphic selfmaps.
3. Hadamard Spaces In light of the preceding section, we would like to present in this section a description of the underlying metric space principles that we use in the study of determining sets. Let (X, d) be a metric space, equipped with the distance function d : X x X > lR. By an isometry we mean a selfmapping f : X > X satisfying the condition:
d(J(p), f(q)) = d(p, q), Vp, q E X. We denote by Isom(X) the collection of isometries of (X, d). Imitating the concept of "length spaces" that is commonly encountered in geometry (cf. [BUS]), we give the following definition. DEFINITION 3.1. Let"(: (a,b)
>
X be a continuous curve in X. We call it
minimal if d("((x), ,,((y)) = t  x + d("((t), ,,((y)), for every t,x,y with a < x:::; t:::; y < b. DEFINITION 3.2. A metric space (X, d) is called a length space if, for every pair of points p, q EX, there exists a minimal curve "( : [a, b] > X such that "((a) = P and "((b) = q. Furthermore, we call (X, d) a Hadamard space if the minimal curve joining each pair of points is unique up to a reversal of parametrization. Notice that any convex subset of Euclidean space is a Hadamard space with respect to the standard Euclidean distance. A strongly convex domain in en, equipped with the Kobayashi distance, is also a Hadamard space. Every complete,
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simply connected Riemannian manifold with nonpositive curvature is also an example of a Hadamard space. These are often called Hadamard manifolds; from this derives our terminology of Hadamard (length) space. Now. for the study of determining sets, we present this lemma. LEMMA 3.3. Let (X, d) be a Hadamard space and let p, q E X be two distinct points. If an isometry f : X > X fixes P and q, tllen f fixes every point on the minimal curve passing through P and q. PROOF. Let'Y : [O,f] > X be a minimal curve with 'Y(O) = p,'Y(f) = q. It is immediate to see that the isometry f of (X, d) has the property that f 0 'Y is also a minimal curve. Since P and q are fixed by f, and since (X, d) is Hadamard, it follows that f 0 'Y(t) = 'Y(t) for every t E [0, fl. Now consider the minimal curve passing through P and q. So far, we have shown that the portion of this minimal curve between P and q is pointwise fixed by f. We still must show that the other portion of the minimal curve is fixed pointwise by f. But this is a simple matter using the uniqueness of the minimal curve passing through any two given points and the minimalcurvepreserving property of isometries. This completes the proof. D We remark that it is possible to derive the same conclusions for locally Hadamard spaces but, in order to keep our exposition concise, we do not introduce the concept here. LEMMA 3.4. Let (X, d) be a Hadamard space and let U be an open subset of X. Then any isometry f fixing every point in U must be the identity map. PROOF. Let P E U. Then, for every minimal curve 'Y emanating from p, f fixes a point in 'Y n (U \ p). The preceding lemma implies now that f fixes every point of 'Y. Since every point in X can be joined to P by a minimal curve, this completes the proof. D Now we consider the concept of convex hull in a Hadamard space. We say that a set Q in a Hadamard space X is convex if every minimal curve joining P and q in X is contained in Q. For a subset A of a Hadamard space X, its convex hull W (A) is the smallest convex subset of X containing A. DEFINITION 3.5. Let Po, . .. ,Pm be points in a Hadamard space (X, d) with minimal geodesics 'Yl .... ,'Ym such that 'Yj passes through Po and Pi for every j = 1, .... m. We call the points Po, ... ,Pm spanning if the convex hull Whl U ... U'Ym) has nonempty interior. Now we have the following general result. PROPOSITION 3.6. If a Hadamard space (X, d) admits m + 1 points Po, PI, ... , Pm ill X that are spanning, then these m + 1 points constitute a determining set
for the isometries of( X, d). PROOF. Notice that the convex hull we obtain from the minimal curves through Po and Pi is fixed pointwise by any isometry that fixes the points Po,··· ,Pm. Then
the preceding lemma finishes the proof.
D
Observe that the full isometry condition is not really needed to prove the conclusion of the proposition. In fact, any distancedecreasing map will satisfy the
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same conclusion (just use the unique continuation principle). Notice that this offers an underlying principle for the determining set theorem for the holomorphic endomorphisms of strongly convex domains in the preceding section.
4. CHSubsets and Automorphisms In this section we demonstrate how the principles of the preceding section reflect upon the main theorem of [FKKM] and its proof. Let M be a connected, complex manifold that admits a smooth invariant Hermitian metric. Here the invariance refers to the property that every holomorphic automorphism of M is an isometry with respect to the Hermitian metric. For a moment, we take the real part of the Hermitian metric, and consider everything in terms of lliemannian geometry. Let P EM. Then we call q E M a cut point of P if there are at least two distinct geodesics joining P and q with the same minimal length. The collection of cut points for P is called the cut locus of p, which we denote by Cpo In [FKKMJ, a subset X of M was called CarlanHadamard ('CH' for short) if there exists Xo E X so that X does not intersect the cut locus C(xo) of Xo in M. Furthermore, we call such a CHset X generating if the set
Ip(X) := {'Y~(O) I 'Y is the unique normal geodesic from Xo to p, Vp
E
Z}
spans TxoM over C. Suppose now that X is a set of finitely many points, and that a certain holomorphic automorphism f fixes every point of X. Then, by complex differentiability, one picks up more geodesics than just the geodesics joining Xo and the other points of X. [That is to say, each geodesic tangent may be multiplied by i.] If x E X \ Xo and if 'Yx is the unit speed geodesic from Xo to x, then 9x == expxo(i"(~(O)) is also fixed, point by point, by f. Now it is not hard to see, using the exponential map and the tangent space, that the convex hull W = Wbl U 91 U ... U 'Ym U 9m) has nonempty interior. Notice that every point of the hull W is fixed by f pointwise. We obtain the following result as a consequence of Proposition 3.6. PROPOSITION 4.1. (Fridman/Kim/Krantz/Ma [FKKM]) Let M be a connected, complex manifold with an invariant Hermitian metric. Let X be a generating CHsubset of M. Then, whenever an automorphism fixes every point of X, it is in fact the identity map. In other words, every generating CHsubset is a determining set for automorphisms. The method of choosing a smallest (in the sense of inclusion of sets) generating CHsubset in a complex manifold with a complete invariant Hermitian metric has been explained in detail in [FKKM]. We briefly describe the paradigm. Choose an arbitrary p E M. Then the cut locus C(p) is a nowhere dense subset of M. Thus choose PI E M \ (C (p) U {p}). Then choose P2 away from C (P) and the complete geodesic through p and Pl. Then P3 will be chosen away from C(p) and the geodesic cone generated by P,Pl and P2. An inductive construction lets us choose p, PI, ... ,Pn which compose a spanning CHsubset of M. Thus we arrive at THEOREM 4.2. (FridmanKimKrantzMa [FKKM]) Let M be a connected ndimensional complex manifold admitting a complete invariant Hermitian metric. Then a determining set, consisting of n + 1 points, exists for the automorphisms of M. Furthermore, the choice of such a determining set is generic.
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Here, by "generic", we mean that the the collection of (n + I)tuples of points in Al x ... x M that satisfy our conclusion form a dense, open set. From the discussion above, if the metric happens to be distancedecreasing, in the sense that all holomorphic endomorphisms are distancedecreasing with respect to the given metric, then this theorem will hold for holomorphic endomorphisms. This result of course uses the idea developed in the preceding section about the distancedecreasing property together with the unique continuation property. We remark at this point that the collection of complex manifolds admitting a complete invariant Hermitian metric is rather large. For instance, every bounded pseudoconvex domain in is equipped with the complete KiihlerEinstein metric. See [MOY] (also [CHY], [OHS], [YAU]) for instance.
en
5. Examples, Counterexamples, and the Cut Locus One might have the impression that some transversality condition for m + 1 points might be sufficient for the determining set problem for holomorphic automorphisms. However, it is shown in [FKKM] that a simplistic topological transversality assumption will not be sufficient; consider the following statement. THEOREM 5.1. ([FKKM]) Fix a finite set K = {PI, ... , Pk} in n > 1. There exists a bounded domain containing K, and a subgroup H C Aut(D) isomorphic to the unitary group U(n  1) of enI, such that each element of H fixes every point of K. Moreover, unlike the onedimensional planar domain case, the consideration of the cut locus seems essential even for onedimensional Riemann surfaces. If one considers the torus coming from the lattice generated by {I, i}, then the map z +  z of e generates an automorphism on the torus. It is easy to see that it has 4 fixed points, and yet is different from the identity map. If one considers a twoholed torus with a wellbalanced fundamental domain centered at 0 in the Poincare disc, then the same map z + z of the disc will generate a nontrivial automorphism with 6 fixed points. In this way, one can generate arbitrarily many fixed points for a nontrivial automorphisms of compact Riemann surfaces of high enough genus. Since our discussion has not depended upon the completeness of manifolds, simple puncturing will create an arbitrary number of fixed points. Notice that all these examples have fixed points in the cut loci.
en,
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References [ALK] G. Aladro and S. G. Krantz, A criterion for normality in Cn, Jour. Math. Anal. and Appl. 161(1991), 18. [BED] E. Bedford and J. Dadok, Bounded domains with prescribed group of automorphisms., Comment. !'v'lath. Helv. 62 (1987), 561572. [BUS] H. Busemann, The geometry of geodesics, Academic Press, New York, NY, 1955. [CHY] S.Y. Cheng and S.T. Yau, On the existence of a compact Kahler metric, Comm. Pure App!. Math., 33 (1980), 507544. [FIF] S. D. Fisher and John Franks, The fixed points of an analytic selfmapping, Proc. AMS, 99(1987), 7678. [FKKM] B. Fridman, KT. Kim, S. G. Krantz, and D. Ma, On Fixed Points and Determining Sets for Holomorphic Automorphisms, Michigan Math. Jour., to appear. [FP] B. L. Fridman and E. A. Poletsky, Upper semicontinuity of automorphism groups, Math. Ann., 299(1994), 615628. [GRK] R. E. Greene and S. G. Krantz, Stability properties of the Bergman kernel and curvature properties of bounded domains, Recent Developments in Several Complex Variables (J. E. Fornress, ed.), Princeton University Press (1979),179198. [GRW] R. E. Greene and H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Mathematics, 699, Springer, Berlin, 1979. [GKM] D. Gromoll, W. Klingenberg, and W. Meyer, Riemannsche Geometrie im Grossen, 2nd ed., Lecture Notes in Mathematics, v. 55, SpringerVerlag, New York, 1975. [ISK] A. Isaev and S. G. Krantz, Domains with noncompact automorphism group: A survey, Advances in Math. 146(1999), 138. [KLI] W. Klingenberg, Riemannian Geometry, 2nd ed., de Gruyter Studies in Mathematics, Berlin, 1995. [KOB] S. Kobayashi, Hyperbolic complex spaces, Springer, 1999. [LEM] L. Lempert, La metrique Kobayashi et las representation des domains sur la boule, Bull. Soc. Math. Prance 109(1981), 427474. [LES] K Leschinger, Uber fixpunkte holomorpher Automorphismen, Manuscripta Math., 25 (1978), 391396. [MA] D. Ma, Upper semicontinuity of isotropy and automorphism groups, Math. Ann., 292(1992), 533545. [MAS] B. Maskit, The conformal group of a plane domain, Amer. J. Math., 90 (1968), 718722. [MOY] N. Mok and S. T. Yau, Completeness of the KahlerEinstein metric on bounded domains and the characterization of domains of holomorphy by curvature conditions, Symposia in Pure Math. The mathematical heritage ofH. Poincare, Amer. Math. Soc., 39, Part I. (1983),4160. [OHS] T. Ohsawa, On complete Kahler domains with C 1 boundary, Pub!. Res. Inst. Math. Sci., RIMS (Kyoto), 16 (1980), 929940. [PEL] E. Peschl and M Lehtinen, A conformal selfmap which fixes 3 points is the identity, Ann. Acad. Sci. Fenn., Ser. A I Math., 4 (1979), no. 1, 8586. [SUI] N. Suita, On fixed points of conformal selfmappings, Hokkaido Math. J., 10(1981), 667671. [Vll] J.P. Vigue, Fixed points of holomorphic mappings in a bounded convex domain in Cn, Proceedings of Symposia in Pure Mathematics, 52(1991), Amer. Math. Soc., 579582. [VI2] J.P. Vigue, Fixed points of holomorphic mappings, Complex Geometry and Analysis (Pisa, 1988), Lecture Notes in Mathematics, v. 1422, Springer, Berlin, 1990, pp. 101106. [YAU] S. T. Yau, A survey on KahlerEinstein metrics. Complex analysis of several variables (Madison, Wis., 1982), 285289, Proc. Sympos. Pure Math., 41, Amer. Math. Soc., Providence, RI, 1984. KANGTAE KIM, DEPARTMENT OF MATHEMATICS, POHANG UNIVERSITY OF SCIENCE AND TECHNOLOGY, POHANG 790784, KOREA STEVEN G. KRANTZ, DEPARTMENT OF MATHEMATICS, CAMPUS Box 1146, WASHINGTON UNIVERSITY, ST. LOUIS, MISSOURI 63130 U.S.A. Email address:kimkt«lpostech.ac.kr Email address: sklDmath. wustl. edu
Contemporary Mathenlatics Volume 328, 2003
Localization in the Spectral Theory of Operators on Banach Spaces T. L. Miller, V. G. Miller, and M. M. Neumann ABSTRACT. In the first two sections of this article, we survey some of the recent progress in the local spectral theory of operators on Banach spaces with emphasis on the local spectrum and on restrictions and quotients of decomposable operators. In particular, the problem of characterizing restrictions and quotients of generalized scalar operators with spectrum in the unit circle in terms of suitable growth conditions is addressed in detail, with emphasis on [11], [22], and [23]. The last two sections center around certain localized versions of the singlevalued extension property, Bishop's property (13), and the decomposition property (8), mainly in the spirit of [2], [5], [6], and [13]. For each of these properties, we find a smallest closed set modulo which it holds. For these residual sets, we establish a spectral mapping theorem with respect to the Riesz functional calculus. We also obtain precise information about the extent to which Bishop's property «(3) holds on the essential or the Kato resolvent set. Our results are exemplified in the case of weighted shifts. Moreover, several of the outstanding open questions of the field are mentioned in their natural context.
1. Decomposable operators and the local spectrum
Among the various aspects and levels of localization in spectral theory, we choose decomposability as our starting point. Let X be a complex Banach space, and let L(X) denote the Banach algebra of all bounded linear operators on X. For T E L(X), let, as usual, a(T), ap(T), aap(T), r(T), and p(T) denote the spectrum, point spectrum, approximate point spectrum, spectral radius, and resolvent set of T, and let Lat(T) stand for the collection of all Tinvariant closed linear subspaces of X. From [18J and [29J we recall that an operator T E L(X) is said to be decomposable provided that, for each open cover {U, V} of C, there exist Y, Z E Lat(T) for which X = Y + Z, a(T Iy) ~ U, and a(T I Z) ~ V. By [18, 1.2.23J or [29, 4.4.28J, this simple definition is equivalent to the original notion of decomposability, as introduced by Foi~ in 1963 and discussed in the classical book by Colojoara and Foi~ [IOJ. 2000 Mathematics Subject Classification. Primary 47All, 47B40; Secondary 47B37. © 247
2003 American Mathematical Society
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T. L. MILLER, V. G. MILLER, AND M. M. NEUMANN
As witnessed, for instance, by the monographs [10], [16], [18], and [29], the theory of decomposable operators is now richly developed with many interesting applications and connections. Evidently, the class of decomposable operators contains all normal operators on Hilbert spaces and, more generally, all spectral operators in the sense of Dunford on Banach spaces. Moreover, a simple application of the Riesz functional calculus shows that all operators with totally disconnected spectrum are decomposable. In particular, all compact and all algebraic operators are decomposable. An important subclass of the decomposable operators is formed by the generalized scalar operators, defined as those operators T E L(X) for which there exists a continuous unital algebra homomorphism
(Tu f)(Jt) := (T  Jt)f(Jt)
for all f E H(U, X) and Jt E U.
It turns out that this operator dominates large parts of spectral theory. The local resolvent set PT(X) of T at a vector x E X is defined to consist of all >. E C for which there exists some f E H(U, X) on an open neighborhood U of >. for which Tuf = x. Clearly, f(Jt) = (T  Jt)lX for all Jt E Un p(T), so that PT(X) is open and contains p(T). Hence the local spectrum aT(x) := C \ pT(X) is a closed subset of a(T). In general, the various analytic functions that occur in the definition of pT(X) need not be consistent. This issue is addressed by the following definition. The operator T E L(X) is said to possess the singlevalued extension property (SVEP), if Tu is injective for all open sets U £; IC. By [18, 3.3.2], T has SVEP precisely when, for each x E X, there exists a unique function f E H(PT(X), X) for which
(T  Jt)f(Jt) = x
for all Jt E PT(X).
This function is then called the local resolvent function for T at x. In remarkable contrast to the usual resolvent function, such functions may well be bounded; this recent discovery of Bermudez and Gonzalez will be exemplified below. One might expect a(T) to be the union of the local spectra aT(x) over all x E X, but this is not true in general. In fact, this union coincides with the surjectivity spectrum asu(T) of T, the set of all >. E C for which T  >. fails to be surjective. However, if T has SVEP, then asu(T) = a(T), and aT(x) is nonempty for all nonzero x E X, [18, 1.2.16 and 1.3.2]. As a powerful application, we obtain
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that every surjective operator with SVEP is actually bijective, [18, 1.2.10]. A more precise version of this result will be discussed in Section 3. For arbitrary T E L(X) and F ~ C, let XT(F) := {x EX: aT(x) ~ F} denote the corresponding local spectml subspace. Evidently, XT(F) is a linear subspace of X, but need not be closed. The following classical result illustrates that these subspaces playa basic role for spectral decompositions, see [18, 1.2] and [29,4.4]. THEOREM 1. Suppose that T E L(X) is decomposable. Then T has SVEP, and, for each closed set F ~ C, the space XT(F) is closed and satisfies a(T I XT(F)) ~ F. In fact, XT(F) is the largest among all spaces Y E Lat(T) for which aT(T IY) ~ F. Moreover, XT(F) ~ XT(Ut}+·· +XT(U n ) for every finite open cover {U1 , ... , Un} ~F
The following examples may illuminate how spectral decompositions work in some important cases; for details see [18] and, for the last assertion of Example 4, also [26, Th.16]. The extent to which the compactness of the group is essential here remains a challenging open problem. EXAMPLE 2. Let T E L(X) be a normal opemtor with spectml measure ~ on a complex Hilbert space X. Then T is decomposable, and XT(F) = ran ~(F) for every closed set F ~ Co Moreover, there exists a nonzero bounded local resolvent function for T precisely when int a(T) =f:. 0. 0 EXAMPLE 3. Let X := C(O) be the space of continuous functions on a compact Hausdorff space 0, and let T E L(X) denote the opemtor of multiplication by a given function g E X. Then T is decomposable, and XT(F) = {f E C(O) : g(supp f) ~ F} for every closed set F ~ Co Also in this case, there exists a nonzero bounded local resolvent function for T precisely when int a(T) =f:. 0. 0 EXAMPLE 4. Let X := Ll(G) be the group algebm of a locally compact abelian group G, and let T E L(X) denote the opemtor of convolution by a given function g E X. Then T is decomposable, and XT(F) = {f E Ll(G) : g(suppj) ~ F} for every closed set F ~ C, where j denotes the Fourier tmnsform. Moreover, at least when G is compact, there exists a nonzero bounded local resolvent function for T precisely when int a(T) =f:. 0. 0 On the other hand, there are important classes of operators which are not covered by decomposability. For instance, by [18, 1.6.14] and [22], a unilateral weighted right shift on the sequence space fP(N o) for arbitrary 1 ::::; p < 00 is decomposable, or, equivalently, the quotient of a decomposable operator, only in the trivial case when it is quasinilpotent, while unilateral weighted right shifts are never generalized scalar. Moreover, as we shall see, there are many examples of unilateral and bilateral weighted left shifts without SVEP. Another illuminating case is that of isometries. By [18, 1.6.7], an arbitrary Banach space isometry is decomposable, or, equivalently, the quotient of a decomposable operator or generalized scalar, precisely when it is invertible. On the other hand, every isometry may be extended, by a classical result due to Douglas, recorded in [18, 1.6.6], to an invertible isometry, and hence has a decomposable extension. In the next section, we shall discuss a more general version of this result.
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2. Moving beyond decomposability Several years before decomposability was formally introduced by Foi~, Bishop [9] investigated a number of spectral decomposition properties in an attempt to extend some of the features of the theory of normal operators to the general setting of Banach spaces. Among these properties, one turned out to be particularly important. We now say that an operator T E L(X) on a complex Banach space X has Bishop's property ((J) provided that, for each open set U ~ C, the operator Tu on H(U, X) is injective with closed range, equivalently, if, for each sequence (fn)nEN in H(U, X) with (T  )..)fn(>\) > 0 asn>oo, uniformly on each compact subset of U, it follows that fn()..) > 0 as n > 00, again uniformly on the compact subsets of U, [18, 1.2.6]. Actually, by [18, 3.3.5], the injectivity condition in this definition is redundant. Obviously, property ({J) implies SVEP. It was shown a long time ago by Foi~ that all decomposable operators share property ((J), but the precise relationship was discovered only recently by Albrecht and Eschmeier [6]. THEOREM 5. An operator T E L(X) has Bishop's property ((J) precisely when T is similar to the restriction of a decomposable operator to a closed invariant subspace. Moreover, in this case, there exists a decomposable extension 8 for which aCT) ~ a(8). 0
The result was, in part, inspired by the work of Putinar [27] who proved that every hyponormal operator is subscalar, in the sense that it has a generalized scalar extension. Thus all hyponormal operators have property ((J). In particular, all unilateral weighted right shifts on f2(N o) with an increasing weight sequence w have property ({J), but a characterization of ({J) in terms of w seems to be an intriguing open problem. For partial results, see [11], [18], [22], and [23]. To discuss the dual notion of Bishop's property ((J), we need a slight variant of the local spectral subspaces. For arbitrary T E L(X) and a closed subset F of
XT(F)
:=
{x
EX:
x E ran TC\F }.
In this definition, the point is that the local resolvent function is defined globally on the entire complement of F. Clearly, XT(F) is a linear subspace contained in XT(F). Moreover, by [18, 3.3.2], the identity XT(F) = XT(F) holds for all closed sets F ~
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THEOREM 6. Property (8) chamcterizes, up to similarity, the quotients of decomposable opemtors by closed invariant subspaces. Moreover. the properties (13) and (8) are dual to each other. in the sense that an opemtor T E L(X) has one of these properties precisely when the adjoint T* E L(X*) on the dual space X*has the other one. D
Here the hardest assertion to prove is that property (13) for T* implies property (8) for T. The construction of decomposable extensions and liftings uses two powerful functional models for operators on Banach spaces of independent interest. These models are in the spirit of functiontheoretic operator theory, and involve the operator of multiplication by the independent variable on certain Sobolevtype spaces together with the theory of topological tensor products. The complete duality between the properties (/3) and (8) employs the GrothendieckK6the duality for spaces of vectorvalued analytic functions. All of this is described, in considerable detail, in [18, Ch.2]. There are interesting applications to the invariant subspace problem. Indeed, if the operator T E L(X) has either property (/3) or property (8), then Eschmeier and Prunaru [14] established that Lat(T) is nontrivial provided that a(T) is thick, and that Lat(T) is rich in the sense that it contains the lattice of all closed subspaces of some infinitedimensional Banach space provided that the essential spectrum ae(T) is thick. Here we skip the formal definition of thick subsets of the complex plane, but note that all compact sets with nonempty interior are thick. A streamlined approach to this result and further references may be found in [18, 2.6]. Since all hyponormal operators have, by Putinar's result [27], property (/3), the preceding result subsumes, in particular, Brown's celebrated invariant subspace theorem for hyponormal operators with thick spectrum. In light of Read's recent construction of a quasinilpotent, and hence decomposable, operator on a Banach space without nontrivial invariant subspaces, it is clear that the condition of thick spectrum cannot. be dropped in general. However, the invariant subspace problem for operators on Hilbert spaces remains open, even for the class of hyponormal operators. As discussed in the monograph by Eschmeier and Putinar [16], there are also interesting connections between property (/3) and the theory of analytic sheaves. These connections are not only important for the spectral theory of several commuting operators, but they are also at the heart of some of the recent developments in the case of single operators. Although, as witnessed by the exposition of local spectral theory in [18], the explicit use of sheaf theory can be avoided in the case of single operator theory, the reader should be aware of these connections. The basic idea is sketched in [18, 2.2]. A classical issue of local spectral theory is to derive spectral decomposition properties from growth conditions on the powers or the resolvent function of a given operator. For instance, Levinson's loglog theorem from complex analysis may be used to show that, for operators with spectrum in the real line or the unit circle 'll', a very weak logarithmic growth condition on the resolvent function suffices to ensure decomposability. The short approach from [18, 1.7] to this classical result due to Radjabalipour is based on the fact that, by Theorem 6, an operator T E L(X) is decomposable precisely when both T and T* have property (13). A very attractive account of the local spectral t.heory for operators with thin spectrum may be found in a recent survey article by Albrecht and Ricker [7].
T. L. MILLER, V. G. MILLER, AND M. M. NEUMANN
252
Here we focus only on one aspect that leads to interesting open problems. For arbitrary T E L(X), let K(T) := inf{IITxll : Ilxll = I} denote the lower bound of T. Evidently, K(T)l = II Tlil when T is invertible. Similar to the case of the spectral radius, it is known that the sequence of numbers K(Tn) lin converges to its supremum, denoted by i(T), and that aap(T) ~ {>' E C : i(T) ~ 1>'1 ~ r(T)}, [18, 1.6.1 and 1.6.2]. By a classical result due to Colojoara and Foi~, [10, 5.1] or [18, 1.5.12], a generalized scalar operator T satisfies a(T) ~ T precisely when T is £(T) scalar, in the sense that T admits a continuous functional calculus on the Frechet algebra £(T) of all COOfunctions on T. Moreover, T is £(T)scalar if and only if T is invertible and satisfies the condition of polynomial growth (P), in the sense that there exist constants c, s > 0 such that 1
s ~
K(Tn) ~ IITnl1 ~ cn s for all n E Nj cn indeed, in this case, a functional calculus cP for T is given by the formula 00
cpU):=
2:
!(n) Tn
for all
f
E £ (T) ,
n=oo
where !(n) denotes the nth Fourier coefficient of f. Evidently, all invertible isometries have property (P), and hence are £ (T)scalar. Also, it follows from the preceding characterization that an operator T E L(X) is £(T)scalar precisely when its adjoint T* is £(T)scalar. Moreover, since property (P) implies that i(T) = r(T) = 1 and consequently aap(T) ~ T, and since aap(T) = a(T) when T has property (6), we are led to the following result. PROPOSITION
7. For every T E L(X) with property (P), the following equiva
lences hold: T is invertible ¢:} a(T)
~
T
¢:}
T has (6)
¢:}
T is decomposable
¢:}
T is £(T)scalar.
Moreover, ifT is not invertible, then aap(T) = T and a(T) is the closed unit disc.o
Evidently, every restriction of an £(T)scalar operator has property (P), but the converse is open in general. The preceding proposition shows that this problem is equivalent to the problem of extending an arbitrary operator with property (P) to an invertible operator with property (P) for possibly larger constants c, s > O. Since the extension provided by the AlbrechtEschmeier functional model in Theorem 5 increases the spectrum, a different approach is needed here. As noted above, for isometries, the desired extension is possible by a result of Douglas. Also, for a certain class of operators that includes all unilateral weighted right shifts, a positive solution was recently provided by Didas [11] and the authors [23]. While Didas exploits the theory of topological tensor products in the spirit of Eschmeier and Putinar [16], the more elementary approach from [23] uses a modification of a construction provided by Bercovici and Petrovic [8] to characterize compressions of £(T)scalar operators. For unilateral weighted right shifts on fP(N o), the method developed in [23] leads to extensions as bilateral weighted shifts on fP(Z) with sharp growth estimates. To reduce the case of quotients of £(T)scalar operators to that of restrictions, we recall that the minimum modulus 'Y(T) of a nonzero operator T E L(X) is defined as 'Y(T):= inf{IITxll/dist(x,kerT): x (j kerT}. Clearly, 'Y(T) = K(T)
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when T is injective. It is also well known that 'Y(T) = 'Y(T*), and that 'Y(T) > 0 precisely when T has closed range. Standard duality theory now leads to the following result. PROPOSITION 8. An operator T E L(X) is the restriction of an £(1l')scalar operator if and only if its adjoint T* is the quotient of an £(1l')scalar operator. Moreover, ifT is the quotient of an £(1l')scalar operator, then T* is the restriction of an £(1l')scalar operator, and hence there exist constants c, s > 0 for which
~::;'Y(Tn)::;IITnll::;cns cn
forallnEN.
In general, it is not known if the last growth condition characterizes the quotients of £(1l')scalar operators, but, by Proposition 8 and [23, Prop.5], this is the case for the class of all unilateral weighted left shifts on fP(N o ) for arbitrary 1 < p < 00. More precisely, a unilateral weighted left shift on fP(N o) satisfies the growth condition of Proposition 8 if and only if it admits a bilateral weighted shift lifting on fP(Z) that is £(1l')scalar. Similar results hold for more general growth conditions; see [22] and [23]. For instance, by another classical result due to Colojoara and Foi~, an invertible operator T E L(X) is decomposable provided that T satisfies Beurling's condition (B), in the sense that
L 00
1 n 2 (llogK(Tn)1
+ IlogliTnll1) < 00,
n=l
[10, 5.3.2] and [18,4.4.7]. Clearly, property (B) is inherited by restrictions, but it remains open, if every operator with property (B) has an invertible extension with property (B). In fact, it is not known, if property (B) implies property ((3). For certain unilateral weighted right shifts, a positive answer was recently given in [22] and [23].
3. Localization of the singlevalued extension property For an arbitrary operator T E L(X) on a complex Banach space X, here the spaces K(T) := XT(C\ {O}) and Ho(T) := X T ( {O}) will be of particular importance. Both spaces were, in some disguise, studied by Mbekhta and also by Vrbova; see [19], [20], and [30]. By [18, 3.3.7], K(T) coincides with the analytic core of T, defined to consist of all x E X for which there exist a constant c > 0 and elements Xn E X such that for all n E N. By this characterization and the open mapping theorem, K(T)
=X
if and only if
T is surjective. In terms of local spectral theory, this follows also from the fact that asu(T) is the union of all local spectra of T. On the other hand, by [18, 3.3.13], Ho(T) is the quasinilpotent part of T, defined as the set of all x E X for which
IITnx11 1/ n _ 0
as n 
00.
In general, neither K(T) nor Ho(T) need to be closed, but, if 0 is isolated in a(T), then, by [19, 1.6], both spaces are closed and X = K(T) EB Ho(T). For more on operators with closed K(T) and Ho(T), see [1], [2], and [24]. For instance, by [24, Cor.6], for any noninvertible decomposable operator T, the point 0 is isolated in
T. L. MILLER, V. G. MILLER, AND M. M. NEUMANN
254
a(T) precisely when K(T) is closed. In particular, the analytic core of a compact or, more generally, a Riesz operator T is closed exactly when T has finite spectrum, [24, Cor.9]. The spaces Ho(T) and K(T) are related to the kernel N(T) and the range R(T) of T as follows. With the notation oc
N(T) :=
U N(Tn)
n 00
and
R(T):=
R(Tn)
n=l
n=l
for the generalized kernel and range of T, there is an increasing chain of kerneltype spaces
N(T) ~ N(Tn) ~ N(T) ~ Ho(T) and a decreasing chain of rangetype spaces
~ X T ({O})
R(T) 2 R(Tn) 2 R(T) 2 K(T) 2 X T (0) for arbitrary n E N. [18, 1.2.16 and 3.3.1]. The geometric position of the kerneltype spaces versus the rangetype spaces turns out to be intimately related to a certain localized version of SVEP for the operator T and its adjoint T*. An operator T E L(X) is said to have SVEP at a point A E C, if, for every open disc U centered at A, the operator Tu is injective on H(U, X). This notion dates back to Finch [17], and was pursued further, for instance, in [1], [2], [3], [4], [5], and [20]. Evidently, T has SVEP at A precisely when T  A has SVEP at 0, while SVEP for T is equivalent to SVEP for T at A for each A E C. Local spectral theory leads to a variety of characterizations of this localized version of SVEP that involve the kerneltype and rangetype spaces introduced above. Our starting point is the following characterization from [3, 1.9]. The result shows, in particular, that every injective operator T E L(X) has SVEP at 0, and may be viewed as a local version of the classical fact that T has SVEP if and only if X T (0) = {O}, [18, 1.2.16]. For completeness, we include a short new proof that uses nothing but local spectral theory. THEOREM
9. For every operator T
T has SVEP at 0 <=} N(T)
E
L(X), the following equivalences hold:
n X T (0) = {O} <=} aT(x) = {O}
for all 0 =f. x E N(T).
Proof. First suppose that T has SVEP at 0, and consider an arbitrary x E N(T) for which aT(x) is empty. Then 0 E pT(X) so that there exists an f E H(U, X) on some open disc U with center 0 for which (TA)f(A) = x for all A E U. It follows that (T  A)Tf(A) = Tx = 0 for all A E U, and therefore Tf(A) = 0 for all A E U, since T has SVEP at O. Thus x = Tf(O) = 0, and hence N(T) n X T (0) = {O}. Next observe that, for each x E N(T), the definition f(A) := xl A yields an analytic function for which (T  A)f(A) = x for all nonzero A E C. Thus aT(x) ~ {O} for all x E N(T). Consequently, the second and third assertions are equivalent. Finally suppose that N(T) n XT(0) = {O}, let U be an open disc with center 0, and consider a function f E H(U, X) for which Tu f = O. By [18, 1.2.14], aT(f(A)) = aT(O) = 0 for all A E U. Now, for the power series representation f(A) = L:~=o an An for all A E U, our task is to show that each of the coefficients an E X is zero. For the case n = 0, this is immediate, since ao = f(O) E N(T) nXT (0) = {O}. But then it follows that for all A E U,
THE SPECTRAL THEORY OF OPERATORS ON BANACH SPACES
255
and therefore (T  >.) (at + a2 >. + a3 >.2 + ... ) = 0 first for all nonzero>. E U, and then, by continuity, also for>. = O. Exactly as before, we conclude that at = 0 and hence, by induction, an = 0 for all n ~ O. Thus f == 0 on U, as desired. 0 Since N(T) n K(T) ~ X T ( {O}) n XT(C \ {O}) = X T (0), it clearly follows that N(T) n K(T) = N(T) n X T (0) for every T E L(X). Thus, by Theorem 9, T has SVEP at 0 if and only if N(T) n K(T) = {O}. In particular, if T is surjective, then, as noted above, K(T) = X, so that T has SVEP at 0 precisely when T is injective. This characterization from [3, 1.11] extends a classical result due to Finch [17]. As another immediate consequence of Theorem 9, we obtain the following result. COROLLARY 10. An operator T E L(X) has SVEP at 0 provided that either Ho(T) n K(T) = {O} or N(T) n R(T) = (0). 0 Recent counterexamples in [2] show that, in general, none of the latter conditions is equivalent to SVEP of T at 0, thus disproving a claim made in [20, 1.4]. However, by [1, 2.7], [5, 1.3], and Theorem 12 below, equivalences do hold for certain classes of operators. We now describe how the localized SVEP behaves under duality. For a linear subspace M of X, let Ml. := {cp E X* : cp(x) = 0 for all x E M}, and for a linear subspace N of X*, let l.N := {x EX: cp(x) = 0 for all cp E N}. By the bipolar theorem, l.(Ml.) is the normclosure of M, and (l.N)l. is the weak*closure of N. Moreover, for every T E L(X), it is well known that N(T*) = R(T)l. and N(T) = l.R(T*), while R(T) is a normdense subspace of l.N(T*), and R(T*) is a weak*dense subspace of N(T)l.. An elementary short proof of the following result may be found in [2,4.1]. PROPOSITION 11. For every operator T E L(X), the following assertions hold: (a) K(T) ~ l.Ho(T*) and K(T*) ~ Ho(T)l.; (b) if Ho(T) + R(T) is normdense in X, then T* has SVEP at 0; (c) if Ho(T*) + R(T*) is weak*dense in X*, then T has SVEP at o. 0 Even in the Hilbert space setting, the inclusions in part (a) of Proposition 11 need not be identities, and the implications of parts (b) and (c) cannot be reversed in general; see [2] for counterexamples in the class of weighted shifts. However, for suitable classes of operators, the results can be improved. As usual, an operator T E L(X) is said to be a semiFredholm operator, if either N(T) is finitedimensional and R(T) is closed, or R(T) is of finite codimension in X. Also, an operator T E L(X) is said to be semiregular, if R(T) is closed and N(T) ~ R(T); see [18], [19], and [21] for a discussion of these operators. THEOREM 12. Suppose that the operator T E L(X) is either semiFredholm or semiregular. Then the following assertions hold: (a) R(T) = K(T) = l.Ho(T*) = l.N(T*); (b) R(T*) = K(T*) = Ho(T)l. = N(T)l.; (c) N(T) n R(T) = {O} <=> T has SVEP at 0; (d) N(T*) n R(T*) = {O} <=> T* has SVEP at 0; (e) N(T)
+ R(T) = X<=>
(f) N;;:;:(T=*""7"")+:R~(T=*";) w'
+ R(T) = X<=> T* has SVEP at 0; <=> Ho(T*) + R(T*) w' = X* <=> T has SVEP
Ho(T)
= X*
where w* indicates the closure with respect to the weak*topology.
at 0,
o
256
T. L. MILLER,
V. G.
MILLER, AND M. M. NEUMANN
Theorem 12 was recently obtained in [2], see also [5]. An important ingredient of the proof is the fact that T has SVEP at 0 if and only Tn has SVEP at 0 for arbitrary n E N. This equivalence is a special case of a spectral mapping formula for the set 6(T) of all A E C at which T fails to have SVEP, namely 6(f(T)) = f(6(T)) for every analytic function f on some open neighborhood of a(T); see [2, 3.1] and also Theorem 18 below. Further developments may be found in [1], [3], [4], [5], [17], and [20]. Here we mention only one simple consequence of Theorem 12 for semiregular operators from [3, 2.13]. For T E L(X), let PK(T) consist of all A E C for which T  A is semiregular. The Kato spectrum aK(T) := C \ PK(T) is a closed subset of a(T) and contains oa(T); see [18, 3.1] and [21] for details. We include a short proof of the following result, since the dichotomy for the connected components of the Kato resolvent set PK (T) with respect to the localized SVEP will play an essential role in Section 4. THEOREM 13. Let T E L(X) be semiregular. Then T has SVEP at 0 precisely when T is injective, or, equivalently, when T is bounded below, while T* has SVEP at 0 precisely when T is surjective. Moreover, for arbitrary T E L(X), each connected component n of PK(T) satisfies either n ~ 6(T) or n n 6(T) = 0. The inclusion n ~ 6(T) OCC1J.rs precisely when n ~ ap(T), or, equivalently, when n n aap(T) i= 0, while the identity n n 6(T) = 0 occurs precisely when n n ap(T) = 0, or, equivalently, when n \ aap(T) i= 0.
Proof. If T is semiregular, then N(T) n n(T) = N(T) and N(T) + R(T) = R(T) = R(T). Hence the first assertions follow from parts (c) and (e) of Theorem 12. For the last claim, it suffices to see that injectivity of T  A for some A E n entails that T  f.J. is injective for every f.J. E n. But this is clear, since, by part (b) of Theorem 12, N(T  f.J.) = J..n(T* f.J.) and, by [18, 3.1.6 and 3.1.11], n(T*  f.J.) = n(T* A) for all f.J. E n. 0 It is well known that the approximate point spectrum and the surjectivity spectrum of an arbitrary operator T E L(X) are related by the duality formulas aap(T) = asu(T*) and asu(T) = aap(T*), [18, 1.3.1]. Moreover, by [18, 1.3.2 and 3.1.7], asu(T) = a(T) and aap(T) = aK(T) if T has SVEP, and aap(T) = a(T) and asu(T) = aK(T) if T* has SVEP. The following local version of these results is immediate from Theorem 13. PROPOSITION 14. For every operator T E L(X), the following assertions hold: (a) If A E a(T) \ aap(T), then T has SVEP at A, but T* fails to have SVEP at A; (b) if A E a(T) \ asu(T), then T* has SVEP at A, but T fails to have SVEP at A.
o The next result from [2, 5.2] is a straightforward consequence of Proposition 14. For instance, it follows that 6(T*) is the open unit disc for every noninvertible operator T with property (P) or (B). Further examples including analytic Toeplitz operators, composition operators on Hardy spaces, and weighted shifts may be found in [2]. COROLLARY 15. If aap(T) ~ oa(T), then T has SVEP and 6(T*) = int a(T). Similarly, if asu(T) ~ oa(T), then T* has SVEP and 6(T) = int a(T). 0
THE SPECTRAL THEORY OF OPERATORS ON BANACH SPACES
257
4. Localization of the properties ((:J) and (8) There is a natural extension of the class of decomposable operators for which spectral decompositions are only required with respect to a given open subset U of the complex plane. These operators were introduced by Vasilescu as residually decomposable operators in 1969, shortly after the publication of the seminal monograph [10]. They became also known as Sdecomposable operators with S = c \ U, and were studied by Bacalu, Nagy, Vasilescu, and others; see [29, eh.4]. As in [6] and [13], we now say that an operator T E L(X) on a complex Banach space X is decomposable on an open subset U of C provided that, for every finite open cover {Vl , ... , Vn } of C with C \ U ~ Vl , there exist Xl,"" Xn E Lat(T) with the property that X
= Xl + ... + Xn and
a(T I X k ) ~
Vk
for k
= 1, ... , n.
It is known, although certainly not obvious, that, in this definition, it suffices to consider the case n = 2; see [6] and [29]. Evidently, classical decomposability occurs when U = C. On the other hand, every operator T E L(X) is at least decomposable on its resolvent set p(T). Among the remarkable early accomplishments of the theory is the following result due to Nagy [25] from 1979: For every T E L(X), there exists a largest open set U ~ C on which T is decomposable. The complement of this set is Nagy's spectral residuum Sr(T), a closed, possibly empty, subset of a(T). In the present section, we shall employ the recent results of Albrecht and Eschmeier [6] to obtain a short proof for the existence and a useful description of Nagy's spectral residuum. In particular, we shall see how Sr(T) is related to the Kato spectrum aK(T) and the essential spectrum ae(T). For this, we shall work with certain localized versions of property ((:J) and property (8) from [6]. An operator T E L(X) is said to possess Bishop's property ((:J) on an open set U ~ C, if, for every open subset V of U, the operator Tv is injective with closed range, equivalently, if, for every sequence of analytic functions In: V + X for which (T>')In(>\) + 0 as n + 00 locally uniformly on V, it follows that In(>') + 0 as n + 00, again locally uniformly on V. It is straightforward to check that this condition is preserved under arbitrary unions of open sets. This shows that there exists a largest open set on which T has property ((:J), denoted by U{3(T). Its complement S{3(T) := C \ U{3(T) is a closed, possibly empty, subset of a(T). In fact, T satisfies Bishop's classical property ((:J) precisely when S{3(T) = 0. Moreover, the operator T is said to have property (8) on U, if X
= XT(C \ V) + XT(W)
for all open sets V, W ~ C for which C \ U ~ V ~ V ~ W; see [6] and [13]. Quite remarkably, as shown in [6, Th.3], this condition holds precisely when, for each closed set F ~ C and every finite open cover {VI"'" Vn } of F with F \ U ~ VI, it follows that XT(F) ~ XT(V d + ... + XT(V n); see also [18, 2.2.2] for the case U=C, These localized versions of ((:J) and (8) already proved to be useful in the theory of invariant subspaces for operators on Banach spaces, [14]. The following result summarizes the main accomplishments from [6].
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T. L. MILLER, V. G. MILLER, AND M. M. NEUMANN
THEOREM 16. For every operator T following equivalences hold:
E
L(X) and every open set U
~
C, the
(a) T has (13) on U {::} T* has (8) on U; (b) T has (8) on U {::} T* has (/3) on U;
(c) T is decomposable on U {::} T has both (13) and (8) on U; (d) T has (13) on U {::} T is the restriction of a decomposable operator on U; (e) T has (15) on U {::} T i,~ the quotient of a decomposable operator on U.
It is not at all obvious from the definition of (8) that there exists a largest open set, say Uc5(T), on which the operator T has property (8), but this now follows from the corresponding result for (13) by duality. In fact, Uc5(T) = U{3(T*) by part (b) of the preceding result. More precisely, Theorem 16 leads to the following result. COROLLARY 17. For every operator T E L( X), there exists a smallest closed set Sc5(T) so that T has property (8) on its complement. Moreover, Sc5(T) = S{3(T*), S{3(T) = Sc5(T*), and Sr(T) = S{3(T) U Sc5(T) = S{3(T) U S{3(T*) = Sr(T*). 0
Perhaps somewhat surprisingly, it will be possible to obtain general information about the location of S{3(T), and hence of Sc5(T) and Sr(T). For this, another localized version of SVEP will play a crucial role. For consistency, we say that the operator T E L(X) has SVEP on an open set U ~ C, if, for every open subset V of U, the operator Tv is injective, [13]. It is straightforward to check that T has SVEP on U precisely when T has SVEP at each point >. E U, as defined in the previous section. Obviously, there exists a largest open set on which T has SVEP, and the analytic spectral residuum S(T) is defined to be the complement of this set; see [29, 4.3.2] and [30]. Clearly, 6(T) ~ S(T), but, since 6(T) is open and S(T) is closed, equality occurs only in the trivial case when T has SVEP. Nevertheless, as noted in [2], a simple verification shows that 6(T) = S(T) ~ S{3(T). It is interesting to observe that all these residual sets behave canonically with respect to the Riesz functional calculus. As usual, for T E L(X) and any analytic complexvalued function f on an open neighborhood 0 of a(T), the operator f(T) E L(X) is defined by
~
r
f(>.)(>.  T)l d>', 2m where r denotes an arbitrary contour in 0 that surrounds a(T), [12] or [18, A.2]. The standard spectral mapping theorem asserts that a(f(T)) = f(a(T)). The next result has a similar flavor, and may be viewed as an extension of the fact that the classical versions of SVEP, property (/3), property (8), and decomposability are all preserved under the Riesz functional calculus, [18, 3.3.6 and 3.3.9]. The fact that the Riesz functional calculus respects Bishop's classical property (13) was established by Eschmeier and Putinar [15]. The following proof involves a different approach to this result. f(T) :=
lr
THEOREM 18. Let T E L(X) be an arbitrary operator, let f : 0 + C be an analytic function on an open neighborhood 0 of a(T), and suppose that f is nonconstant on each connected component ofO. If E denotes any of the symbols 6, S, S{3, Sc5, or S., then E(f(T)) = f(E(T)).
THE SPECTRAL THEORY OF OPERATORS ON BANACH SPACES
259
Proof. In the case of 6(T), the spectral mapping formula was recently obtained in [2, 3.1]. The result for S(T) is a standard fact that may be found in [29, 4.3.14] and [30, 1.6]. Note, however, that the formula for S(T) is also an immediate consequence of that for 6(T), because 6(T) = S(T). While the existence of the residual set So(T) was most conveniently established by using S{3(T) and the AlbrechtEschmeier duality between the localized versions of (f3) and (8), for the issue at hand it seems appropriate to switch the order. Indeed, since f(T*) = f(T) *, Corollary 17 ensures that it suffices to prove the claim for So(T). For this, fortunately, we may proceed as in the proof of [18,3.3.6 and 3.3.9], where property (8) is shown to be stable under the Riesz functional calculus. First, consider arbitrary open sets V, W ~ C for which f(So(T)) ~ V ~ V ~ W. Then {fl(C \ V), f 1(W)} is an open cover of a(T) for which So(T) ~ fl(W). Thus, by the characterization of the localized property (8) mentioned above, we obtain that
X = XT(a(T))
= XT
(11(C \ V)
n a(T)) + XT (11(W) n a(T)) .
Clearly, f 1(C \ V) n a(T) ~ f 1(C \ V) n a(T) and, similarly, fl(W) n a(T) ~ fl(W) n a(T). Since, by [18,3.3.6], the formula XT (fl(F) n a(T)) = Xf(T) (F) holds for every closed set F ~ C, we conclude that X = Xf(T) (C \ V)
+ Xf(T) (W).
This shows that f(T) has (8) on C \ f(So(T)), thus C \ f(So(T)) ~ Uo(f(T)), and hence So(f(T)) ~ f(So(T)). Note that this inclusion even holds without the requirement that f be nonconstant on each of the connected components of its domain. The reverse inclusion is less obvious, but may be obtained by a suitable modification of the proof of [18, 3.3.9]. Let S := f 1 (So(f(T))) n a(T). Then the desired inclusion f(So(T)) ~ So(f(T)) means precisely that the decomposition X = XT(C \ V) + XT(W) holds for all open sets V, W ~ C for which S ~ V ~ V ~ W. Evidently, it suffices to show that X = XT(G) + XT(H) for every open cover {G,H} of a(T) for which S ~ G, SnH = 0, and both G and H are compact subsets of n. Ignoring momentarily the exceptional set S, we may proceed word by word along the lines of the proof of [18, 3.3.9] to obtain a finite open cover {WI, ... , W n } of a(T) in n for which fork=I, ... ,n. To handle the residual set, we note that the identity SnH = 0 may be reformulated in the form So(f(T)) n f(a(T) n H) = 0. Hence, by continuity and compactness, there exists an open neighborhood V of So (f(T)) for which V n f(a(T) n H) = 0. This implies that f 1 (V) n a(T) n H = 0, hence f 1 (V) n a(T) ~ G, and therefore, by [18, 3.3.6], Xf(T)(V) = XT (Jl(V)
n a(T))
~ XT(G).
Now, since f(T) has (8) on C \ So(f(T)), and since {V, f(W1), ... , f(Wn )} is an open cover of a(f(T)) = f(a(T)) for which So(f(T)) ~ V, we conclude that X
= Xf(T)(a(T)) = Xf(T)(V) + Xf(T)
(f(Wd)
+ ... + Xf(T)
(f(Wn)) '
T.
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L. MILLER, V. G. MILLER, AND M. M. NEUMANN
again by the basic characterization of the localized version of (8) provided in [6, Th.3]. Thus X = XT(G) + XT(H), as desired. 0 In the following results, we show how to verify property ({3) on the Kato resolvent set and its Fredholm counterpart. THEOREM 19. For an arbitrary operator T E L(X) and every connected component n of PK(T), the following equivalences hold:
T has ({3) on
n {:}
T has SVEP on
n {:} n n O'p(T) = 0 {:} n \ O'ap(T) =F 0j
in particular, T has property ({3) on PK(T) precisely when T has SVEP on PK(T). Proof. Clearly, the first of the displayed conditions implies the second one, and the equivalence of the last three conditions follows from Theorem 13. Conversely, if these three conditions hold, then T  A is injective with closed range for all A E n. Thus f'i,(T  A) = "((T  A) > 0 for all A E n. Moreover, by [18, 3.1.10], for every compact subset K of n, there exists a constant c > 0 such that f'i,(T  A) > 0 for all A E Kj in fact, as shown, for instance, in [21,4.1], the function A t+ "((T  A) is continuous and strictly positive on PK(T). From this it is immediate that T has ({3) on n. 0 The next result is clear from Corollary 17, Theorem 19, and the wellknown identity PK(T) = PK(T*), [18, 3.1.6]. Part (c) of Corollary 20 may be viewed as an extension of the fact that, by [18, 3.1.7], p(T) = PK(T) whenever both T and T* have SVEP. COROLLARY 20. For every operatorT E L(X), the following assertions hold:(a) T has SVEP on PK(T) {:} S{3(T) ~ O'K(T)j (b) T* has SVEP on PK(T) {:} SIi(T) ~ O'K(T)j (c) T and T* have SVEP on PK(T) {:} Sr(T) ~ O'K(T). 0 To derive the companion result for the essential spectrum, we employ the fact that, for every operator T E L(X) and every open subset V of the essential resolvent set Pe(T) := C\O'e(T), the operator Tv has closed range (but need not be injective). This interesting result was recently obtained by Eschmeier [13, 3.1], based on sheaftheoretic methods developed by Putinar [28] to show that quasisimilar operators with property ({3) have the same essential spectrum. In tandem with Corollary 17, we obtain the following extension of [13, 3.9]. COROLLARY 21. For every operator T E L(X), the following assertions hold: (a) T has SVEP on Pe(T) {:} S{3(T) ~ O'e(T)j (b) T* has SVEP on Pe(T) {:} SIi(T) ~ O'e(T)j (c) T and T* have SVEP on Pe(T) {:} Sr(T) ~ O'e(T).
o
We close with an application to the spectral theory of weighted shifts. EXAMPLE 22. Let W := (Wn)nEN"o be a bounded sequence of strictly positive real numbers, and let T E L(X) denote the corresponding unilateral weighted right shift on the sequence space X := fP(N o) for some 1 ::; p < 00. Clearly,
i(T) = lim inf
n+oo k~O
(Wk· .. Wk+n_d 1 / n
and
r(T) = lim sup (Wk· n+oo k~O
.. Wk+n_d 1 / n .
THE SPECTRAL THEORY OF OPERATORS ON BANACH SPACES
Since T has no eigenvalues, T has SVEP and asu(T) = a(T) = {A E C : IAI by [18, 1.3.2 and 1.6.15J. Moreover, as noted in [18,3.7.7],
261
s r(T)},
siAl s r(T)} , and therefore, by Corollaries 20 or 21, the annulus {A E C : i(T) siAl s r(T)} contains S(3(T). On the other hand, by [2, 6.1J, 6(T*) = {A E C : IAI < c(T)} , where ae(T) = aK(T) = aap(T) = {A E C : i(T)
c(T):= liminf(wl" n>oo
'Wn)l/n.
Thus, by Corollary 17, it follows that S5(T) = S(3(T*) 2 {A E C : IAI S c(T)}. We finally note that, by [22, 2.7J, condition (f3) on T implies that i(T) = r(T) and aT(x) = a(T) for all nonzero x E X, while, by [22, 3.3J or [23, Prop.5], a certain growth condition of exponential type for the weight sequence w suffices to ensure that T has (f3). 0
References [IJ P. Aiena, M. L. Colasante, and M. Gonzalez, Opemtors which have a closed quasinilpotent part, Proc. Amer. Math. Soc. 130 (2002), 27012710. [2J P. Aiena, T. L. Miller, and M. M. Neumann, On a localized singlevalued extension property, to appear in Proc. Royal Irish Acad. [3J P. Aiena and O. Monsalve, Opemtors which do not have the single valued extension property, J. Math. Anal. Appl. 250 (2000),435448. [4J P. Aiena and O. Monsalve, The single valued extension property and the genemlized Kato decomposition property, Acta Sci. Math. (Szeged) 67 (2001), 791807. [5J P. Aiena and F. Villafane, Components of resolvent sets and local spectml theory, submitted to this volume. [6J E. Albrecht and J. Eschmeier, Analytic functional models and local spectml theory, Proc. London Math. Soc. (3) 75 (1997), 323348. [7J E. Albrecht and W. J. Ricker, Local spectml theory f01· opemtors with thin spectrum, preprint, University of Saarbriicken, 2002. [8J H. Bercovici and S. Petrovic, Genemlized scalar opemtors as dilations, Proc. Amer. Math. Soc. 123 (1995), 21732180. [9J E. Bishop, A duality theory for an arbitmry opemtor, Pacific J. Math. 9 (1959), 379397. [lOJ I. Colojoara and C. Foi~, Theory of Genemlized Spectml Opemtors, Gordon and Breach, New York, 1968. [11J M. Didas, E(]"n )subscalar ntuples and the Cesaro opemtor on HP, Annales Universitatis Saraviensis, Series Mathematicae 10 (2000), 285335. [12J N. Dunford and J. T. Schwartz, Linear Opemtors III, WileyInterscience, New York, 1971. [13J J. Eschmeier, On the essential spectrum of Banachspace opemtors, Proc. Edinburgh Math. Soc. (2) 43 (2000), 511528. [14J J. Eschmeier and B. Prunaru, Invariant subspaces for opemtors with Bishop's property ({3) and thick spectrum, J. Funct. Anal. 94 (1990), 196222. [15J J. Eschmeier and M. Putinar, Bishop's condition ({3) and rich extensions of linear opemtors, Indiana Univ. Math. J. 37 (1988), 325348. [16J J. Eschmeier and M. Putinar, Spectml Decompositions and Analytic Sheaves, Clarendon Press, Oxford, 1996. [17J J. K. Finch, The single valued exten.9ion property on a Banach space, Pacific J. Math. 58 (1975),6169. [18J K. B. Laursen and M. M. Neumann, An Introduction to Local Spectml Theory, Clarendon Press, Oxford, 2000. [19J M. Mbekbta, Genemlisation de la decomposition de Kato aux opemteurs pamnormaux et spectmux, Glasgow Math. J. 29 (1987), 159175. [20J M. Mbekhta, Sur la theorie spectmle locale et limite des nilpotents, Proc. Amer. Math. Soc. 110 (1990), 621631.
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T. L. MILLER, V. G. MILLER, AND M. M. NEUMANN
[21] M. Mbekhta and A. Ouahab, Operateurs sregulier dans un espace de Banach et theorie spectrale, Acta Sci. Math. (Szeged) 59 (1994), 525543. [22] T. L. Miller, V. G. Miller, and M. M. Neumann, Local spectral properties of weighted shifts,
to appear in J. Operator Theory. [23] T. L. Miller, V. G. Miller, and M. M. Neumann, Growth conditions and decomposable exten
sions, to appear in Contemp. Math. [24] T. L. Miller, V. G. Miller, and M. M. Neumann, On operators with closed analytic core, to appear in Rend. Cire. Mat. Palermo (2) 51 (2002). [25] B. Nagy, On Sdecomposable operators, J. Operator Theory 2 (1979),277286. [26] M. M. Neumann, Recent developments in local spectral theory, Rend. Circ. Mat. Palermo (2) Suppl. 68 (2002), 111131. [27] M. Putinar, Hyponormal operators are subsealar, J. Operator Theory 12 (1984), 385395. [28] M. Putinar, Quasisimilarity of tuples with Bishop's property (,8), Integral Equations Operator Theory 15 (1992), 10471052. [29] F.H. Vasilescu, Analytic FUnctional Calculus and Spectral Decompositions, Editura Aeademiei and D. Reidel Publishing Company, Bucharest and Dordreeht, 1982. [30] P. Vrbova., On local spectral properties of operators in Banach spaces, Czechoslovak Math. J. 23 (98) (1973),483492. DEPARTMENT OF MATHEMATICS AND STATISTICS, MISSISSIPPI STATE UNIVERSITY, MISSISSIPPI STATE, MS 39762, USA Email address: neumannOmath.msstate.edu
Contemporary Mathematics Volume 328, 2003
Abstract harmonic analysis, homological algebra, and operator spaces Volker Runde ABSTRACT. In 1972, B. E. Johnson proved that a locally compact group G is amenable if and only if certain Hochschild cohomology groups of its convolution algebra Ll(G) vanish. Similarly, G is compact if and only if Ll(G) is biprojective: In each case, a classical property of G corresponds to a cohomological propety of Ll(G). Starting with the work of Z.J. Ruan in 1995, it has become apparent that in the noncommutative setting, i.e. when dealing with the Fourier algebra A(G) or the FourierStieltjes algebra B(G), the canonical operator space structure of the algebras under consideration has to be taken into account: In analogy with Johnson's result, Ruan characterized the amenable locally compact groups G through the vanishing of certain cohomology groups of A(G). In this paper, we give a survey of historical developments, known results, and current open problems.
1. Abstract harmonic analysis, ...
The central objects of interest in abstract harmonic analysis are locally compact groups, i.e. groups equipped with a locally compact Hausdorff topology such that multiplication and inversion are continuous. This includes all discrete groups, but also all Lie groups. There are various function spaces associated with a locally compact group G, e.g. the space Co(G) of all continuous functions on G that vanish at infinity. The dual space of Co(G) can be identified with M(G), the space of all regular (complex) Borel measures on G. The convolution product * oftwo measures is defined via (1,11* v):= LLf(XY)dJ.L(X)V(Y)
(J.L,V E M(G), f E Co(G))
and turns M(G) into a Banach algebra. Moreover, M(G) has an isometric involution given by
(I,J.L*):= Lf(x1)dJ.L(X)
(J.L E M(G), f E Co(G)).
1991 Mathematics Subject Classification. 22D15, 22D25, 43A20, 43A30, 46H20 (primary), 46H25, 46L07, 46M18, 46M20, 47B47, 47L25, 47L50. Key words and phrases. locally compact groups, group algebra, Fourier algebra, FourierStieltjes algebra, Hochschild cohomology, homological algebra, operator spaces. Financial support by NSERC under grant no. 22704300 is gratefully acknowledged. © 263
2003 American Mathematical Society
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The most surprising feature of an object as general as a locally compact group is the existence of (left) Haar measure: a regular Borel measure which is invariant under left translation and unique up to a multiplicative constant. For example, the Haar measure of a discrete group is simply counting measure, and the Haar measure of ]RN, is Ndimensional Lebesgue measure. The space Ll(G) of all integrable functions with respect to Haar measure can be identified with a closed *ideal of M(G) via the RadonNikodym theorem. Both M(G) and Ll(G) are complete invariants for G: Whenver Ll(G 1 ) and £1(G 2 ) (or M(Gt} and M(G 2 )) are isometrically isomorphic, then G 1 and G 2 are topologically isomorphic. This means that all information on a locally compact group is already encoded in Ll (G) and M(G). For example, Ll(G) and M(G) are abelian if and only if G is abelian, and Ll (G) has an identity if and only if G is discrete. References for abstract harmonic analysis are [Fol], [HR], and [RSt]. The property of locally compact groups we will mostly be concerned in this survey is amenability. A a mean on a locally compact group G is a bounded linear functional m: LOC(G) + C such that (1, m) = Ilmil = 1. For any function I on G and for any x E G, we write LxI for the left translate of I by x, i.e. (Lxf)(y) := I(xy) for y E G. DEFINITION 1.1. A locally compact group G is called amenable if there is a (left) translation invariant mean on G, i.e. a mean m such that
(¢, m) = (L x ¢, m)
(¢ E LOC(G), x E G).
EXAMPLE 1.2. (1) Since the Haar measure of a compact group G is finite, LOC(G) C £1(G) holds. Consequently, Haar measure is an invariant mean on G. (2) For abelian G, the MarkovKakutani fixed point theorem yields an invariant mean on G. (3) The free group in two generators is not amenable ([Pat, (0.6) Exanlple]). Moreover, amenability is stable under standard constructions on locally compact groups such as taking subgroups, quotients, extensions, and inductive limits. Amenable, locally compact groups were first considered by J. v. Neumann ([Neu]) in the discrete case; he used the term "Gruppen von endlichem MaB". The adjective amenable for groups satisfying Definition 1.1 is due to M. M. Day ([Day]), apparently with a pun in mind: They are amenable because they have an invariant mean, but also since they are particularly pleasant to deal with and thus are truly amenable  just in the sense of that adjective in everyday speech. For more on the theory of amenable, locally compact groups, we refer to the monographs [Gre], [Pat], and [Pie].
2. homological algebra, ... We will not attempt here to give a survey on a area as vast as homological algebra, but outline only a few, basic cohomological concepts that are relevant in connection with abstract harmonic analysis. For the general theory of homological algebra, we refer to [CE], [MacL], and [Wei]. The first to adapt notions from homological algebra to the functional analytic context was H. Kamowitz in [Kam]. Let 2l be a Banach algebra. A Banach 2lbimodule is a Banach space E which is also an 2lbimodule such that the module actions of 2l on E are jointly continuous.
ABSTRACT HARMONIC ANALYSIS A derivation from 2l to E is a (bounded) linear map D: 2l D(ab) = a . Db + (Da) . b
265
+
E satisfying
(a, bE !2l);
the space of all derivation from 2l to E is commonly denoted by ZI(!2l, E). A derivation D is called inner if there is x E E such that Da = a·xx·a
(a E !2l).
The symbol for the subspace of ZI (!2l, E) consisting of the inner derivations is B 1 (2l,E); note that B 1 (!2l,E) need not be closed in ZI(!2l,E). DEFINITION 2.1. Let !2l be a Banach algebra, and let E be a Banach 2lbimodule. Then then the first Hochschild cohomology group 'HI (2l, E) of 2l with coefficients in E is defined as 'H 1 (!2l, E) := ZI(!2l, E)/B 1 (2l, E). The name Hochschild cohomology group is in the honor of G. Hochschild who first considered these groups in a purely algebraic context ([Hoch 1] and [Hoch 2]). Given a Banach !2lbimodule E, its dual space E* carries a natural Banach 2lbimodule structure via (x,a· ¢) := (x· a,¢)
and
(x,¢· a) := (a· x,¢)
(a E !2l, ¢ E E*, x E E).
We call such Banach !2lbimodules dual. In his seminal memoir [Joh 1], B. E. Johnson characterized the amenable locally compact groups G through Hochschild cohomology groups of Ll(G) with coefficients in dual Banach £l(G)bimodules ([Joh 1, Theorem 2.5]): THEOREM 2.2 (B. E. Johnson). Let G be a locally compact group. Then G is amenable if and only if'Hl(Ll(G),E*) = {O} for each Banach Ll(G)bimodule E. The relevance of Theorem 2.2 is twofold: First of all, homological algebra is a large and powerful toolkit  the fact that a certain property is cohomological in nature allows to apply its tools, which then yield further insights. Secondly, the cohomological triviality condition in Theorem 2.2 makes sense for every Banach algebra. This motivates the following definition from [Joh 1]: DEFINITION 2.3. A Banach algebra 2l is called amenable if 'Hl(!2l, E*) = {O} for each Banach 2lbimodule E. Given a new definition, the question of how significant it is arises naturally. Without going into the details and even without defining what a nuclear C* algebra is, we would like to only mention the following very deep result which is very much a collective accomplishment of many mathematicians, among them A. Connes, M. D. Choi, E. G. Effros, U. Haagerup, E. C. Lance, and S. Wassermann: THEOREM 2.4. A C* algebra is amenable if and only if it is nuclear. For a relatively selfcontained exposition of the proof, see [Run, Chapter 6]. Of course, Definition 2.3 allows for modifications by replacing the class of all dual Banach 2lbimodules by any other class. In [BCD], W. G. Bade, P. C. Curtis, Jr., and H. G. Dales called a commutative Banach algebra !2l weakly amenable if and only if 'HI (2l, E) = {O} for every symmetric Banach !2lbimodule E, i.e. satisfying (a E !2l, x E E). a·x=x·a
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This definition is of little use for noncommutative 21. For commutative 21, weak amenability, however, is equivalent to 'Jtl(21, 21*) = {O} ([BCD, Theorem 1.5]), and in [Joh 2], Johnson suggested that this should be used to define weak amenability for arbitrary 21: DEFINITION 2.5. A Banach algebra 21 is called weakly amenable if 'Jtl(21, 21*) =
{O}. REMARK 2.6. There is also the notion of a weakly amenable, locally compact group ([CH]). This coincidence of terminology, however, is purely accidental. In contrast to Theorem 2.2, we have: THEOREM 2.7 ([Joh 3]). Let G be a locally compact group. Then £l(G) is weakly amenable. For a particularly simple proof of this result, see [DGh]. For M(G), things are strikingly different: THEOREM 2.8 ([DGhH]). Let G be a locally compact group. Then M(G) is weakly amenable if and only if G is discrete. In particular, M (G) is amenable if and only if G is discrete and amenable. Sometime after Kamowitz's pioneering paper, several mathematicians started to systematically develop a homological algebra with functional analytic overtones. Besides Johnson, who followed Hochschild's original approach, there were A. Guichardet ([Gui]), whose point of view was homological rather than cohomological, and J. A. Taylor ([Tay]) and  most persistently  A. Ya. Helemskil and his Moscow school, whose approaches used projective or injective resolutions; Helemskil's development of homological algebra for Banach and more general topological algebras is expounded in the monograph [He} 2]. In homological algebra, the notions of projective, injective, and flat modules play a pivotal role. Each of these concepts tranlates into the functional analytic context. Helemskil calls a Banach algebra 21 biprojective (respectively biflat) if it is a projective (respetively flat) Banach 21bimodule over itself. We do not attempt to give the fairly technical definitions of a projective or a flat Banach 21bimodule. Fortunately, there are equivalent, but more elementary characterizations of biprojectivity and biflatness, respectively. We use ®y to denote the completed projective tensor product of Banach spaces. If 21 is a Banach algebra, then 21 ®y 21 has a natural Banach 21bimodule structure via a·(x®y):=ax®y and (x®y)·a=:x®ya (a, x, y E 21). This turns the multiplication operator
A: 21 ®y 21
+
21,
a ® b f+ ab
into a homomorphism of Banach 21bimodules. DEFINITION 2.9. Let 21 be a Banach algebra. Then: (a) 21 is called biprojective if and only if A has bounded right inverse which is an 21bimodule homomorphism. (b) 21 is called biftat if and only if A * has bounded left inverse which is an 21bimodule homomorphism.
ABSTRACT HARMONIC ANALYSIS
267
Clearly, biflatness is a property weaker than biprojectivity. The following theorem holds ([Hell, Theorem 51]): THEOREM 2.10 (A. Ya. Helemskil). Let G be a locally compact group. Then Ll (G) is biprojective if and only if G is compact. Again, a classical property of G is equivalent to a cohomological property of Ll(G). The question for which locally compact groups G the Banach algebra Ll(G) is biflat seems natural at the first glance. However, any Banach algebra is amenable if and only if it is biflat and has a bounded approximate identity ([Hel 2, Theorem Vii.2.20]). Since Ll(G) has a bounded approximate identity for any G, this means that Ll (G) is biflat precisely when G is amenable. Let G be a locally compact group. A unitary representation of G on a Hilbert space jj is a group homomorphism 7r from G into the unitary operators on jj which is continuous with respect to the given topology on G and the strong operator topology on B(jj). A function G+C,
with
~, TJ
Xf+(7r(x)~,1J)
E jj is called a coefficient function of 7r.
EXAMPLE 2.11. The left regular representation A of G on L2(G) is given by A(X)~ := LXl~
(x E G, ~ E L2(G)).
DEFINITION 2.12 ([Eym]). Let G be a locally compact group. (a) The Fourier algebra A(G) of G is defined as A(G) := {f: G
+
C : f is a coefficient function of A}.
(b) The FourierStieltjes algebra B( G) of G is defined as B( G) := {f: G
+
C : f is a coefficient function of a unitary representation of G}.
It is immediate that A(G) c B(G), that B(G) consists of bounded continuous functions, and that A(G) C Co(G). However, it is not obvious that A(G) and B(G) are linear spaces, let alone algebras. Nevertheless, the following are true ([Eym]): • Let C*(G) be the enveloping C*algebra of the Banach *algebra Ll(G). Then B(G) can be canonically identified with C*(G)*. This turns B(G) into a commutative Banach algebra. • Let VN(G) := A(G)" denote the group von Neumann algebra of G. Then A(G) can be canonically identified with the unique predual of VN(G). This turns A( G) into a commutative Banach algebra whose character space is G. • A(G) is a closed ideal in B(G). If G is an abelian group with dual group r, then the Fourier and FourierStieltjes transform, respectively, yield isometric isomorphisms A( G) ~ Ll (r) and B(G) ~ M(r). Consequently, A(G) is amenable for any abelian locally compact group G. It doesn't require much extra effort to see that A(G) is also amenable if G has an abelian subgroup of finite index ([LLW, Theorem 4.1] and [For 2, Theorem 2.2]). On the other hand, every amenable Banach algebra has a bounded approximate identity, and hence Leptin's theorem ([Lep]) implies that the amenability of A(G) forces G to be amenable. Nevertheless, the tempting conjecture that A( G) is amenable if and only if G is amenable is false:
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THEOREM 2.13 ([Joh 4]). The Fourier algebra of SO(3) is not amenable. This leaves the following intriguing open question: QUESTION 2.14. Which are the locally compact groups G for which A(G) is amenable? The only groups G for which A( G) is known to be amenable are those with an abelian subgroup of finite index. It is a plausible conjecture that these are indeed the only ones. The corresponding question for weak amenability is open as well. B. E. Forrest has shown that A( G) is weakly amenable whenever the principal component of G is abelian ([For 2, Theorem 2.4]). One can, of course, ask the same question(s) for the FourierStieltjes algebra: QUESTION 2.15. Which are the locally compact groups G for which B(G) is amenable? Here, the natural conjecture is that those groups are precisely those with a compact, abelian subgroup of finite index. Since A( G) is a complemented ideal in B( G), the hereditary properties of amenability for Banach algebras ([Run, Theorem 2.3.7]) yield that A( G) has to be amenable whenever B( G) is. It is easy to see that, if the conjectured answer to Question 2.14 is true, then so is the one to Question 2.15. Partial answers to both Question 2.14 and Question 2.15 can be found in [LLW] and [For 2]. 3. and operator spaces Given any linear space E and n E N, we denote the n x nmatrices with entries from E by Mn(E); if E = C, we simply write Mn. Clearly, formal matrix multiplication turns Mn(E) into an Mnbimodule. Identifying Mn with the bounded linear operators on ndimensional Hilbert space, we equip Mn with a norm, which we denote by I· I· DEFINITION 3.1. An operator space is a linear space E with a complete norm II· lin on Mn(E) for each n E N such that (R 1) II
~ I~
Iln+m
= max{llxll n, IIYllm}
(n, mEN, x E Mn(E), Y E Mm(E))
and
(R 2) EXAMPLE 3.2. Let fJ be a Hilbert space. The unique C*norms on Mn(13(SJ)) 13(fJn) turn 13(SJ) and any of its subspaces into operator spaces.
~
Given two linear spaces E and F, a linear map T: E + F, and n E N, we define the the nth amplification T(n) : Mn(E) + Mn(F) by applying T to each matrix entry. DEFINITION 3.3. Let E and F be operator spaces, and let T E 13(E, F). Then: (a) T is completely bounded if IITllcb := sup nEN
IIT(n) IIB(Mn(E),Mn(F))
<
00.
ABSTRACT HARMONIC ANALYSIS
269
(b) T is a complete contraction if IITlicb ~ 1. (c ) T is a complete isometry if T( n) is an isometry for each n EN. The following theorem due to Z.J. Ruan marks the beginning of abstract operator space theory: THEOREM 3.4 ([Rna 1]). Let E be an operator space. Then there is a Hilbert space Sj and a complete isometry from E into B(Sj). To appreciate Theorem 3.4, one should think of it as the operator space analogue of the elementary fact that every Banach space is isometrically isomorphic to a closed subspace of C(O) for some compact Hausdorff space O. One could thus define a Banach space as a closed subspace of C(O) some compact Hausdorff space O. With this definition, however, even checking, e.g., that £1 is a Banach space or that quotients and dual spaces of Banach spaces are again Banach spaces is difficult if not imposssible. Since any C* algebra can be represented on a Hilbert space, each Banach space E can be isometrically embedded into B(Sj) for some Hilbert space Sj. For an operator space, it is not important that, but how it sits inside B(Sj). There is one monograph devoted to the theory of operator spaces ([ER]) as well as an online survey article ([Wit et al.]). The notions of complete boundedness as well as of complete contractivity can be defined for multilinear maps as well ([ER, p. 126]). Since this is somewhat more technical than Definition 3.3, we won't give the details here. As in the category of Banach spaces, there is a universallinearizer for the right, i.e. completely bounded, bilinear maps: the projective operator space tensor product ([ER, Section 7.1]), which we denote by ®. DEFINITION 3.5. An operator space 2l which is also an algebra is called a completely contractive Banach algebra if multiplication on 2l is a complete (bilinear) contraction. The universal property of ® ([ER, Proposition 7.1.2]) yields that, for a completely contractive Banach algebra 2l, the multiplication induces a complete (linear) contraction ~: 2l®2l + 2l. EXAMPLE 3.6. (1) For any Banach space E, there is an operator space maxE such that, for any other operator space F, every T E B(E,F) is completely bounded with IITlicb = IITII ([ER, pp. 4754]). Given a Banach algebra 2l, the operator space max2l is a completely contractive Banach algebra ([ER, p. 316]). (2) Any closed subalgebra of B(Sj) for some Hilbert space Sj is a completely contractive Banach algebra. To obtain more, more interesting, and  in the context of abstract harmonic analysis  more relevant examples, we require some more operator space theory. Given two operator spaces E and F, let
CB(E, F) := {T: E
+
F : T is completely bounded}.
It is easy to check that CB(E, F) equipped with 11·llcb is a Banach space. To define an operator space structure on CB(E, F), first note that Mn(F) is, for each n E N, an operator space in a canonical manner. The (purely algebraic) identification
Mn(CB(E, F)) := CB(E, Mn(F))
(n E N)
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then yields norms 1I·lln on the spaces Mn(CB(E, F)) that satisfy (R 1) and (R 2), which is not hard to verify. Since, for any operator space E, the Banach spaces E* and CB(E, C) are isometrically isomorphic ([ER, Corollary 2.2.3]), this yields a canonical operator space structure on the dual Banach space of an operator space. In partiuclar, the unique predual of a von Neumann algebra is an operator space in a canonical way. We shall see how this yields further examples of completely contractive Banach algebras. We denote the W* tensor product by ®. DEFINITION 3.7. A Hop/von Neumann algebra is a pair (rot, V), where rot is a von Neumann algebra, and V is a comultiplication: a unital, injective, w*continuous *homomorphism V: rot + rot®rot which is coassociative, i.e. the diagram rot
v
rot®rot
+
1
vl rot®rot
V®id!lJl
I
id!lJl®V
rot®rot®rot
commutes. EXAMPLE 3.8. Let G be a locally compact group. (1) Define V: £oo(G) + £oo(G x G) by letting
(V»(xy) := >(xy)
(> E £oo(G), x, y E G).
Since £oo(G)®£oo(G) ~ £oo(G x G), this turns £oo(G) into a Hopfvon Neumann algebra. (2) Let W*(G) be the enveloping von Neumann algebra of C*(G). There is a canonical w* continuous homomorphism w from G into the unitaries of W* (G) with the following universal property: For any unitary representation 7r of G on a Hilbert space, there is unique w* continuous *homomorphism (J: W*(G) + 7r(G)" such that 7r = (J 0 w. Applying this universal property to the representation G
+
W*(G)®W*(G),
x
yields a comultiplication V: W*(G)
1+
+
w(x) ® w(x) W*(G)®W*(G).
Given two von Neumann algebras rot and 1)1 with preduals rot* and 1)1*, their W*tensor product rot®1)1 also has a unique predual (rot®I)1)*. Operator space theory allows to identify this predual in terms of rot* and 1)1* ([ER, Theorem 7.2.4]): (rot®I)1)* ~ rot*ci~m*. Since VN(G)® VN(H) ~ VN(G x H) for any locally compact groups G and H, this implies in particular that A(G x H) ~ A(G)®A(H).
Suppose now that rot is a Hopfvon Neumann algebra with predual rot*. The comultiplication V : rot + rot®rot is w· continuous and thus the adjoint map of a complete contraction V. : rot*®rot. + rot.. This turns rot. into a completely contractive Banach algebra. In view of Example 3.8, we have:
ABSTRACT HARMONIC ANALYSIS
271
EXAMPLE 3.9. Let G be a locally compact group. (1) The multiplication on L1(G) induced by \7 as in Example 3.8.1 is just the usual convolution product. Hence, L1 (G) is a completely contractive Banach algebra. (2) The multiplication on B( G) induced by \7 as in Example 3.8.2 is pointwise multiplication, so that B( G) is a completely contractive Banach algebra. Since A (G) is an ideal in B (G) and since the operator space strucures A ( G) has as the predual of VN(G) and as a subspce of B(G) coincide, A(G) with its canonical operator space structure is also a completely contractive Banach algebra. REMARK 3.10. Since A(G) fails to be Arens regular for any nondiscrete or infinite, amenable, locally compact group G ([For 1]), it cannot be a subalgebra of the Arens regular Banach algebra B(f)). Hence, for those groups, A(G) is not of the form described in Example 3.6.2. We now return to homological algebra and its applications to abstract harmonic analysis. An operator bimodule over a completely contractive Banach algebra Il is an operator space E which is also an !!bimodule such that the module actions of Il on E are completely bounded bilinear maps. One can then define operator Hochschild cohomology groups 01t 1 (!!, E) by considering only completely bounded derivations (all inner derivations are automatically completely bounded). It is routine to check that the dual space of an operator !!bimodule is again an operator Ilbimodule, so that the following definition makes sense: DEFINITION 3.11 ([Rua 2]). A completely contractive Banach algebra Il is called operator amenable if 01t 1 (Il,E*) = {O} for each operator Ilbimodule E. The following result ([Rua 2, Theorem 3.6]) shows that Definition 3.11 is indeed a good one: THEOREM 3.12 (Z.J. Ruan). Let G be a locally compact group. Then G is amenable if and only if A( G) is operator amenable. REMARK 3.13. A Banach algebra!! is amenable if and only if max!! is operator amenable ([ER, Proposition 16.1.5]). Since L1(G) is the predual of the abelian von Neumann algebra LOO(G), the canonical operator space structure on L1(G) is maxL1(G). Hence, Definition 3.11 yields no information on L1(G) beyond Theorem 2.2. The following is an open problem: QUESTION 3.14. Which are the locally compact groups G for which B(G) is operator amenable? With Theorem 2.8 and the abelian case in mind, it is reasonable to conjecture that B(G) is operator amenable if and only if G is compact. One direction is obvious in the light of Theorem 3.12; a partial result towards the converse is given in [RSp]. Adding operator space overtones to Definition 2.5, we define: DEFINITION 3.15. A completely contractive Banach algebra Il is called operator weakly amenable if 01t 1 (!!, Il *) = {O}.
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In analogy with Theorem 2.7, we have: THEOREM 3.16 ([Spr]). Let G be a locally compact group. Then A(G) is operator weakly amenable.
One can translate Helemskir's homological algebra for Banach algebras relatively painlessly to the operator space setting: This is done to some extent in [Ari] and [Woo 1]. Of course, appropriate notions of projectivity and flatness play an important role in this operator space homological algebra. Operator biprojectivity and biflatness can be defined as in the classical setting, and an analogue  with ® instead of Q$)y  of the characterization used for Definition 2.9 holds. The operator counterpart of Theorem 2.10 was discovered, independently, by O. Yu. Aristov and P. J. Wood: THEOREM 3.17 ([Ari], [Woo 2]). Let G be a locally compact group. Then G is discrete if and only if A(G) is operator biprojective.
As in the classical setting, both operator amenability and operator biprojectivity imply operator biflatness. Hence, Theorem 3.17 immediately supplies examples of locally compact groups G for which A( G) is operator biflat, but not operator amenable. A locally compact group is called a [SIN]group if Ll(G) has a bounded approximate identity belonging to its center. By [RX, Corollary 4.5], A(G) is also operator biflat whenever G is a [SIN]group. It may be that A( G) is operator biflat for every locally compact group G: this question is investigated in [ARSp]. All these results suggest that in order to get a proper understanding of the Fourier algebra and of how its cohomological properties relate to the underlying group, one has to take its canonical operator space structure into account.
References O. Yu. Aristov, Biprojective algebras and operator spaces. J. Math. Sci. (to appear). O. Yu. Aristov, V. Runde, and N. Spronk. Operator biflatness of the Fourier algebra. In preparation. [BCD] W. G. Bade, P. C. Curtis, Jr., and H. G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras. Proc. London Math. Soc. (3) 55 (1987), 359377. [CE] H. Cartan and S. Eilenberg, Homological algebra. Princeton University Press, Princeton, 1956. M. Cowling and U. Haagerup, Completely bounded multipliers of the Fourier algebra [CH] of a simple Lie group of real rank one. Invent. Math. 96 (1989), 507549. [DGhH] H. G. Dales, F. Ghahramani, and A. Va. HelemskiY, The amenability of measure algebras. J. London Math. Soc. 66 (2002), 213226. M. M. Day, Means on semigroups and groups. Bull. Amer. Math. Soc. 55 (1949), [Day] 10541055. [DGh] M. Despic and F. Ghahramani, Weak amenability of group algebras of locally compact groups. Ganad. Math. Bull. 37 (1994), 165167. E. G. Effros and Z.J. Ruan, Operator spaces. Clarendon Press, Oxford, 2000. [ER] [Eym] P. Eymard, L'algebre de Fourier d'un groupe localement compact. Bull. Soc. Math. France 92 (1964), 181236. G. B. Folland, A course in abstract harmonic analysis. CRC Press, Boca Raton, [Fol] Florida, 1995. B. E. Forrest, Arens regularity and discrete groups. Pacific J. Math. 151 (1991), [For 1] 217227. B. E. Forrest, Amenability and weak amenability of the Fourier algebra. Preprint [For 2] (2000). [Gre] F. P. Greenleaf, Invariant means on locally compact groups. Van Nostrand, New YorkTorontoLondon, 1969. [Ari] [ARSp]
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A. Guichardet, Sur I'homologie et la cohomologie des algebres de Banach. C. R. Acad. Sci. Paris, Ser. A 262 (1966), 3842. [Her] C. S. Herz, Harmonic synthesis for subgruops. Ann. Inst. Fourier (Grenoble) 23 (1973),91123. [HR] E. Hewitt and K. A. Ross, Abstract harmonic analysis, I and II. Springer Verlag, BerlinHeidebergNew York, 1963 and 1970. [Hell] A. Ya. Helemskil', Flat Banach modules and amenable algebras. Trans. Moscow Math. Soc. 47 (1985), 199224. [HeI2] A. Ya. Helemskil, The homology of banach and topological algebras (translated from the Russian). Kluwer Academic Publishers, Dordrecht, 1989. G. Hochschild, On the cohomology groups of an associative algebra. Ann. of Math. [Hoch 1] (2) 46 (1945), 5867. [Hoch 2] G. Hochschild, On the cohomology theory for associative algebras. Ann. of Math. (2) 47 (1946), 568579. B. E. Johnson, Cohomology in Banach algebras. Mem. Amer. Math. Soc. 127 (1972). [Joh 1] B. E. Johnson, Derivations from Ll(G) into Ll(G) and LOO(G). In: J. P. Pier (ed.), [Joh 2] Harmonic analysis (Luxembourg, 1987), pp. 191198. Lectures Notes in Mathematics 1359. Springer Verlag, BerlinHeidelbergNew York, 1988. [Joh 3] B. E. Johnson, Weak amenability of group algebras. Bull. London Math. Soc. 23 (1991),281284. [Joh 4] B. E. JOHNSON, Nonamenability of the Fourier algebra of a compact group. J. London Math. Soc. (2) 50 (1994),361374. [Kam] H. Kamowitz, Cohomology groups of commutative Banach algebras. Trans. Amer. Math. Soc. 102 (1962), 352372. [LLW] A. T.M. Lau, R. J. Loy, and G. A. Willis, Amenability of Banach and C"algebras on locally compact groups. Studia Math. 119 (1996), 161178. H. Leptin, Sur l'algebre de Fourier d'un groupe localement compact. C. R. Acad. Sci. [Lep] Paris, Ser. A 266 (1968), 11801182. S. MacLane, Homology. Springer Verlag, BerlinHeidelbergNew York, 1995. [MacL] J. von Neumann, Zur allgemeinen Theorie des MaBes. Fund. Math. 13 (1929), 73116. [Neu] A. L. T. Paterson, Amenability. American Mathematical Society, Providence, 1988. [Pat] J. P. Pier, Amenable locally compact groups. WileyInterscience, New York, 1984. [Pie] H. Reiter and J. D. Stegeman, Classical harmonic analysis and locally compact [RSt] groups. Clarendon Press, Oxford, 2000. [Rua 1] Z.J. Ruan, Subspaces of C"algebras. J. Funct. Anal. 76 (1988), 217230. [Rua2] Z.J. Ruan, The operator amenability of A(G). Amer. J. Math. 117 (1995), 14491474. Z.J. Ruan and G. Xu, Splitting properties of operator bimodules and operator ame[RX] nability of Kac algebras. In: A. Gheondea, R. N. Gologan and D. Timotin, Operator theory, operator algebras, and related topics, pp. 193216. The Theta Foundation, Bucharest, 1997. v. Runde, Lectures on amenability. Lecture Notes in Mathematics 1774. Springer [Run] Verlag, BerlinHeidelbergNew York, 2002. V. Runde and N. Spronk, Operator amenability of FourierStieltjes algebras. Preprint [RSp] (2001). N. Spronk, Operator weak amenability of the Fourier algebra. Proc. Amer. Math. [Spr] Soc. 130 (2002), 36093617. [Tay] J. A. Taylor, Homology and cohomology for topological algebras. Adv. in Math. 9 (1970), 137182. [Wei] C. A. Weibel, An introduction to homological algebra. Cambridge University Press, Cambridge, 1994. [Wit et al.] G. Wittstock et al., What are operator spaces?  An online dictionary. URL: http://wwv.math.unisb.de/~agwittstock/projekt2001.html (2001). [Woo 1] P. J. Wood, Homological algebra in operator spaces with applications to harmonic analysis. Ph.D. thesis, University of Waterloo, 1999. P. J. Wood, The operator biprojectivity of the Fourier algebra. Canadian .1. Math. [Woo 2] (to appear).
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VOLKER RUNDE
DEPARTMENT OF MATHEMATICAL AND STATISTICAL SCIENC MONTON,
AB,
CANADA
T6G 2Gl
Email address:vrundeClualberta.ca
Contemporary Mathematics Volume 328, 2003
Relative Tensor Products for Modules over von Neumann Algebras David Sherman ABSTRACT. We give an overview of relative tensor products (RTPs) for von
Neumann algebra modules. For background, we start with the categorical definition and go on to examine its algebraic formulation, which is applied to Morita equivalence and index. Then we consider the analytic construction, with particular emphasis on explaining why the RTP is not generally defined for every pair of vectors. We also look at recent work justifying a representation of RTPs as composition of unbounded operators, noting that these ideas work equally well for LP modules. Finally, we prove some new results characterizing preclosedness of the map (~, 7) f> ~ 181",7).
1. Introduction
The purpose of this article is to summarize and explore some of the various constructions of the relative tensor product (RTP) of von Neumann algebra modules. Alternately known as composition or fusion, RTPs are a key tool in subfactor theory and the study of Morita equivalence. The idea is this: given a von Neumann algebra M, we want a map which associates a vector space to certain pairs of a right Mmodule and a left Mmodule. If we write module actions with subscripts, we have (XM,M!i)) f> X 0M!i). This should be functorial, covariant in both variables, and appropriately normalized. Other than this, we only need to specify which modules and spaces we are considering. In spirit, RTPs are algebraic; a ringtheoretic definition can be found in most algebra textbooks. But in the context of operator algebras, the requirement that the output be a certain type of space  typically a Hilbert space  causes an analytic obstruction. As a consequence, there are domain issues in any vectorbased construction. Fortunately, von Neumann algebras have a sufficiently simple representation theory to allow a recasting of RTPs in algebraic terms. The analytic study of RTPs can be related nicely to noncommutative £P spaces. Indeed, examination of the usual (£2) case reveals that the technical difficulties 2000 Mathematics Subject Classification. Primary: 46LIO; Secondary: 46M05. Key words and phrases. relative tensor product, von Neumann algebra, bimodule. © 275
2003 American Mathematical Society
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DAVID SHERMAN
come from a "change of density". (We say that the density of an LPtype space is lip.) Once this is understood, it is easy to handle LP modules [JS] as well. Modular algebras ([Y], [S]) provide an elegant framework, so we briefly explain their meaning. The final section of the paper investigates the question, "When is the map (~, 1]) f+ ~ ®
2. Notations and background The basic objects of this paper are von Neumann algebras, always denoted here by M, N, or P. These can be defined in many equivalent ways: • C*algebras which are dual spaces. • stronglyclosed unital *subalgebras of B(i)). B(i)) is the set of bounded linear operators on a Hilbert space i); the strong topology is generated by the seminorms x f+ Ilx~ll, ~ E i); the * operation is given by the operator adjoint. • *closed subsets of B(i)) which equal their double (iterated) commutant. The commutant of a set S c B(i)) is {x E B(i)) I xy = yx, 'l:/y E S}. As one might guess from the definitions, the study of von Neumann algebras turns on the interplay between algebraic and analytic techniques. Finitedimensional von Neumann algebras are direct sums of full matrix algebras. At the other extreme, commutative von Neumann algebras are all of the form Loo(X, Jl) for some measure space (X, Jl)' so the study of general von Neumann algebras is considered "noncommutative measure theory." Based on this analogy, the (unique) predual M* of M is called Ll(M); it is the set of normal (= continuous in yet another topology, the aweak) linear functionals on M c B(i)), and can be thought of as "noncommutative countably additive measures". A functional r.p is positive when x> 0 =} r.p(x) 2: 0; the set of positive normal functionals is denoted M;t. The support s( r.p) of a positive normal linear functional r.p is the smallest projection q E M with r.p(l  q) = O. So if M is abelian, r.p corresponds to a measure and q is the (indicator function of the) usual support. For simplicity, all modules in this paper are separable Hilbert spaces (except in Section 6), all algebras have separable predual, all linear functionals are normal, and all representations are normal and nondegenerate (MSJ or SJM is all of SJ). Two projections p, q in a von Neumann algebra are said to be (Murrayvon Neumann) equivalent if there exists v E M with v*v = p, vv* = q. Such an element v is called a partial isometry, and we think of p and q as being "the same size". Subscripts are used to represent actions, so XM indicates that X is a right Mmodule, i.e. a representation of the opposite algebra MOP. It is implicit in the term "bimodule", or in the notation Mi)N, that the two actions commute. The phrase "left (resp. right) action of' is frequently abbreviated to L (resp. R) for operators or entire algebras, so that we speak of L(x) or R(M). Finally, we often write Moo for the von Neumann algebra of all bounded operators on a separable infinitedimensional
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Hilbert space, and Moc(M) for the von Neumann tensor product Moc®M. One can think of this as the set of infinite matrices with entries in M; we will denote by eij the matrix unit with 1 in the ij position and 0 elsewhere. The (left) representation theory of von Neumann algebras on Hilbert spaces is simple, so we recall it briefly. (Most of this development can be found in Chapters 1 and 2 of [JoS].) First, there is a standard construction, due to GelfandNeumark and Segal (abbreviated GNS), for building a representation from 'P E Mt. To each x E M we formally associate the vector x'P 1/ 2 (various notations are in use, e.g. 7]",(x) or A",(.'1:), but this one is especially appropriate ([C2] V.App.B, [S])). We endow this set with the inner product
< x'P1/2, Y'P 1/ 2 >= 'P(Y* x), and set fJ", to be the closure in the inherited topology, modulo the null space. The left action of M on fJ", = M'P1/2 is bounded and densely defined by left composition. When 'P is faithful (meaning x > 0 => 'P(x) > 0), the vector 'P 1/ 2 = I'P1/2 is cyclic (M'P1/2 = fJ",) and separating (x =f. 0 => X'P1/2 =f. 0). Now all representations with a cyclic and separating vector are isomorphic  a sort of "left regular representation"; we will denote this by ML 2 (M). It is a fundamental fact that the commutant of this action is antiisomorphic to M, and when we make this identification we call ML2(M)M the standard form of M. If 'P is not faithful, the GNS construction produces a vector 'Pl/2 which is cyclic but not separating, and a representation which is isomorphic to ML2(M)s('P) ([T2], Ch. VIII, IX). Now let us examine an arbitrary (separable, so countably generated) module MfJ. Following standard arguments (e.g. [TI] I.9), fJ decomposes into a direct sum of cyclic representations M(M~n), each of which is isomorphic to the GNS representation for the associated vector state w~n (=< ·~n, ~n ». With qn = s(w~J, we have MfJ ~ EBMM~n ~ EBMfJw~n ~ EB M L 2(M)qn. (Here and elsewhere, "~" means a unitary equivalence of (bi)modules.) Since this is a left module, it is natural to write vectors as rows with the nth entry in L 2(M)qn:
We will call such a decomposition a row representation of MfJ. Here e nn are diagonal matrix units in Moo, so (Eqn ®e nn ) is a diagonal projection in Moc(M). The left action of M is, of course, matrix multiplication (by 1 x 1 matrices) on the left. The module (L2(M)L2(M) ... ) will be denoted R2(M) (for "row"). Since the standard form behaves naturally with respect to restriction  L2(q,Nq) ~ qL 2(,N)q as bimodules  it follows that L2(Moo(M)) is built as infinite matrices over L2(M) (see (3.3)). Thus R2(M) can be realized as ellL 2(Moo (M)). PROPOSITION 2.1. Any countably generated left representation of M on a Hilbert space is isomorphic to R 2(M)q for some diagonal projection q E Moc(M). Any projection ,in Moc(M), diagonal or not, defines a module in this way, and two such modules are isomorphic exactly when the projections are equivalent. In fact
(2.2)
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DAVID SHERMAN
So isomorphism classes correspond to equivalence classes of projections in lVIoo(M), which is the monoid V(Moo(M)) in K theoretic language [WO}. The direct sum of isomorphism ciasses of modules corresponds to the sum of orthogonal representatives in V(Moo(M)), giving a monoidal equivalence.
We denote the category of separable left Mmodules by Left L 2(M). For us, the most important consequence of (2.2) is that (2.3)
C.(MR2(M)q) = R(qMoo(M)q),
where "c." stands for the commutant of the Maction. (In particular, the case = ell is just the standard form.) The algebra qMoo(M)q is called an amplification of M, being a generalization of a matrix algebra with entries in M. Of course everything above can be done for right modules  the relevant abbreviations are C 2(M), for "column," and Right L 2(M). Example. Suppose M = M3(C). In this case the standard form may be taken as M3L2(M3hf3; L2(M3) ~ (M3, < .,. », where < x, y >= Tr(y*x). Note that this norm, called the HilbertSchmidt norm, is just the e2 norm of the matrix entries, and that the left and right multiplicative actions are commutants. (If we had chosen a nontracial positive linear functional, we would have obtained an isomorphic bimodule with a "twisted" right action ... this is inchoate TomitaTakesaki theory.) The module R2(M3) is M 3xoo , again with the HilbertSchmidt norm, and the commutant is Moo(M3) ~ Moo. According to Proposition 2.1, isomorphism classes of left M 3modules should be parameterized by equivalence classes of projections in Moo. These are indexed by their rank n E (1:+Uoo); the corresponding isomorphism class of modules has representative M 3xn . In summary, we have learned that any left representation of M3 on a Hilbert space is isomorphic to some number of copies of C 3. The same argument shows that V(Moo(Mk)) ~ (1:+ U 00) for any k. Properties of the monoid V ( Moo (M)) determine the socalled type of the algebra. For a factor (a von Neumann algebra whose center is just the scalars), there are only three possibilities: (1:+ U 00), (lR.+ U 00), and {O,+oo}. These are called types I, II, III, respectively; a fuller discussion is given in Section 7. q
3. Algebraic approaches to RTPs When R is a ring, the algebraic Rrelative tensor product is the functor, covariant in both variables, which maps a right Rmodule A and left Rmodule B to the vector space (A ®alg B)/N, where N is the subspace generated algebraically by tensors of the form ar ® b  a ® rb. In functional analysis, where spaces are usually normed and infinitedimensional, one obvious amendment is to replace vector spaces with their closures. But in the context of Hilbert modules over a von Neumann algebra M, this is still not enough. Surprisingly, a result of Falcone ([FJ, Theorem 3.8) shows that if the RTP L2(M) ®M L2(M) is the closure of a continuous (meaning III(~ ® 7])11 < ClI~IIII7]11) nondegenerate image ofthe algebraic Mrelative tensor product, M must be atomic, Le. M ~ EBnB(f.>n). We will discuss the analytic obstruction further in Section 5. For now, we take Falcone's theorem as a directive: do not look for a map which is defined for every pair of vectors. If we give up completely on a vectorlevel construction, we can at least make the functorial
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DEFINITION 3.1 (Sa). Given a von Neumann algebra M, a relative tensor product is a junctor, covariant in both variables,
RightL 2(M) x LeJtL2(M)
(3.1)
>
Hilber·t:
(n,.ft)
f+
n ®M.ft,
which satisfies (3.2)
as bimodules. Although at first glance this definition seems broad, in fact we see in the next proposition that there is exactly one RTP functor (up to equivalence) for each algebra. The reader is reminded that functoriality implies a mapping of intertwiner spaces as well, so it is enough to specify the map on representatives of each isomorphism class. In particular we have the bimodule structure '£:(j) .••ll(n
®M .ft)C(M.ll)·
PROPOSITION 3.2. Let n ~ P C 2(M) E Right L2 M od(M) and.ft ~ R 2(M)q E Left L2Mod(M) Jor some projections p,q E Moo(M). Then
n ®M.ft ~ P L2(Moo(M))q with natural action oj the commutants. PROOF. By implementing an isomorphism, we may assume that the projections are diagonal: p = LPi ®eii, q = Lqj ®ejj. Using (3.2) and functoriality, we have the bimodule isomorphisms
n ®M.ft ~ (EBPi L2(M)) ®M (EBL2(M)qj) ~
E9Pi L2(M) ®M L2(M)qj ~ E9Pi L2(M)qj ~ p L2(Moo(M))q. i,j
i,j D
Visually, (3.3) where of course the £2 sums of the norms of the entries in these matrices are finite. After making the categorical definition above, Sauvageot immediately noted that it gives us no way to perform computations. We will turn to his analytic construction in Section 5; here we discuss an approach to bimodules and RTPs due to Connes. In his terminology a bimodule is called a correspondence. (The best references known to the author are [C2l and [Pl, but there was an earlier unpublished manuscript which is truly the source of Connes fusion.) Consider a correspondence MnN. Choosing a row representation R2(M)q for n, we know that the full commutant of L(M) is isomorphic to R(qMoo(M)q). This gives us a unital injective *homomorphism p : N '+ qMoo(M)q, and from the map p one can reconstruct the original bimodule (up to isomorphism) as M(R 2(M)q)p(N)'
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DAVID SHERMAN
What if we had chosen a different row representation R 2 (M)q' and obtained P' : N + q'Moo(M)q'? By Proposition 2.1, the module isomorphism MR 2(M)q ~ MR 2(M)q'
(3.4)
is necessarily given by the right action of a partial isometry v between q and q' in Moo(M). Then P and P' differ by an inner perturbation: p(x) = v*p'(x)v. We conclude that the class of M  N correspondences, modulo isomorphism, is equivalent to the class of unital injective *homomorphisms from N into an amplification of M, modulo inner perturbation. (These last are called sectors in subfactor theory.) The distinctions between bimodules, morphisms, and their appropriate equivalence classes are frequently blurred in the literature; our convention here is to use the term "correspondence" to mean a representative *homomorphism for a bimodule. Notice that a unital inclusion N eM gives the bimodule ML 2(M)N. The RTP of correspondences is extremely simple. PROPOSITION 3.3. Consider bimodules MYJN and N.ftP coming from correspondences PI : N '> qMoo(M)q and P2 : P '> q'Moo(N)q'. The bimodule M(YJ®N.ft)P is the correspondence PI 0 P2, where we amplify PI appropriately.
We pause to mention that it is also fruitful to realize correspondences in terms of completely positive maps. We shall have nothing to say about this approach; the reader is referred to [P] for basics or [A2] for a recent investigation. 4. Applications to Morita equivalence and index An invertible *functor from Left L2 M od(N) to Left L2 M od(M) is called a Morita equivalence [R]. Here a *functor is a functor which commutes with the adjoint operation at the level of morphisms. One way to create *functors is the following: to the bimodule MYJN, we associate (4.1) FSj: Left L2 Mod(N)
+
Left L2 Mod(M);
N.ft 1+ (MYJN) ®N (N.ft).
The next theorem is fundamental. THEOREM 4.1. When L(M) and R(N) are commutants on YJ, the RTP functor FSj is a Morita equivalence. Moreover, every Morita equivalence is equivalent to an RTP functor.
This type of result  the second statement is an operator algebraic analogue of the EilenbergWatts theorem  goes back to several sources, primarily the fundamental paper of Rieffel [R]. His investigation was more general and algebraic, and his bimodules were not Hilbert spaces but rigged selfdual Hilbert C*modules, following Paschke [Pal. From a correspondence point of view, rigged selfdual Hilbert C*modules and Hilbert space bimodules give the same theory; the equivalence is discussed nicely in [A1]. (And the former is nothing but an L oo version of the latter, as explained in [JS].) Our Hilbert space approach here is parallel to that of Sauvageot [Sa], though modeled more on [R], and is streamlined by our standing assumption of separable preduals. We will need DEFINITION 4.2. The contragredient of the bimodule MYJN is the bimodule NfJM' where fJ is conjugate linearly isomorphic to fj (the image of ~ is written (J, and the actions are defined by n~m = m*~n*.
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LEMMA 4.3. Suppose L(M) and R(N) are commutants on fl. Then
NfJM ®M MflN ~ NL2(N)N. PROOF. If Mfl ~ MR 2(M)q, then N ~ qMoo(M)q by (2.3), and fJM qC 2 (M)M. By Proposition 3.2 and the comment preceding Proposition 2.1,
NfJM ®M MflN ~ N(qL2(Moo(M))q)N ~ NL2(qMoo(M)q)N ~ NL2(N)N. D Lemma 4.3 was first proven by Sauvageot (in another way). In our situation it means
FfJ
FSj(Nfi) ~ L2(N) ®N Nfi ~ Nfi.
(Here we have used the associativity of the RTP, which is most easily seen from the explicit construction in Section 5.) We conclude that Fi) 0 FSj is equivalent to the identity functor on Left L2Mod(N), and by a symmetric argument FSj 0 Fi) is equivalent to the identity functor on Left L2Mod(M). Thus FSj is a Morita equivalence, and the first implication of Theorem 4.1 is proved. Now let F be a Morita equivalence from Left L2 M od(N) to Left L2 M od(M). Then F must take NR2(N) to a module isomorphic to MR2(M), because each is in the unique isomorphism class which absorbs all other modules. (This is meant in the sense that NR2(N) ffiNfl ~ NR2(N); R2(N) is the "infinite element" in the monoid V (Moo (N)).) Being an invertible *functor, F implements a *isomorphism of commutants  call it a, not F, to ease the notation: (4.2) Apparently we have (4.3) Before continuing the argument, we need an observation: isomorphic algebras have isomorphic standard forms. Specifically, we may write L2(Moo(N)) as the GNS construction for tp E Moo(N)t and obtain the isomorphism (a1)t : L2(Moo(N)) ..::. L2(Moo(M)), (a1)t : xtp1/2 ........ a(x)(tp 0 a 1)1/2. Note that (a 1 )t(x~y) = a(x)[(a 1 )t(~)la(y). Now consider the RTP functor for the bimodule
MflN =
u 1 (M)a 1(et'{)C 2(N)N.
By Proposition 3.2 and the comment preceding Proposition 2.1, its action is
R2(N)q ........
171 (M)a 1 (et'{)L2(Moo (N))q (17;:)'
Met'{ L 2(Moo (M))a(q)
~ MR 2(M)a(q) ~ F(R 2(N)q).
We conclude that F is equivalent to FSj, which finishes the proof of Theorem 4.1. Notice that (4.2) and (4.3) can also be used to define a functor; this gives us COROLLARY 4.4. For two von Neumann algebras M and N, the following are equivalent: (1) M and N are Morita equivalent;
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(2) Moo(N) ~ Moo(M); (3) there is a bimodule Mf)N where the actions are commutants of each other; (4) there is a projection q E Moo(M) with central support 1 such that
qMoo(M)q
~
N.
(The central support of x E M is the least projection z in the center of M satisfying x = zx.) Example continued. M3 and M5 are Morita equivalent. This can be deduced easily from condition (2) or (4) of the corollary above. It also follows from the (Hilbert) equivalence bimodule M3HS(M3X5)Ms, where "HS" indicates the HilbertSchmidt norm; this bimodule gives us an RTP functor which is a Morita equivalence. Regardless of the construction, the equivalence will send (functorially) n copies of C 5 to n copies of C 3 . Apparently Morita equivalence is a coarse relation on type I algebras  it only classifies the center of the algebra (up to isomorphism). At the other extreme, Morita equivalence for type III algebras is the same as algebraic isomorphism; Morita equivalence for type II algebras is somewhere in the middle ([RJ, Sec. 8). For a bimodule Mf)N where the algebras are not necessarily commutants, the functor (4.1) still makes sense. To get a more tractable object, we may consider the domain and range to be isomorphism classes of modules:
(4.4)
71'1) :
V(Moo(N)) . V(Moo(M));
F1)(R 2(N)q) = Mf)N ®N R2(N)q ~ R2(M)7I'1) ([q]). We call this the bimodule morphism associated to f), a sort of "skeleton" for the correspondence. It follows from Proposition 3.3 that if the bimodule is p : N '+ qMoo(M)q, then 71'1) is nothing but poo, the amplification of p to Moo(N), restricted to equivalence classes of projections. This has an important application to inclusions of algebras. We have seen in M is equivalent to a bimodule ML2(M)N. Section 3 that a unital inclusion N When the correspondence p is surjective, the module generates a Morita equivalence via its RTP functor, and the induced bimodule morphism is an isomorphism of monoids. When N i= M, it is natural to expect that the bimodule morphism gives us a way to measure the relative size, or index, of N in M. (For readers unfamiliar with this concept, the index of an inclusion N c M is denoted [M : N] and is analogous to the index of a subgroup. The kernel of this idea goes back to Murray and von Neumann, but the startling results of Jones [J] in the early 1980's touched off a new wave of investigation. We recommend the exposition [K] as a nice starting point.) For algebras of type I or II, the index can be calculated in terms of bimodule morphisms. (There are also broader definitions of index which require a conditional expectation (=normdecreasing projection) from M ontoN.) This amounts largely to rephrasing and extension of the paper [Jol] , and we do not give details here. Very briefly, let 71' : V(Moo(M)) . V(Moo(M)) be the bimodule morphism for
t.
(4.5)
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When M is a factor, 11" is a monoidal morphism on the extended nonnegative integers (type I) or extended nonnegative reals (type II). It must be multiplication by a scalar, and this scalar is the index. If M is not a factor, the index is the spectral radius of 11", provided that V(Moo(M)) is endowed with some extra structure (at present it is not even a vector space). Example. Consider the correspondence
The image of M6L2(M6) under the RTP functor for (4.5) is
~ M6 L2 (M6)M3 ®M3 M3 L2 (M6) (now counting the dimensions of the Hilbert spaces) ~
M6HS(M12X3)M3 ®M3 M3 HS (M3xd ~ M6HS(M12X12) ~ M6HS(M6X24).
We have gone from 6 copies of (:6 to 24 copies; that is, 61+24. Apparently the index is 4, which is also the ratio of the dimensions of the algebras. 5. Analytic approaches to RTPs
As mentioned in Section 3, we cannot expect the expression ~ ®M T/ to make sense for every pair of vectors ~,T/. In essence, the problem is that the product of two L2 vectors is Ll, and an Ll space does not lie inside its corresponding L2 space unless the underlying measure is atomic. Densities add, even in the noncommutative setting, and so the product in (3.3) "should" be an Ll matrix. To make this work at the vector level, we need to decrease the density by 1/2 without affecting the "outside" action of the commutants ... and the solution by Connes and Sauvageot ([Sa], [C2]) is almost obvious: choose a faithful cp E Mt and put cpl/2 in the middle of the product. That is, (5.1) This requires some explanation. Fix faithful cp E Mt and row and column representations of 5) and .It as in (2.1). We define
D(S;, 1") ~ {
(::~:;:)
E S; ,
~>~Xn ex;"" in M}
V(5), cp) is dense in 5), and its elements are called cpleft bounded vectors [C1]. Now by (5.1) we mean the following: for ~ E V(5), cp), we simply erase the symbol cpl/2 from the right of each entry, then carry out the multiplication. The natural domain is V(5), cp) x .It. Visually, .
284
(5.2)
DAVID SHERMAN
( ( ::::;:) ,(" "' '" ))
~ (::::;:) (,"') (" "' '" ) ~ (X;"»,
For cp =I 1jJ E Mt, we cannot expect € ®cp TJ = € ®,p TJ even if both are defined, although the reader familiar with modular theory will see that (5.3)
€ ®cp TJ = (€cpl/2)TJ = (€cpl/21jJl/21jJl/2)TJ = €(Dcp : D1jJ)i/2 ®,p TJ.
(An interpretation of the symbols cpl/2, cpl/2 as unbounded operators will be discussed in the next section.) Now we define S) ®cp .!'t to be the closed linear span of the vectors € ®cp TJ inside L2(Moo(M)). Up to isomorphism, this is independent of cp. (We know this because of functoriality; the "change of weight" isomorphism is densely defined by (5.3).) The given definition for V(S), cp) c S) makes it seem dependent on the choice of column representation. That this is not so can be seen by noting (as in (3.4)) that the intertwining isomorphism is given by L( v) for some partial isometry v E Moo(M). But let us also mention a method of defining the same RTP construction without representing S) and.!'t. First notice that V(S), cp) can also be defined as the set of vectors € for which 7rf(€) : L 2(M)M + S)M. cpl/2x 1+ €x, is bounded. (A more suggestive (and rigorous) notation would be L(€cpl/2).) Now we consider an inner product on the algebraic tensor product V(S), cp) ®.!'t, defined on simple tensors by (5.4) The important point here is that 7rt(€2)*7rf(6) E .c(L2(M)M) = M. The closure of V(S), cp) ®.!'t in this inner product, modulo the null space, is once again S) ®cp .!'t. (If we do choose a row representation as in (5.2), we have
The paper [F] contains more exposition of this approach, including some alternate constructions. 6. Realization of the relative tensor product as composition of unbounded operators
In this section we briefly indicate how (5.1) can be rigorously justified and extended. Readers are referred to the sources for all details. In his pioneering theory of noncommutative LP spaces, Haagerup [H] estab. lished a linear isomorphism between Mt and a class of positive unbounded operators affiliated with the core of M. (The core, welldefined up to isomorphism, is the crossed product of M with one of its modular automorphism groups.) We will denote the operator corresponding to the positive functional cp by cp also. These operators are rmeasurable (see the next section), where r is the canonical trace
RELATIVE TENSOR PRODUCTS
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on the core, and so they generate a certain graded *algebra: positive elements of LP(M) are defined to be operators of the form '{)1/ p . The basic development of this theory can be found in [Tej; our choice of notation is influenced by [Yj, where it is called a modular algebra. The composition of two L2 operators is an Ll operator, and it turns out that (5.1) can be rigorously justified [Sj as an operator equation. (This is not automatic, as operators like ,{)1/2 are not 7measurable and require more delicate arguments.) In fact, there is nothing sacred about halfdensities. With the recent development of noncommutative LP modules [JS], one can allow relative tensor products to be bifunctors on Right U(M) x Left Lq(M), with range in a certain L'" space. The mapping is denselydefined by
~ ®cp 1/ ~ (~,{)~*~)1/. In the case of an RTP of L oo modules (or more generally, Hilbert C*modules), the middle term is trivial and there is no change of density. This explains why there are no domain issues in defining a vectorvalued RTP of Hilbert C*modules [Rj. Let us mention that the recent theory of operator bimodules, in which vectors can be realized as bounded operators, allows a variety of relative tensor products over C*algebras [APj. This can be naturally viewed as a generalization of the theory of Banach space tensor products, which corresponds to a C*algebra of scalars. 7. Preclosedness of the map
(~, 1/)
1+
~
®cp 1/
Our purpose in this final section is to study when the relative tensor map is preclosed. This is a weaker condition than that of Falcone, who studied (effectively) when the map was bounded. We begin with a base case: a fa.ctor, two standard modules, and a simple product. With the usual notation Bcp for V(L 2 (M), '{)), the relevant map is
Bcp x L2(M) :3 (~, 1/) 1+ ~ ®cp 1/ E L 2(M). This is bilinear: we take "preclosed" to mean that if ~n + ~ E Bcp, 1/n + 1/, ~n ®cp 1/n + (, then necessarily ( = ~ ®cp 1/. We will also consider several variations: changing the domain to an algebraic tensor product, allowing nonfactors, and allowing arbitrary modules. Readers unfamiliar with von Neumann algebras will find this section more technical, and any background we can offer here is sure to be insufficient. Still, we introduce the necessary concepts in hopes that the nonexpert will at least find the statements of the theorems accessible. A weight is an "unbounded positive linear functional": a linear map from M+ to [0, +ooj. We will always assume that weights are normal, so Xc< / x strongly => '{)(xc<) / '{)(x); and semifinite, so {x E M+ I '{)(x) < oo} is aweakly dense in M+. We can still define RTPs for faithful weights, but now Bcp = {x'{)I/2 I '{)(x*x) < oo} C L 2(M). For details of the representations associated to weights, see [T2j. A weight 7 which satisfies 7(XY) = 7(YX) on its domain of definition will be called a trace (more properly called a "tracial weight"). An algebra which admits a faithful trace 7 is semifinite; if in addition we can have 7(1) < 00, it is finite. This facilitates the following classification of factors: a factor with n orthogonal
DAVID SHERMAN
286
minimal projections is type In (possibly n = 00); a semifinite factor without minimal projections is type III if finite and type 1100 if not; a factor which is not semifinite is type III. The reader will note that this refines our previous definitions of type, as a trace is exactly the object which orders the equivalence classes of projections. Obviously, there is much more to be said, and most of it can be found in [Tl]. For a faithful trace T on semifinite M, it is useful to consider the Tmeasure topology [N]. This is a uniform topology with neighborhoods of 0 given by
N(6, f) = {x E M 13p E P(M) with T(P.L) < 6, Ilxpll < fl. The closure of M in this topology can be identified as a space of closed, denselydefined operators affiliated with M. It is denoted VJ1(M) and actually forms a *algebra to which T extends naturally. (The Tmeasurability of an operator T is equivalent to the assertion that T(e(A).L) < 00 for some spectral projection e(A) of ITI, so we get that VJ1(M) = M if M is atomic.) It follows from modular theory that every weight on (M, T) is of the form Th ="T(h·)" for some closed, denselydefined, and positive operator h. In case h is not Tmeasurable, this is to be interpreted as 1 / 2) where h = h(1 lim T(h e1/ 2 . he ' e
e+ O
Finally, the presence of a faithful trace
LP(M, T) = {T
E
T
+ ch)l .
allows us to introduce the spaces
VJ1(M) I T(ITIP) =
IITIIP < oo},
which are antecedent to Haagerup's. Exposition can be found in [N]. Here we will only need L2(M, T), which is a standard form and in particular isomorphic as a left module to any faithful GNS representation SJ",. It is easy to check that the norm topology in L2(M, T) is stronger than the Tmeasure topology. THEOREM 7.1. Let M be a factor. The map
(7.1) is preclosed iff M
~'" x L2(M)  L2(M) :
= (M, T)
(~, TJ)
1+
~ 0'" TJ
is semifinite and h 1 is Tmeasurable, where
= Th.
PROOF. The proof is by consideration of cases.
M is type III: Choose a projection eo so
But
1I(I/n)Vn
RELATIVE TENSOR PRODUCTS
287
M = (M,r) is semifinite and h 1 is not rmeasurable: First note that the measurability of h 1 does not depend on the choice of r. Writing h = J >.de(>.) , the hypothesis is that r(e(>.)) = 00, V>.. Choose a projection eo with cp(eo) < 00 and r(eo) < 00. Then e(1/n 3) has a subprojection en which is equivalent to eo. The above construction again shows that the map is not preclosed, except that
M = (M, r) is semifinite and h 1 is rmeasurable: We assume (7.2) and want to show ( n = {x E M
= xTJ. Set
I xh 1/ 2 E L2(M,r)};
nIP
= {x
E
M
I cp(x·x) < oo},
both of which are strongly dense in M. (The bar stands for graph closure.) Then 7r : ncpl/2 ~ L2(M, r);
xcpl/2
1+
Xhl/2
densely defines a left module Hilbert space isomorphism from fJ IP to L2(M, r); denote its extension by 7r as well. Recalling that h 1 / 2 is rmeasurable by assumption,
p: nIP ~ !m(M); x 1+ 7r(Xcpl/2)h 1/ 2 is welldefined and the identity map on n. It is also strongmeasure continuous: Xa
~ X '* Xa cpl/2 ~ Xcpl/2
'* 7r(xacpl/2) £ 7r(xcpl/2)
'* 7r(Xacpl/2) ~ 7r(Xcpl/2) '* 7r(xacpl/2)hl/2 ~ 7r(Xcpl/2)h1/2, where we used that multiplication is jointly continuous in the measure topology. We may conclude that p is the identity on all of nIP' Implementing the isomorphism 7r, (7.2) becomes
(7.3)
7r(Xncpl/2) ~ 7r(Xcpl/2),
7r(TJn) ~ 7r(TJ),
Xn7r(TJn) ~ 7r(().
The convergences in (7.3) are also in measure; by the foregoing discussion we have
Xn7r(TJn) = 7r(x ncpl/2)h 1/ 27r(TJn) ~ 7r(xcpl/2)h 1/ 27r(TJ) = X7r(TJ) in measure as well. The measure topology is also Hausdorff, so 7r(() = X7r(TJ) and therefore ( = XTJ. D The map p suggests a schematic recovery of the "operators" in fJ IP : (7.4) Such operators are denselydefined but in general not closable (or may have multiple closed extensions [Sk]). Not surprisingly, then, the righthand side of (7.4) may be only formal. The condition on h in Theorem 7.1 makes the equality (7.4) rigorous, as the products on the righthand side are welldefined rmeasurable operators. Note that hand h 1 are automatically rmeasurable when M is finite, and in
DAVID SHERMAN
288
this case all multiplications and isomorphisms between GNS representations stay within W1(M), and all operators are closed  a version, somewhat oblique, of the Ttheorem of Murray and von Neumann. THEOREM 7.2. Let M be a factor, and consider 23"" 0alg L2(M) as a subspace of the Hilbert space tensor product L2(M) 0 L 2(M). The linear map (7.5)
23"" 0
al g
L2(M)
+
L 2(M):
L
~n 0 TIn
is preclosed iff M = (M, T) is atomic and T(h 1 ) < it is actually a bounded map, with norm T(h 1 )1/2.
1+
00,
~n 0"" TIn
L
where c.p
= Th.
In this case
PROOF. When M is type III, the map is not preclosed by the previous theorem. We will therefore fix a trace T, set c.p = Th, use the decomposition h = f Ade(A), and view all vectors as elements of L2(M, T). (When M is type I, we assume that T is normalized so that T(p) = 1 for any minimal projection p.) The rest of the proof is again by cases. M is type II: Choose p < e(A) for some A with T(p) = C < 00. For each k, break up p into equivalent orthogonal projections as L~=l p~. Consider the tensors
Tk
= LP~hl/2 0
p~
1+
LP~
= p.
Since the p~ are orthogonal,
JJTkJJ2 =
LT(p~h)T(p~) ~ L (~c) (~) = A~2
+
and the map is not preclosed. M is type leo and T(e(A)) = 00 for some A: Fix an orthogonal sequence of minimal projections {Pn}, Pn < e(A). The equivalence gives partial isometries with v~vn = PI, VnV~ = Pn· Then
2
1~
1~
JJTkJJ = k 2 L...J T(Pnh)T(Pl) ~ k2 L...JA =
A
k
+
and the map is not preclosed. In the only remaining situation, M is type I and h is diagonalizable. Let {An} be the eigenvalues (with repetition), arranged in nondecreasing order. We will write all matrices with respect to the basis of eigenvectors. If Sk
(7.6)
= L~=l
)..In /'00:
Consider
RELATIVE TENSOR PRODUCTS
289
and the map is not preclosed.
If Sk = L~=l >L / C < 00; that is, T(h 1 ) < 00: We show that the map is bounded on finite tensors of the form T = L i j eij ® yij. We have
T
1+
S =
~
~eij
h 1/ 2 Y ij =
'J
~
\1/2 Y ij
~eijl'\j
=~ ~
(~\1/2 ~I'\j
,k
'J
ij ) Yjkeik
.
J
By CauchySchwarz,
:s cl: IY~{12 :s Cl: ly;tl 2= CIITI12. ijk
ijkl
Since such tensors are dense in the Hilbert space tensor product, we may conclude that the norm of the map is C 1 / 2 • But the tensors Tk from (7.6) show that the norm is at least Cl/2. 0
:s
We now extend Theorem 7.1 to the nonfactor case. A general von Neumann algebra is a direct integral of factors (see [Tl] for details), and weights on the algebra disintegrate as well. PROPOSITION
7.3. Let M be a von Neumann algebra with central decomposition
frtf! M(w)dIL(W). The map (7.7)
23
+
L 2(M):
(~, 11)
1+
~ ®
is preclosed iff M = (M, T) is semijinite and (*) h(W)1 is T(w)measurable for JLa.e. w, where 'P = Th· PROOF. If M contains a summand of type III, the construction from Theorem 7.1, with the added restriction that fn and gn are chosen with equal central support, demonstrates that the map is not preclosed. If there is a trace T for which 'P = Th and h 1 is Tmeasurable, then the argument in Theorem 7.1 still shows that the map is preclosed. We will see that this possibility is equivalent to (*). First note that (*) is independent of the trace chosen, as the choice of a different trace changes a.e. h(w) by a constant factor. If (*) does not hold, fix any trace T, write 'P = Th, and let {e(A)} be the spectral projections of h. By hypothesis, we can find a nonzero central projection z with ze(A) a properly infinite projection for all A. The second construction of Theorem 7.1 shows that the map is not preclosed, where we choose all en, including eo, with central support z. Now suppose that (*) holds. We may choose a trace T which factors as 70 q" where q, is an extended centervalued trace and 7 is a trace on the center with 7(1) < 00. Let hand {e(A)} be as before. Now by assumption, the function
z(w) = max{l/n I T(w)(e(l/n)(w)) < I} is a.e. defined, nonzero, and finite. It is measurable by construction, so z and z1 represent elements of the extended center. Now write 'P = (Tz ) z  1 h. Let f be the
290
DAVID SHERMAN
spectral projection of zlh for [0,1]. We have f(w) = e(z(w))(w), so T(W)(f(W)) < 1. Then
o PROPOSITION
7.4. Let M be a factor with left module Mit and right module
SjM' The map
(7.8) is preclosed only under the same conditions as in Theorem 7.1; i.e. M = (M, T) is semijinite and h 1 is Tmeasurable, where
~ (:~~~~:), (11~ 11~ ~ ( :~::~:) · . ... )
·
·
k+oo.
(111712 ... ),
k+oo
(X:"7j)
k~ (ij), +00
.
we also have L2 convergence in each coordinate. By Theorem 7.1, (ij = Xi"7j' When h 1 is not Tmeasurable, M must be 100 or 1100 , In this case M Moo(M), and we do not need row and column matrices: Sj ~ ql L2(M) and it ~ L2(M) q2 for appropriate projections ql, q2' Fix equivalent finite projections ft ~ ql, h ~ q2 with v*v = ft, vv* = h. By assumption e(1/n 3 ) is infinite for all n; let gn be a subprojection equivalent to the fi with vin Vin = fi, vinvin = 9n. Then
11(1/n)v2nI1 2 = (1/n 2 )T(h) > 0, and the map is not preclosed.
o References [AI] [A2] [AP] [C1] [C2] [F] [H] [Jol]
C. AnantharamanDelaroche, Atomic correspondences, Indiana Univ. Math. J. 42 (1993), no. 2, 505531. C. AnantharamanDelaroche, Amenability of bimodules and opemtor algebms, in Opemtor algebms and quantum field theory, Internat. Press, Cambridge, MA, 1997, 225235. C. AnantharamanDelaroche and C. Pop, Relative tensor products and infinite C*algebras, J. Operator Theory 47 (2002), 389412. A. Connes, On the spatial theory of von Neumann algebms, J. Funct. Anal. 35 (1980), 153164. A. Connes, Noncommutative geometry, Harcourt Brace & Co., San Diego, 1994. T. Falcone, L2von Neumann modules, their relative tensor products and the spatial derivative, Illinois J. Math. 44 (2000), no. 2,407437. U. Haagerup, LPspaces associated with an arbitmry von Neumann algebm, Algebres d'operateurs et leurs applications en physique mathematique, CNRS 15 (1979), 175184. P. Jolissaint, Index for pairs of finite von Neumann algebms, Pac. J. Math. 146 (1990), 4370.
RELATIVE TENSOR PRODUCTS
[J] [JoS] [JS] [KR] [K] [N] [Pal [P] [R] [Sa] [S] [Sk] [Tl] [T2] [Te] [W] [y]
291
V. Jones, Index for subfactors, Invent. Math. 72 (1983), 125. V. Jones and V. Sunder, Introduction to subfactors, London Mathematical Society Lecture Note Series 234, Cambridge University Press, Cambridge, 1997. M. Junge and D. Sherman, Noncommutative LP modules, J. Operator Theory, to appear. R. Kadison and J. Ringrose, FUndamentals of the theory of operator algebras I,ll, Graduate Studies in Mathematics 15, 16, AMS, Providence, 1997. H. Kosaki, Index theory for operator algebras, Sugaku Expositions 4 (1991), no. 2, 177197. E. Nelson, Notes on noncommutative integration, J. Funct. Anal. 15 (1974), 103116. W. Paschke, Inner product modules over B*algebras, Trans. AMS 182 (1973),443468. S. Popa, Correspondences, notes, 1986. M. Rieffel, Morita equivalence for C*algebras and W*algebras, J. Pure and Appl. Algebra 5 (1974), 5196. J.L. Sauvageot, Sur Ie produit tensoriel relatif d'espaces de Hilbert, J. Operator Theory 9 (1983), 237252. D. Sherman, Applications of modular algebras, in preparation. C. Skau, Positive selfadjoint extensions of operators affiliated with a von Neumann algebra, Math. Scand. 44 (1979), 171195. M. Takesaki, Theory of Operator Algebras I, SpringerVerlag, New York, 1979. M. Takesaki, Theory of Operator Algebras II, SpringerVerlag, to appear. M. Terp, LPspaces associated with von Neumann algebras, notes, Copenhagen University, 1981. N. E. WeggeOlsen, K theory and C*algebras, Oxford University Press, Oxford, 1993. S. Yamagami, Algebraic aspects in modular theory, Publ. RIMS 28 (1992), 10751106. DEPARTMENT OF MATHEMATICS, UNIVERSITY Email address: dashermalDmath. ui uc . edu
OF
ILLINOIS, URBANA, IL 618012975
Contemporary ~1athematics
Volume 328. 2003
Uniform Algebras Generated by Unimodular Functions Stuart J. Sidney ABSTRACT. Our main result is the following reduction theorem: If A is a uniform algebra on X that is generated by unimodular functions, in order to verify the strong corona property for A on its spectrum E(A), it suffices to verify it when the corona data are unimodular functions from A. This is a step in the direction of finding a simpler proof of Carleson's Corona Theorem [C], and of extending it to higher dimensions. The main tool is a proof that the algebra of bounded sequences from A, regarded as a Banach algebra of continuous functions on the StoneCech compactification of ]\I! x X. is itself a uniform algebra generated by unimodular functions.
1. Introduction
One of the jewels of twentiethcentury analysis is Lennart Carleson's Corona Theorem, which asserts that the open unit disc lDl in the complex plane is dense in the maximal ideal space or spectrum of H OO = HOO(lDl), the Banach algebra of bounded analytic functions on lDl in the supremum norm; here each point ( of lDl is identified with the complex homomorphism "evaluation at (." The publication of this result in 1962 [C] generated a search for a more transparent proof, and for comparable theorems with other domains (both in complex dimension 1 and in higher dimensions) in place of the disc. Progress has been made in both directions, but all proofs of Carleson's theorem are still fairly involved, and the other domains to which it has been extended are Idimensional. One approach to the problem was introduced in the 1960s by a group at the Institut Fourier in Grenoble, France under the leadership of Alain Bernard. It involves a sequence algebra naturally associated to a uniform algebra. In this paper we shall revisit the Grenoble approach and add to it some new results that we hope will lead toward its eventual sucess. Our main result is the following: THEOREM 1 (Reduction Theorem). Let A be a uniform algebra on a compact Hausdorff space X. Assume that A is generated as a Banach algebra by the unimodular (on X) functions in A. Then in order to verify the strong corona property for A on its spectrum ~(A), it suffices to verify it for unimodular corona data. 2000 Mathematics Subject Classification. Primary 46JlO; Secondary 46E15, 46E25, 46J15. Key words and phrases. Uniform algebra, corona property, corona data, unimodular.
© 293
2003 American Mathematical Society
294
s. J. SIDNEY
The terms strong corona property and corona data will be defined below. Note that functions in A that we are calling unimodular take values of modulus 1 on X, but (in general) only of modulus:::; 1 on E(A). Such functions are often called inner. Observe that for the disc algebra (see below), the reduction theorem says that the strong corona property need only be verified for corona data consisting of
finite Blaschke products. In section 2 we recall the relationship between corona problems of density in spectra, and corona problems of solving systems of equations in a uniform algebra. Section 3 introduces sequence algebras and presents a proof that the property of being generated by unimodular functions passes from a uniform algebra to its associated sequence algebra. In section 4 we prove the reduction theorem.
2. Background on corona problems The key abstract result that is just about always used in tackling corona problems is the following easy consequence of elementary Gelfand theory. PROPOSITION 2. Let A be a uniform algebra and let E be a subset of its spectrum E(A). Then E is dense in E(A) if and only if whenever h, ... , fn are finitely many functions in A and there is a positive constant 8 such that max{lh(x)I, ... , Ifn(x)l} ~ 8 for every x E E, it follows that there are functions gl, ... ,gn in A such that hgl + ... + fngn = l.
In this proposition, the fJ are known as corona data (for A on E), and the assertion equivalent to density of E is the corona property (again, of A on E). In particular, if A is a uniform algebra on X, then X = E(A) if and only if A enjoys the following property: whenever h, ... , fn are finitely many functions in A that do not all vanish simultaneously at any point of X, there exist functions gl, ... ,gn in A for which hgl + ... + fngn = 1. Carleson actually proved a stronger version of the corona property for HOO on Jl)), one in which there are bounds on the gj: DEFINITION 1. A uniform algebra A has the strong corona property on a subset E of E(A) if for all positive integers n and positive numbers 8 < 1 there are finite constants C(n, 8) such that whenever h, ... ,fn are functions in A satisfying IlfJ II :::; 1 and maxj IfJ(x)1 ~ 8 for all x E E, there exist functions gl, . .. , gn in A that satisfy E j fJgj = 1 and Ilgjll :::; C(n,8).
Clearly if A has the strong corona property on E then A has the corona property on E, so E is dense in E(A). In this definition, the fJ are strong corona data for A on E. Consider now the disc algebra A(Jl))) consisting of all continuous complexvalued functions on the closed unit disc ii} that are analytic on Jl)). It is standard, and not hard to prove, that E(A(Jl)))) = ii}, and so A(Jl))) has the corona property on ii} (equivalently, on Jl))). Suppose one can show that the disc algebra actually has the strong corona property on Jl)) with constants C(n,8). If h, ... , fn are strong corona data for H OO on Jl)) for some 8, we can for each natural number k produce strong corona data hk, ... fnk for A(Jl))) and this same 8 such that for each j, fJk + fJ pointwise on Jl)) as k + 00 (for instance, fJk(() = fJ((l  k 1 )()). By assumption there are functions glk, ... , gnk in A(Jl))) such that E j fJkgjk = 1 and Ilgjkll :::; C(n,8). By a normal families argument, we may assume that for each j
UNIMODULAR FUNCTIONS
295
there is gj in Hoc such that gjk  gj pointwise as k  00, and clearly I:j hgj = 1 and Ilgjll :::; C(n,8). We see that HOC enjoys the strong corona property with the same constants as the disc algebra. (Conversely, the reader may show that if Hoc has the strong corona property on J]), then A(J]}) also has it on J]}, with any choice of constants greater than those for HOC.) The above argument works just as well on many complex domains S1 other than the unit disc J]}, in particular on unit balls and polydiscs in Cd. For these domains it is elementary that the spectrum of A(S1), the algebra of continuous functions on Ii that are analytic on S1, is just fl, and the same argument as for S1 = J]} shows that if A(S1) enjoys the strong corona property on S1 then so does HOC(S1), the algebra of bounded analytic functions on S1. Clearly the reduction theorem applies to the polydisc algebra A = A(S1) when S1 is the unit polydisc (and X is taken to be the torus consisting of points in Cd all of whose coordinates have modulus 1). Thus the reduction theorem aims directly at the goal of proving the corona theorem for polydisc algebras, namely, that the polydisc S1 is dense in the spectrum of HOO(S1). One last word before moving on to Bernard's technique. A natural question proposed by Walter Rudin ([Bil, page 347) asks whether, inasmuch as every uniform algebra has the corona property on its spectrum, perhaps it also has the strong corona property on its spectrum. That is, in fact, the conclusion that the sequence spaces we are about to study was designed to prove. Unfortunately, an ingenious example produced by another member of the Grenoble team, JeanPierre Rosay [RJ, shows that not every uniform algebra has the strong corona property. Furthermore, Brian Cole ([Gal, chapter 4) has exhibited an open Riemann surface R that is not dense in the spectrum of HOO(R). 3. Bernard's sequence algebras and unimodular functions To any uniform algebra A we associate the unital Banach algebra A consisting = (h) with h E A and IIJII == sUPk Ilhll < 00. If A is of those sequences a uniform algebra on X (where X is any compact subset of E(A) that contains the Silov boundary of A), then every J E A may be naturally identified with the bounded continuous function on N x X that takes the value h(x) at the point (k, x) of N x X, and so with a continuous function (also denoted J) on X = ,B(N x X), the Stonetech compactification of N x X. In this way, A becomes a uniformly closed algebra of continuous functions on X that contains the constant functions; in general, A need_not separate the points of so is a uniform algebra on some quotient space of X, but not necessarily on X itself. These sequence algebras and corona problems are related by the following result, the original raison d 'etre for the study of A:
J
x,
PROPOSITION 3. If A is a uniform algebra, A has the strong corona property on E(A) if and only ifN x E(A) is dense in E(A). PROOF. One direction is trivial: If A has the strong corona property on E(A), then A has the strong corona property on N x E(A) with the same constants. To go in the opposite direction, suppose that A does not have the strong corona property on E(A), so for some nand 8 no appropriate constant C(n,8) exists. Each natural number k cannot serve as C(n,8), so there are flk,"" fnk in A for which Ilhkll :::; 1 and maxj Ihkl ~ 8 throughout E(A), but if gl, ... ,gn are
296
S . .1. SIDNEY
functions in A for which Lj /jkgj = 1, then maxj !!gj!! > k. Let ij = (fjk)k, so i j E A, lIijll ::; 1, and maxj !ij! ~ 8 throughout N x E(A). There can be no ih (glk)k, ... , 9n (gnk)k in A for which Lj i j 9j 1, for this equality would mean that for each k we would have Lj fjkgjk = 1, and for k > maxj IIgjll this would yield maxj IIgjk II > k > maxj 119j 1/, which is impossible. Thus N x E( A) cannot be dense in E(A). D
=
=
=
Proposition 3 offers the potential to prove strong corona theorems by proving density of N x E(A) in E(A). It was hoped that a "soft" Banach algebra argument might accomplish this. However, such an argument never materialized, as (in view of Rosay's example) it could not in complete generality. Instead, A found a central role in the theory of functions that operate on function spaces. The seminal document here is Bernard's paper [Be], and a recent introductory account of both the relation to corona problems and the applications to functions that operate may be found in [HS]. That the Grenoble program cannot work in general does not imply that it cannot work in special situations. Our goal in this paper is to begin movement toward a positive outcome in one important special situation, namely, that in which the uniform algebra is generated by unimodular functions. Let us now establish some notation. If A is a uniform a~ebra on X, we let U denote the set of unimodular functions in A and we let U denote the set of all functions i = (ik) such that ik E U for every k; viewed as functions on X, ii consists of precisely the unimodular (on X) functions in A. We shall need the following "trick" developed by Bernard in a context related to ours but not involving sequence algebras. LEMMA 4 (Bernard trick). [BGM] Suppose v is in the subalgebra (equivalently, linear subspace) of A generated algebraically by U, and that IIvll < 1. There are functions u E U s'u.ch that u'iJ E A. Take such a u, and for real numbers 0 let v(8) Then 0
I>
= 'U'

i8
uv  e . . 1  u'iJe t8
v(8) is a continuous mapping from the real line into U, and v
=.2.27r
r
21r
io
v(8) dO.
This lemma, an immediate consequence of the Cauchy integral formula, was used in [BGM] to obtain easily the fact that if U generates A as a Banach algebra then the closed unit ball of A is the closed convex hull of U. We shall use it in pretty much the same way to prove the second part of the following result: THEOREM 5. Let A be a uniform algebra on X, and suppose that U generates A as a Banach algebra. Then A separates the points of X (and so is a uniform algebra on X), and is generated as a Banach algebra by indeed, the closed unit ball of A is the closed convex hull of
ii.
ii;
PROOF. The hypotheses imply that U separates the points of X, and then an easy argument (see for instance Lemma 4.12 in [GI]) shows that the absolute values of the functions in the algebra generated algebraically by U, and so the absolute
UNIMODULAR FUNCTIONS
297
values of the functions in A, are uniformly dense in the set of nonnegative realvalued continuous functions on X. An argument used by Bernard in [Be] for the Dirichlet algebra case trivially works here as well to give the fact that A separates the points of X. To tackle the second part of the statement, first note that if 0 < r < 1, then continuity of 0 1+ v(9) and convergence of the integral in the lemma are uniform over all v as in the lemma that satisfy Ilvll < r. Now suppose W = (Wk) is in the open unit ball of A, and take r < 1 such that Ilwll < r. By the hypotheses we may approximate w as closely as we wish on N x X, and so on X, by a function ii = (Vk) E A such that for each k, Vk is in the subalgebra of A generated algebraically by U, and Ilvk II < r. For each k choose Uk and define vi9) as in the lemma, so 0 1+ vi9) is a continuous mapping of the real line into U, and Vk
=
1 211'
10r
21r
(9) Vk
dO,
the continuity and convergence being uniform in k. For each 0 we have ii(9) (vi9») E U, the mapping 0 1+ ii(9) is continuous, and most important,
ii =
r 211' 10
~
21r
ii(9) dO.
Thus ii lies in the closed convex hull of U.
o
4. Proof of the reduction theorem We first require a simple lemma, which will be applied to to A and U.
A and U rather than
LEMMA 6. Let A be a uniform algebra on X, and suppose A is generated as a Banach algebra by its subset U of unimodular (on X) functions. If c.p and 1jJ are distinct elements ofE(A) and c.p ¢. X, then there exists u E U such that u(c.p) = 0 =Iu(1jJ). PROOF. c.p has a representing measure J.L on X which cannot be a point mass, so some v E U must be nonconstant on the support of J.L, hence Iv(c.p)1 < 1. Composing v with a Mobius transformation, we may suppose v(p) =I O. If v(1jJ) =I v(c.p), let W = v; if v(1jJ) = v(c.p), multiply v by an element of U that separates 1jJ and c.p to get w. In either case, W E u, Iw(c.p) I < 1, and w(1jJ) =I w(c.p). Composing w with an appropriate Mobius transformation produces the required u. 0 We now prove the Reduction Theorem. PROOF OF REDUCTION THEOREM. We are given that A is a uniform algebra on X generated by its set U of unimodular elements, and that there are always constants C(n, 0) such that whenever Ul, ... ,Un are elements of U such that maxj IUjl ?: 0 throughout E(A), it follows that there are gl,'" ,gn in A satisfying ~j Ujgj = 1 and Ilgj II ::; C(n,o). According to Proposition 3, we must deduce that N x E(A) is dense in E(A). Suppose c.p E E(A). If c.p E X then there is nothing to prove, so assume c.p ¢. X. By Theorem 5 and Lemma 6, for each 1jJ E E(A) other than c.p there is an element of U that is zero at c.p and nonzero at 1jJ. Thus if W is any neighborhood of c.p in
298
S. J. SIDNEY
I:(A), standard topological arguments provide finitely many functions U and a number 0 < 8 < 1 such that Uj (
{1P
E I:(A) :
ut. ... ,Un in
IUj (1/1) I < 8 \lj} C W. If W n (N x I:(A)) = 0
Let Uj = (Ujk)k where Ujk E U. then for each k, maxj IUjk(1/1) I ~ 8 for every 1/1 E I:(A), so there are 91k, ... ,9nk in A that satisfy Lj Ujk9jk = 1 and 1!9jkll ::; C(n,8). Letting [}J = (9jk)k E A, we have Lj Uj9j = 1 on N x I:(A), so on I:(A); but this is impossible at the point <po Thus W n (N x I:(A)) i= 0 after all, completing the proof of the theorem. 0
References [Be] A. Bernard, Espaces des parties rt~elles des elements d'une algebre de Banach de fonctions, J. Functional Anal. 10 (1972), 387409. [BGM] A. Bernard, J. B. Garnett and D. E. Marshall, Algebras generated by inner functions, J. Functional Anal. 25 (1977), 275285. [Bi] F. Birtel, editor, Function algebras, ScottForesman, Fair Lawn, N. J., 1966. [e] L. Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. 76 (1962), 547559. [Ga] T. W. Gamelin, Uniform algebras and Jensen measures, London Math. Soc. Lecture Note Series 32, Cambridge Univ. Press, Cambridge, England, 1978. [GI] I. Glicksberg, Measures orthogonal to algebras and sets of antisymmetry, Trans. Amer. Math. Soc. 105 (1962), 415435. [HS] S. Hwang and S. J. Sidney, Sequence spaces of continuous functions, Rocky Mountain J. Math. 31 (2001), 641659. [R] J.P. Rosay, Sur un probleme pose par W. Rudin, C. R. Acad. Sci. Paris Ser. AB 267 (1968), A922A925. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CONNECTICUT, STORRS, CONNECTICUT 062693009
Contemporary Mathematics Volume 328, 2003
Analytic Functions on Compact Groups and their applications to almost periodic functions Thomas Tonev and S. A. Grigoryan ABSTRACT. This is a survey on some recent developments in the theory of uniform algebras of continuous functions on compact groups, that are invariant under group shifts. Contents:
I. Analytic Functions on Groups 1. Almost periodic functions 2. Shift invariant algebras on groups II. Shift Invariant Algebras on Groups 1. Rad6's and Riemann's theorems for analytic functions on groups 2. Extension of linear multiplicative functionals of shift invariant algebras on groups 3. Automorphisms of shift invariant algebras on groups 4. Primary ideals of algebras of analytic functions on solenoidal groups 5. Asymptotic almost periodic functions III. Inductive limits and Shift Invariant Algebras on Solenoidal Groups 1. Inductive limits of disc algebras on Gdiscs 2. Inductive limits of algebras on subsets of Gdiscs 3. Gleason parts of inductive limits of disc algebras on Gdiscs 4. Inductive limits of HOC spaces on Gdiscs 5. HOC spaces on solenoidal groups 6. Bourgain algebras and inductive limits of algebras
I. Analytic functions on groups 1. Almost periodic functions. Almost periodic functions were introduced by H. Bohr [4] who has established their basic properties. Other results were obtained by Besicovitch [2] and Jessen [26]. Bohr discovered almost periodicity in the course of his study of Dirichlet series of analytic functions. For a deeper insight on almost periodic functions we refer the reader to the books of Loomis [29], and 1991 Mathematics Subject Classification. Primary 46J15; Secondary 30H05, 46J10. Key words and phrases. Uniform algebra, compact group, shift invariant algebra. The authors acknowledge the support of a NSF Cooperative Research Grant in Modern Analysis @ 2003 American Mathematical Society
299
T. TONEV AND S. GRIGORYAN
300
Corduneanu [I1J. Unless otherwise said, all continuous functions in the sequel will be considered complex valued. A continuous function f on the real line lR. is said to be almost periodic if for every E > 0 there is an L > 0 such that within every interval I c lR., III 2: L there is an x E I such that max If(t)  f(t + x)1 < E (H. Bohr [4]). According to the tEIR
famous theorem of Bochner [3J, f is almost periodic on lR. if and only if the set of all its translates h(x) = f(x + t), t E lR. is relatively uniformly compact in BC(lR.), the space of bounded continuous functions on R Equivalently, f is almost periodic if it can be approximated uniformly on lR. by exponential polynomials, i.e. by functions n
L
akeiskX, where ak are complex, and Sk are real numbers. It is easy to k=l see that the set AP(lR.) of all almost periodic functions on lR. is an algebra over C. Actually, under the uniform norm AP(lR.) is a commutative Banach algebra with unit. of type
Dirichlet coefficients a{, A E lR. of an almost periodic function f(x) on lR. are
l1
the numbers a{ = lim T T+oo
Y T
+ f(x)ei>,xdx, where the limit, and its value in the
Y
right hand side exists independently on y E R Dirichlet coefficients a{ are nonzero for count ably many A'S at most, which are called Dirichlet exponents of f(x). The set sp (I) of all Dirichlet's exponents of f(x) is called the spectrum of f. Hence, sp (I) = {A E lR. : a{ =f. O} is a countable set. It is customary to express the fact that Ak are the Dirichlet exponents of f(x) and the numbers = are the Dirichlet coefficients of f(x) for any k = 1,2, ... by a power series notation, namely
A' at
00
f(x) '"
L A'ei>'k X. This series, not necessarily convergent, is called the Dirichlet
k=l series of f(x). If all Dirichlet coefficients of a f E AP(lR.) are zero, then, as it is easy to see, f == O. Consequently, the correspondence between almost periodic functions and their Dirichlet series is injective. E~ery almost periodic function f on lR. can be extended as a continuous function f on the Bohr compactification J3lR. of R The Fourier coefficients c[ of the extended in this way function 1 on J3lR. equal the Dirichlet coefficients A' of f. Moreover, the maximal ideal space MAP(IR) of the algebra of almost periodic functions on lR. is homeomorphic to the Bohr compactification J3lR. of R For every A C lR., by APA(lR.) we denote the space of all almost periodic Ajunctions, namely, almost periodic functions on lR. with spectrum contained in the set A, i.e.
APA(lR.) = {f E AP(lR.) : sp (I) C A}. Note that every f E APA(lR.) can be approximated uniformly on lR. by exponential n
Apolynomials, i.e. by exponential polynomials of type
L akei8kX, Sk k=l
E A.
ANALYTIC FUNCTIONS ON COMPACT GROUPS
301
2. Shift invariant algebras on groups. Let G be a compact abelian group, and let S be an additive subsemigroup of its dual r = 8, containing the origin. Linear combinations over C of functions of type Xa , a E S are called S polynomials on G. Denote by As the set of all continuous functions on G whose Fourier coefficients c!
=
J
f(g)xa(g) da are zero for any a outside
r\s.
Here a is the normalized Haar
G
measure on G. The functions in As are called Sfunctions on G. Any Sfunction on G can be approximated uniformly on G by Spolynomials, and vice versa. The set As is a uniform algebra on the group G. A uniform algebra A on G is Gshift invariant if, given an f E A and 9 E G, the translated function fg(h) = f(gh) belongs to A. Every algebra of Sfunctions is invariant under shifts by elements of G. Vice versa, every Gshift invariant uniform algebra on G is an algebra of Sfunctions for some uniquely defined subsemigroup S C 8 (Arens, Singer [1]). Algebras As of Sfunctions are natural generalization of polydisc algebras A(']['n), n E N. With G = ']['n, = 8 = zn, and S = Z+.' the algebra As in fact coincides with the algebra Azn = A(']['n) on the torus ']['n, and Z+functions
r
+
are traces on ,][,n of usual analytic functions in n variables in the polydisc continuous up to the boundary ']['n.
W,
The maximal ideal space Ms of As is the set H(S) = Hom (S, ~), and the Shilov boundary BAs is the group G (ArensSinger [1]). H(S) is a semigroup under the pointwise operation (cp1/1)(a) = cp(a)1/1(a), a E S. The Gelfand transforms of elements f E As are continuous functions on Ms, and the space As = {i: f E As} is a uniform algebra on Ms.
i
As shown by Arens and Singer (e.g. Gamelin [14]), As is a maximal algebra if and only if the partial order generated by the semigroup S in 8 is Archimedean. Note that in this case 8 c JR and there is a natural embedding of the real line JR into G so that the restrictions of Sfunctions on this embedding are almost periodic functions that admit analytic extension on the upper halfplane II over JR. Moreover, an algebra of type As is antisymmetric if and only if the semigroup S does not contain nontrivial subgroups, i.e. if S n (S) = {O} (Arens, Singer [1]). A compact group G is said to be solenoidal, if there is an isomorphism of the group JR of real numbers into G with a dense range. Equivalently, a compact group is solenoidal if and only if there is an isomorphism from 8 into R Note that the StoneChech compactification (3T = of T is a solenoidal group for every additive subgroup r of JR. If G is a solenoidal group, then its dual group r = 8 is isomorphic to a subgroup of R If r is not dense in JR, then it is isomorphic to Z. In this case G is isomorphic to the unit circle '][', S c Z+, and therefore the elements of the algebra As can be approximated uniformly on '][' by polynomials. Hence they can be extended on the unit disc lJ)) as analytic functions, and therefore Ms = ~, while As ~ A(lJ))). If r is dense in JR, then the maximal ideal space Ms has a more complicated nature.
r;,
In the case when S c JR+ and S u (S) = r, the Sfunctions in As, are called analytic, or generalized analytic functions in the sense of ArensSinger on G. As
302
T. TONEV AND S. GRIGORYAN
mentioned before, if S = R+ the group G coincides with the Bohr compactification ,BR of R In this case the maximal ideal space of the algebra AIR+ is the set ll}a = ([0,1] x G)/( {O} x G), which is called the Gdisc, or big disc over G. The algebra Ar+ = Ar+ (lI}a) is called also the Gdisc algebra, or the big disc algebra. The points in the Gdisc ll}a are denoted by r· g, where r E [0,1] and 9 E G =,BR We identify the points of type 0 . g, 9 E G, and the resulting point we denote by w. Hence, w = O· 9 for every 9 E G. The points of type 1· g, 9 E G, we denote by g. Since R is dense in G, the set (0,1] x R is dense in the Gdisc ll}a. Equivalently, the upper halfplane n ~ (0,1] x R can be embedded as a dense subset of the Gdisc ll}a. Below we summarize some of the basic properties of the Gdisc algebra Ar+ (lI}a), where r+ = r n [0,00) (cf. Gamelin [14]). (i) Mr+ = ll}a. (ii) 8A r +(lI}a) = G. (iii) A local maximum principle holds on Ar+(lI}a), namely, for every analytic r+function f(r . g) on ll}a, for every compact set U c ll}a, and for each ro . go E U we have If(ro . go)l:::; max If(r. g)l· r·gEbU
(iv) Every f E LP(G, da), 1 :::; p :::; 00 can be approximated in the LP(G, da)norm by sp (f)polynomials. In particular, every f E As can be approximated uniformly on G by Spolynomials. (v) Ar+(lI}a) is an analytic algebra, i.e. every analytic r+function which vanishes on a nonvoid open subset of ll}a vanishes identically on ll}a. (vi) Any realvalued analytic r+function is constant. (vii) Ar+ (lI}a) is a Dirichlet algebra; (viii) A r + (lI}a) is a maximal algebra. (ix) The upper halfplane n can be embedded as a dense subset of the Gdisc ll}a.
Examples 1. (a) Let G be a solenoidal group, and S is an additive subsemigroup ofR, containing the origin. Note that the restriction of a character Xa E Gon R is the function eiax , x E R. As an algebra generated by the characters Xa, a E S on G, the algebra As of analytic Sfunctions is isometrically isomorphic to the algebra APs(R) of almost periodic Sfunctions on R, generated by the functions eiax , x E R, a E S. (b) It is easy to see that As, S C R is isometrically isomorphic to the algebra on '][' \ {I} generated by the singular functions ea :~~, a E S via a Mobius transformation. In the case when S C R+, As is isometrically isomorphic to the subalgebra .!.ll He; of H OO generated by the functions ea . 1 , a E S on II} \ {I}. (c) The portion over Jij \ {O} of the Riemann surface is densely embeddable into the Gdisc ll}a.
Slog
of the function log z
Example 1 b) implies the following PROPOSITION 1. Let G be a solenoidal group, such that its dual group r = is a dense subgroup of R, and let S be an additive subsemigroup of r+ = r
G n
ANALYTIC FUNCTIONS ON COMPACT GROUPS
303
[0,00), containing the origin. Then the algebra As of analytic Sfunctions on G is isometrically isomorphic to the algebra of almost periodic Sfunctions on R DEFINITION 1. Let S be a semigroup of G. The weak enhancement [S]8 of S is the set of elem ts a E G for which there is a rna E N such that na E S for every n ~ rna. The stron enhancement [S]8 of S is the set of elements a E G for which there is a rna EN su that rnaa E S. S is weakly enhanced, or strongly enhanced if [S]w = S, or [S]8 = respectively. Note that S c [S]w C [S]8 C weakly and strongly enhanced.
G.
If S
c G and S U (S)
=
G,
then S is both
PROPOSITION 2 [23]. For an a E G\ S by Sa denote the semigroup Sa = S +Na. Then MSa = Ms if and only if a E [S]w. As an immediate consequence we obtain that M[sJw = Ms for every semigroup S c G. Also, if S, reG are two subsemigroups of G such that S + (S) = r + (r) = G, and if [S]w = [r]w then Ms = Mr· PROPOSITION 3 [23]. Let S e r e G be two subsemigroups of G such that S + (S) = G and Mr = Ms. IE ASa is analytic for some a E r \ S, then Ms = MSa (and therefore a E [S]w according to the previous proposition). In particular, [S]w = [r]w if Ms = Mr·
II. Shift invariant algebras on groups 1. Rad6's and Riemann's theorems for analytic functions on groups. Let U be an open set in the maximal ideal space MA of a uniform algebra A. A continuous function on U is said to be Aholomorphic on U if for every x E U there is a neighborhood V of x so that can be approximated uniformly on V by Gelfand transforms of functions in A. A uniform algebra A is said to be analytic on its maximal ideal space MA if whenever a function f E A vanishes on an open subset of MA \ 8A then f vanishes identically on MA. If a Gshift invariant algebra As is analytic, then S does not contain subgroups other than {O}, i.e. Sn( S) = {O}. Throughout this section we will consider all algebras to be analytic, and that S + (S) = G.
f
DEFINITION 2. A uniform algebra A satisfies Rad6 's property, if every function continuous on MA and Aholomorphic on MA \ Z(f) belongs to A.
The classical theorem of Rad6 asserts that the disc algebra A(j())) possesses Rad6's property. However, it fails for the algebra Ao(j())) of functions f E A(j())) with vanishing at 0 derivatives. Observe that this algebra is of type As with S = {O, 2, 3, 4 ... }, whose weak enhancenment is Z+ =/:. S. THEOREM 1 (Grigoryan, Ponkrateva, Tonev [23]). The algebra As possesses Rad6's property if and only if the semigroup S is weakly enhanced.
T. TONEV AND S. GRIGORYAN
304
DEFINITION 3. A uniform algebra A C C(MA) is integrally closed in C(MA) if every continuous function on MA satisfying a polynomial equation of type xn + alX n  1 + ... + an = 0, aj E A belongs to A. Integrally closed in C(MA) uniform algebras were studied extensively by Glicksberg [15]. Examples of integrally closed in C(MA) algebras are the disc algebra, the polydisc algebra, the algebra of analytic Sfunctions on a Gdisc over a group G with ordered dual. THEOREM 2 [23]. The algebra As is integrally closed in C(MA) if and only if the semigroup S is weakly enhanced. DEFINITION 4. A uniform algebra A possesses Riemann's property if, given a function 9 E A with Z(g) n 8A = 0, then every bounded Aholomorphic function on MA \ Z(g) belongs to A. The classical theorem of Riemann asserts that the disc algebra A(lJ))) possesses Riemann's property. Note that single points in the complex plane are zeros of certain analytic functions. DEFINITION 5. The bounded enhancement [S]b of S is the set of elements a E which there are b, c E S with a = b  c, such that Xb/X c is bounded on Ms \ Z(X C ), where Z(X) = {m E Ms : m(x) = o} is the zeroset of X. A semigroup S is said to be boundedly enhanced if [S]b = S.
G for
THEOREM 3 (Grigoryan, Ponkrateva, Tonev [23]). The algebra As possesses Riemann's property if and only if the semigroup S is boundedly enhanced. A uniform algebra A possesses the weak Riemann property if, given a function n 8A = 0, then every bounded Aholomorphic function on MA \ Z(g) can be extended continuously on MA. One can show in a similar way that a Gshift invariant algebra As possesses the weak Riemann property if and only if the weak and the strong enhancements of S coincide [23].
9 E A with Z(g)
2. Extension of linear multiplicative functionals of shift invariant algebras on groups. Let S be an additive semigroup which contains 0, and possesses the cancellation property, i.e. a = c whenever a + b = c + b for some bE S. In this case S is a subsemigroup of a group. Denote by r = S  S the group generated by S. Consider a subsemigroup P ::::> S of r such that P + (P) = r, and Pn (P) = {O}. P generates a partial (pseudo) order on r, by b» a if and only if b  a E P. Note that every nonnegative semicharacter e E Hom (P, [0, 1]) is monotone decreasing on P with respect to the order generated by P. Indeed, if b » a for some a, bE r+, then b = a + p for some pEP. Therefore, e(b) = e(a)e(p) ::; e(a) since e(c) ::; 1 on P. Consequently, if a nonnegative semicharacter e is extendable on + as an element in Hom (P, [0, 1]), then ~!ly is monotone decreasing onSCP. / .
r
ANALYTIC FUNCTIONS ON COMPACT GROUPS
305
PROPOSITION 4 (Grigoryan, Tonev [25]). A positive semicharacter e E H(S) is extendable on r+ as a positive semicharacter if and only if e is monotone decreasing on S with respect to the order generated by P.
Proof. Let the positive semicharacter eon S be monotone decreasing. If bE r+, then b = a  c for some a,c E S, a > c, and e(b) = e(a)je(c) is a well defined and natural homomorphic extension of (} on r+. Clearly, e(a) ~ e(c) if and only if e(b) ~ 1, i.e. if and only if e is a positive semicharacter on r+. THEOREM 4 [25]. Nonvanishing semicharacters rp on S can be extended as (nonvanishing) semicharacters on r+ if and only if eveIY positive semicharacter e E H+(S) is monotone decreasing on S with respect to the order generated by P.
Proof. Let rp E H (S), rp =I O. The function ')'(a) = {rp(a)jlrp(a)1 ')'( a) = Ih(a) can be extended naturally on the group ;Y(b) =
~~:j
for a E S for a E (S)
r as a character of r by
whenever b = a  c E r, a, c E S.
Thus, rp = Irpl~ = (!')' extends on r+ (as an element of H(r+)) if and only if e = Irpl does. By the above proposition this happens if and only if e is monotone decreasing on S with respect to the order on S generated by P. Let S c JR, and P = JR+. Define eg = Lio} E H+(S) to be the characteristic function of {O} in S, namely eg (0) = 1, eg (a) = 0 for every a E S \ {O}. Note that e~+ is the only vanishing semicharacter on r+. Consequently, if S is an additive subsemigroup of JR+ containing 0 and P = JR+, then a vanishing semicharacter e E H(S) is extendable on r+ if and only if e = eg. Therefore not every vanishing semicharacter e E H (S) possesses a semicharacter extension on a larger semigroup. PROPOSITION 5 [25]. Let S c P = JR+. A nonnegative semicharacter e E H(S) is uniquely extendable on r+ as a nonnegative semicharacter on r+ if and only if e is monotone decreasing on S with respect to the order generated by P.
Proof. Assume that a semicharacter e is monotone decreasing and e( a) = 0 for some a E S. Then e(na) = 0 for every n E N, and the monotonicity argument shows that e(a) = 0 for all a E S \ {O}. In this case e = eg extends naturally on r + to the semicharacter e = e~+ . Recently by S. Grigoryan, and independently  Sherstnev [31], have generalized Proposition 5 for arbitrary semigroups S with cancellation property. Namely, a nonnegative semicharacter r on S can be extended (nonuniquely) as a (nonnegative) semicharacter on a supsemigroup E :::::> S if and only if r is monotone decreasing with respect to the order on S generated by E.
306
T. TONEV AND S. GRIGORYAN
Example 2 (cf. Tonev [32]). Let v > 0 be a positive number. Consider the semigroup rv = {O} U [v,oo) c R Clearly, r = rv  rv = lR., and r+ = lR.+. Since x(a + b) = x(a)x(b) ::; x(a) for every a, b E r v , every semicharacter X on rv is monotone decreasing. Therefore, it is extendable on lR.+, namely as the characteristic function l?~+ of the origin {O}. Example 3. Let a be an irrational number. Consider the 2dimensional semigroup 80. = {n+ma : n, m E Il+} C R Here the group generated by 80. is ro. = 80.80. = {n+ma: n,m Ell}, while (ro.)+ = ro.nlR.+ = {n+ma ~ 0: n,m Ell}. Clearly, 80. =I (ro.)+' For instance the positive number a  [a] E (ro.)+ \ 80.' For a fixed a E (0,1) the function I'(n + ma) = an, n + ma E 80. is a homomorphism from 80. to (0,1] C iTh". However, I' is not monotone decreasing on 8. Indeed, I'(ma) = 0, while I'(n) = an> 0 for every n > ma. The natural (and only possible) homomorphic extension 1 of I' on (ro.)+ is given by 1(n + ma) = an, n,m E Il,n+ma ~ O. However, 1 ¢ H((ro.)+), since, for instance, 1(a [a]) = a[o.] > 1. PROPOSITION 6 (Grigoryan, Tonev [25]). The maximal ideal space Ms of the algebra As of analytic 8functions on G = r = 8  8 with spectrum in 8 C lR.+ is homeomorphic to the maximal ideal space Mr+ = iTh"c of the algebra Ar+ of analytic r+functions on G if and only if all positive semicharacters on 8 are monotone decreasing.
r,
As an immediate consequence we get the following PROPOSITION 7 [25]. The maximalideal space MAPs(JR) ofthe algebra APs(lR.) of almost periodic functions with spectrum in a semigroup 8 C lR.+ is homeomorphic to the Gdisc iTh"c, where G = if and only if all positive semicharacters on 8 are monotone decreasing.
r,
Since the upper half plane II = {z E C : 1m Z ~ O} can be embedded densely in the maximal ideal space Ms of the algebra As (and, together, of APs(lR.)) if and only if MAs = iTh"c, then the upper half plane II is densely embeddable in the maximal ideal space MAPs(JR) of the algebra APs(lR.) of almost periodic functions with spectrum in 8 if and only if all positive semicharacters on 8 are monotone her [6], II is densely embedable in MAPs decreasing. Note that, as shown by B if and only if every additive posit' e function 0 on 8 is of type O(a) = yoa for some Yo E [0,00), or O(a) = 00, for a O. . .!.±!.
For an a E 8 let 'Pa E Hoo be the singular function 'Pa(z) = eW 1. on the unit disc j[)). Recall that HS' is the Banach algebra on j[)) generated by the functions 'Pa(z), a E 8 equipped by the supnorm on j[)). As mentioned in Example 1 b), HS' is a subalgebra of Hoo, which is isometrically isomorphic to the algebra As of analytic 8functions on G = (8  8)~. PROPOSITION 8 (Grigoryan, Tonev [25]). The unit disc j[)) is dense in the maximal ideal space of the algebra HS' if and only if all positive semicharacters on 8 are monotone decreasing.
Let P be a semigroup of r that generates a partial order on r, and suppose that 8 C E are additive subsemigroups of P that contain the origin, and such that
ANALYTIC FUNCTIONS ON COMPACT GROUPS
307
[S] .. :J E, i.e. Na nSf. 0 for every a E E. Then every nonnegative semicharacter E H(S) can be extended naturally on E as a monotone decreasing semicharacter, namely by ~(a) = [(!(na)]l/n. (!
PROPOSITION 9 [25]. If SeE are subsemigroups of P such that E C [S]s, then every semicharacter c.p E H(S) on S is uniquely extendable on E as a semicharacter in H(E), and therefore, Ms = ME.
In particular, if S is a subsemigroup of IR such that [S]s :J r+, then the upper half plane II is densely embedable in the maximal ideal space M APs (lR) of the algebra APs(lR) of almost periodic functions on IR with spectrum in S. PROPOSITION 10 [25]. If S is a subsemigroup of IR such that [S]s :J r+, then the algebra H'S does not have corona, i.e. the unit disc ID> is dense in its maximal ideal space MH:;'. PROPOSITION 11 [25]. Let S be a subsemigroup of R Then ME = Mr+ = ll}c for every semigroup E with SeE c 1R+ if and only if [S]s = r+, i.e. for every a E r+ = r n [0,00) there is an n E N such that na E S.
Note that under the hypotheses of this proposition, the semicharacters on all semigroups E with SeE c 1R+ are uniquely extendable on r+ as semicharacters on r+.
3. Automorphisms of shift invariant algebras on groups. Assume that = {O}, i.e. that S contains no nontrivial subgroups. Under this condition the algebra As is antisymmetric. An element £ E Ms = H(S) is an idempotent homomorphism of S if £2 = £. Let Is be the set of all idempotents in H(S) that are not identically equal to 0 on S. It is easy to see that Is is a subsemigroup of H(S). Clearly, an idempotent homomorphism can take values 0 or 1 only. Denote by Z. the zero set {a E S : £(a) = O}, and by E.  the support set {a E S: £(a) = I} of £ E Is. It is easy to see that if £ is an idempotent homomorphism of S, then E. is a semigroup of S, Z. is a semigroup ideal in S, Z. U E. = S, and Z. n E. = 0.
Sn (S)
12 [20]. Let As be a Gshift invariant algebra on G, where £ E Is possesses a representing measure supported on a subgroup of G. PROPOSITION
SeC. Every idempotent homomorphism
Note that every idempotent homomorphism of S can be extended un~ely to an idempotent homomorphism on the strong saturation [S]8 of S, i.e. I{, '= IIS]. for every subsemigroup SeC. An automorphism on a shiftinvariant algebra As is an isometric isomorphism c.p : As + As that maps As onto itself. The conjugate mapping c.p* of c.p defined by ( c.p* (m) ) (f) = m ( c.p(f) ), is a homeomorphism of the maximal ideal space M s onto itself. For instance, the conjugate mapping c.p* of an automorphism c.p of the disc algebra A(ID» = Az+ is a Mobius transformation of the unit disc, i.e. c.p*(z)
=G
z  Zo
1 ZoZ
,
IGI = 1, Izol <
l.
T. TONEV AND S. GRIGORYAN
308
Note that if the origin 0 is a fixed point of a Mobius transformation cp*, then cp*(z) = Cz for some constant C with ICI = 1. It is easy to see that this is also the case with the automorphisms of the subalgebra Ao(lD») = {J E A(lIJ» : 1'(0) = O} of the disc algebra A(lIJ», i.e. the conjugate mapping of any automorphism of the algebra Ao(lIJ» fixes the origin. Observe that the conjugate mapping of an automorphism cp : As + As maps idempotent homomorphisms of S to idempotent homomorphisms of S, i.e. cp* : Is + Is. Indeed, (cp*(t))2(f) = (t(cp(f)))2 = t(cp(f))) = (cp*(t)) (f), i.e.
(cp*(t))2 = (cp*(t)). An automorphism cp of a Gshift invariant algebra As is said to be inner, if there is aTE Hom (S, S) and an element 90 E G such that cp(X a) = Xa(90) . XT(a) for every Xa E S. Every automorphism cp of the disc algebra A(lIJ» with conjugate of type cp*(z) = Cz, ICI = 1 is inner. Indeed, for every z E IDi we have (cp(f))(z) = f(cp*(z)) = f(Cz). For Xn E Z+ : Xn(z) = zn, n ~ 0 we get (cp(Xn))(z) = (cp*(z)t = (Cz)n = cnXn(z), hence cp(Xn) = cnXn = Xn(c)xn, i.e. cp is an inner automorphism. Arens and Singer [1] have shown that every automorphism cp of the algebra As is inner in the case when G is a solenoidal group and S is a semigroup in JR with SU (S) = G. 5 (Grigoryan, Pankrateva, Tonev [20]). If G is a solenoidal group, then either As ~ A(lIJ», or every automorphism of the algebra As is inner. THEOREM
Proof. If the group S generated by S is not dense in JR, then the algebra As is a subalgebra of the disc algebra A(lIJ». If As =1= A(lIJ», then As C Ao(lIJ». In the same way as for the algebra Ao(lIJ» one can see that in this case every automorphism is the composition by a Mobius transformation, fixing the origin, i.e. every automorphism is inner. If the group S generated by S is dense in JR, then the algebra As is a subalgebra of the S~gebra As. If cp is an automorphism of As then the bounded analytic
il == 1 on JR, since ~(j(z)) = ~(z) =
function cp(Xa)(z) does not have zeros in II. Moreover, Icp(x a) 0
Ixal == 1 on G = BAs. By the Besicovitch theorem [2], Ce is % = CXs(j(z)), where s ~ 0, C E C, ICI = 1. It is easy to see that s E S, and that the mapping
T :
r + S : Xa
1+
XS is a homomorphism from S to S.
4. Primary ideals of algebras of analytic functions on solenoidal groups. Characterizing various types of ideals is an important and interesting topic in uniform algebra theory. A proper ideal of an algebra is said to be a primary ideal if it is contained in only one maximal ideal of the alg a. By f r .g below will be denoted the maximal ideal of functions in nish at the point r . 9 E IDi. Recall that every primary ideal J of the disc algebra A(lIJ» which is contained in some maximal ideal of type f%O' Izol < 1, admits the representation J = un A(lIJ» , where u(z) is the unimodular function
z 
1
Zo
_. ZOZ
ANALYTIC FUNCTIONS ON COMPACT GROUPS
309
6 (Grigoryan [18]). Let r = lR. and S = Il~+ If J is a primary ideal of the algebra As that is contained in I w , then either J = Xs(J) Iw. or, J = Xs(J) As; Every primary ideal I of As that is contained in a maximal ideal of type I r .g , 1'· 9 E Jl))c has a finite codimension in As. THEOREM
Let M{3 = HJ (lR.) . exp( ij3C I ) • l!t t 2 '
13 2:: O.
Note that M{3 :) M{3' for
13 2::
13' 2:: O. 7 [18]. Let J be a primary ideal of As that is contained in Ie = Ije (0)' Then there exists a 13 2:: 0 such that J.l.. = (As).l.. + C80 + M{3. THEOREM
5. Asymptotic Almost. Periodic~Ftmctions. A function f E BC(lR.) is asymptotic almost periodic, i~an almost periodic function j(x) on lR., such that limn~oo If(x n )  j(xn)1 = 0 for every sequence {xn}~=l + ±oo. Since h(x) = f(x)  j(x) E Co (lR.) , we have that for every asymptotic almost periodic function f 011 lR. there are unique E AP(lR.) and h E Co(lR.) such that f = + h. One can show that
J
J
THEOREM 8. Let G = j3lR. be the the Bohr compactification ofR The maximal ideal space M APo (1R) of the algebra of asymptotic almost periodic functions APo(lR.) is homeomorphic to the Cartesian product G x T.
Let r be an additive subgroup of lR., and let APr(lR.) be the set of almost periodic rfunctions. Clearly, APr(lR.) EB Co(lR.) is a uniform sub algebra of APo(lR.) , containing Co (lR.). It is not hard to see that every antisymmetric subalgebra of APo(lR.) that contains Co(lR.) is of this type. THEOREM 9. Let A be an uniform subalgebra of APo(lR.) which is invariant under lR.shifts. Then there is a subgroup r c lR., and a closed subalgebra Ao of Co (lR.) , such that (a) The algebra APr(lR.) of almost periodic rfunctions is a closed subalgebra of A. (b) A = APr(lR.) EB Ao. (c) Ao is an ideal in A. DEFINITION
f
E APo(lR.) and
6. A function f E BC(lR.) is analytic asymptotic almost periodic if f possesses a bounded analytic extension on the upper halfplane
II. Clearly, the set AAPo(lR.) of analytic asymptotic almost periodic functions on lR. is an antisymmetric uniform algebra under the supnorm on lR., and AAPo(lR.) C APo(lR.). Note that AAP(lR.) ~ AIR+ ~ AFlR+ (lR.). Consequently, MIR+ is the Gdisc K»c over the group G = j3R We have also the following results. THEOREM 10. The maximal ideal space phic to the Cartesian product K»{31R x K».
MAAPo(lR)
of AAPo(lR.) is homeomor
T. TONEV AND S. GRIGORYAN
310
11. Let G be a solenoidal group, such that its dual group r = G is a dense subgroup oflR, and let S be an additive subsemigroup of r+ containing the origin, with [S]s = r+. Tlwn there is a continuous projection from MH'X> onto the maximal ideal space M AAPo (lR) ~ ~G x~. THEOREM
12. The maximal ideal space of any subalgebra of AAPo(lR) of type AAPs(lR) EB B, where S c lR+ and B C Co(lR)n HoI (II), is the set M.4APs (IR)EBB THEOREM
=~G
X MB.
In particular, the upper halfplane II is not dense in the maximal ideal space of any subalgebra AAPs(lR) EB B of AAPo(lR) which contains properly AAPs(lR); The unit disc II} is not dense in the maximal ideal space of any subalgebra of the algebra [ea~ ,a E S] EB B c Hoc n A(~ \ {1}), where S is an additive semigroup in lR, and B =f. {a} is a subalgebra of the space {J E C(1l') : 1(1) = a}. A function 1 E BC(lR) is called weakly almost periodic, if the set of alllRshifts, It(x) = I(x + t), t E lR is relatively weakly compact in BC(JR) (e.g. Eberline [13], Burckel [7]). If W AP(lR) denotes the set of weakly almost periodic functions onlR, then AP(lR) C APo(lR) c W AP(lR). In fact, W AP(lR) = AP(lR) EB C([oo, 00])11R. Similarly to Theorem 11, one can show the following THEOREM 13. The maximal ideal space MAWAP(IR) of AW AP(lR) of analytically extendable on II weakly almost periodic functions on lR is homeomorphic to tile Cartesian product ~~IR x {([a, 1] x [0,1])/([0,1] x {a})}.
The space AW AP(lR) orp is isometrically isomorphic to the subalgebra of HOC n =.±! A(II} \ {I}) generated by the functions ea z1, a E lR+ and the set of continuous functions on 1l' \ {I} that possess both one sided limits at 1. 
III. Inductive limits and shift invariant algebras on solenoidal groups 1. Inductive limits of disc algebras on Gdiscs. Let A C lR+ be a basis in lR over Q, and lR = lim r h Il)' where >h.n)EJ
'
Let P h .n ) = r(Y, n)+ = r h .n ) U [0,00). If Ap(Y.n) is the algebra of analytic P h .n )functions on G, one can show that AIR+ = [ lim A p ("Y,n ) (II}G)]. A similar expression + h,n)EJ
holds for the algebra As, S c lR+. Uniform algebras that can be expressed as inductive limits of disc algebras A(II}) are of special interest. Consider the inverse sequence {~kH' T;H }k=I' ~k = ~ and T;H(z) = Zdk on ~k' The limit lim {~k+l,T;+I} of the inverse sequence ......... k ..... oc
{~k+l,T;H}, is the GAdisc ~G,\ = ([0,1] x GA)/({O} x GA) over the group GA = TA . There arises a conjugate inductive sequence {A(~k)' i~H}f of algebras A(~) ~
ANALYTIC FUNCTIONS ON COMPACT GROUPS
A('Jl') with connecting homomorphisms iZ+l: A(Jijk)
+
311
A(Jijk+d defined by
(iZ+1(f))(Z) = (f(z))d k , i.e. iZ+ 1 = (7:+ 1)*. The elements of the component algebras A(Jijd can be interpreted as continuous functions on G J1. The uniform closure A(Jijc,\) = [ {A(Jijk), iZ+l}] in C(JijCA) of
!!!!!
k ....... oo
the inductive limit of the system {A(Jijk), i~+1 }k=1 and the corresponding restricted {A('Jl' k), iZ+l }] are isometrically isomorphic to the GJ1disc algebra algebra
[!!!!!
k ....... oo
A rA +, i.e., to the algebra of analytic r,1+functions on the GJ1disc (e.g. [21]). Consider an inductive sequence of disc algebras
where the connecting homomorphisms iZ+ 1 : A('Jl'k) + A('Jl'k+l) are embeddings with Mi~+l(A(ll'k)) = Jij and 8(iZ+ 1 (A('Jl'k))) = 'Jl'. There are finite Blaschke products
Bk :
JD) + JD),
Bk(Z)
= eiIJk
IT (
1=1
z
~~~) )
1  zl
, Izfk)1 < 1, such that iZ+l
=
Bie for
z
every kEN, i.e. i~+1(f) = ! 0 B k. Let B = {Bdk=l be the sequence of finite Blaschke products corresponding to iZ+ 1 , i.e. (Bk)*(Z) = iZ+1(z). Let A = {dk}~1 be the sequence of orders of Blaschke products {Bdk=l and let rJ1 c IQl be the group generated by l/mk, k = 0,1,2, ... , where mk = I1~=1 dj, mo = 1. Consider the inverse sequence Jijl ~ Jij2 ~ Jij3 ~ Jij4 ~ ...
The inverse limit VB
+
VB·
= lim {Jijk+1, Bd is a Hausdorff compact space. The limit +koo
of the composition system {A(Jijk), ,B~+1 HO of disc algebras A(Jijk) and connecting homomorphisms ,B~+1 = Bie : A(Jijk) + A(Jijk+1): (,BZ+l(f))(zk+d = !(Bk(Zk+d) is an algebra of functions on VB whose closure [lim {A(Jijk),,B~+1}] t
= A(VB )
k ....... oo
in C(V B ) we call a Blaschke inductive limit algebra. It is isometrically isomorphic to the algebra [lim {A('Jl' k), ,B~+1 }]. t
k ....... oo
PROPOSITION 13 (Grigoryan, Tonev [21]). Let B = {Bdk=l be a sequence of finite Blaschke products and let A(V B ) = [lim {A(JD)k), Bn] be the corresponding t k ....... oo
inductive limit of disc algebras. Then (i) A(VB) is a uniform algebra on the compact set VB = lim {Jijk+l, Bd. +k ....... oo
(ii) The maximal ideal space of A(VB) is VB. (iii) A(VB ) is a Dirichlet algebra. (iv) A(VB ) is a maximal algebra.
T. TONEV AND S. GRIGORYAN
312
(v) The Shilov boundary of A('DB) is a group isomorphic to GA, and its dual 00
group is isometric to the group rA
~
U (l/mk)Z c
Q, where mk
k=O
=
THEOREM 14 [21]. Let G be a solenoidal group, i.e. G is a compact abelian group with dual group G isomorphic to a subgroup r of JR.. The Gdisc algebra Ar+ is a Blaschke inductive limit of disc algebras if and only if r is isomorphic to a subgroup ofQ. THEOREM 15 [21]. Let B = {Bdk"=1 be a sequence of finite Blaschke products on ~, each with at most one singular points zak ) and such that Bk(Zak+ 1») = zak ). Then the algebra A('DB ) is isometrically isomorphic to the algebra A(rA)+ with A = {ddk"=1' where dk = ordBk.
In particular, if every Blaschke product Bk in the above theorem is a Mobius transformation, then the algebra A('DB) is isometrically isomorphic to the disc algebra A z = A(lI'). 2. Inductive limits of algebras on subsets of Gdiscs. Let lDJ[I",1J = {z E C : r ~ JzJ ~ I}, and blDJ[r,1J = {z E C : JzJ = r or JzJ = I} is the topological boundary of lDJ[r,1J. Denote by A(lDJ[r,1J) the uniform algebra of continuous functions on the set ~[r,1J that are analytic in its interior. Note that A(lDJ[r,1J) = R(lDJ[r,1J), the algebra of continuous rational functions on lDJ[r,1J. By a well known result of Bishop, the Shilov boundary of A(lDJ[I·,1J) is blDJ[r,lJ, and the restriction of A(lDJ[r,1J) on blDJ[r,1J is a maximal algebra with codim (Re (A(lDJ[r,1 J)JbllJ)lr.!J)) = 1. These results can be extended to the case of analytic r+flllctions on solenoid groups (e.g. S. Grigoryan [19]). Let G be a solenoidal group, and its dual group is denoted as r c JR.. Let lDJ~,1J = [r,l] x G, 0 < r < 1 be the [r,I]annulus in the Gdisc ~G, [I" 1J [I" 1J [r 1J and let A(lDJa' ) = R(lDJa' ) be the Gannulus algebra on lDJa' ,generated by the functions a E r. Let A = {d k } k"=1 be a sequence of natural numbers and T~+1 (z) = zd k , and let r be a fixed number, 0 < r ~ 1. For every kEN consider the sets
xa,
where mk = I17=1 dl, mo = 1, and E1 = iij[r,1 J• There arises an inverse sequence
of compact subsets of iij. Consider the conjugate composition inductive sequence
where the embedings
iZ+ 1 : A(Ek)
+
A(Ek+d are the conjugates of zd k , namely,
(iZ+1 0 J)(z) = J(zd k ). Let GA denote the compact abelian group whose dual group
ANALYTIC FUNCTIONS ON COMPACT GROUPS
rA
=
GA
313
is the subgroup of Q generated by A. The algebra [lim {A(Ek)' iZ+1}] +
k+oo
is isomorphic to A([J)~,11). THEOREM 16 [21]. Let Fn+1 = B:;;l(Fn), Fl = [J)~,11. If the Blaschke products Bn do not have singular points on the sets Fn for any n E N, then D~,11 ~ [J)~,11, and the algebra A(D~,11) = [lim {A(Fn), B~}] is isometrically isomorphic to the +
n+oo
Gannulus algebra A([J)~,11). Below we summarize some of the basic properties of the algebra A(D~,11) (see [21]). (a) The maximal ideal space of A(D~,11) is homeomorphic to the set [J)~,11. (b) The Shilov boundary of A(D~,11) is the set b[J)~,11 = {r, I} x G. (c) A(D~,11) is a maximal algebra on its Shilov boundary. (d) co dim (Re(A(D~,11)lb]IJ)[r"I)) = 1. G
Let B = {B I , B 2, ... , B n , ... } be a sequence of finite Blaschke products on ii} and let 0 < r < 1. Let D n+ l = B:;;l (Dn), Dl = [J)[O,r1 = {z E [J) : Izl ::; r}. Consider Tn\[O,r1
llJI
~
D2
~
D.3
~
D.........fu. 4
n[O,r1
.•. VB
of subsets of [J). The inductive limit A(D~,r1) = [lim {A(Dn), B~}] is a uniform +
n+oo
algebra on its maximal ideal space ~ {Dn, BntlDn} = D~,r1
c DB.
k+oo
PROPOSITION 14 [21]. LetB = {B l ,B2,B3, ... } be a sequenceoffinite Blaschke products on ii} and let 0 < r < 1. Suppose that the set Dn does not contain singular points of B n l for every n E N. Then (i) There is a compact set Y such that D~,r1 = ~ {D n+ l , BnIDn+l} n+oo
M A(D~.rl) is homeomorphic to the Cartesian product [J)[O,r1 x Y. (ii) A(D~,r1) is isometrically isomorphic to an algebra of functions f(x,y) E C([J)[O,r1 x Y), such that f(· ,y) E A([J)[O,r1) for every y E Y.
(iii) A(D~,r1) 1]IJ)[o.rl x {y} ~ A([J)[O,r]) for every y E Y. The proof makes use of the fact that every finite Blaschke product of order n generates an nsheeted covering over any simply connected domain V c [J) free of singular points of B. Proposition 14 implies that the onepoint Gleason parts of the algebra A(D~,r]) are the points of the Shilov boundary bD~,r1 ~ 'll'r x Y. PROPOSITION 15 [21]. Let B = {B I , B 2, B3,"'} be a sequence offinite Blaschke products on ii}, and let 0 < r < 1. Suppose that (a) For every n E N the points of the set :F = (Bl 0 B2 0 ' " 0 Bn_dl(O) are the only singular points for B n l in Dn (b) All points in (a) have one and the same order dn  l > 1.
314
T. TONEV AND S. GRIGORYAN
Then (i) There is a compact Y such that
V~·r] = ~ {Ir»n+1' BnID,,+!} = M A(V~.rl) k+oo
is homeomorphic to the Cartesian product Ir»~:] x Y, where A = {dd~1 is the sequence of the orders of B k . (ii) The algebra A(V~,r]) on V~,r] is isometrically isomorphic to an algebra of functions f(x, y) E C(Ir»~:] x Y), such that f( . ,y) E A(Ir»~:]) for every yEY. (iii) A(V~,r])IIIli[(l.rlx{y}
= A(Ir»~:]) for every y
E Y.
The set Y in Propositions 14 and 15 is homeomorphic to the set {{yn}~=I' Yn E (Bl 0 B2 0 ' " 0 Bn_t}I(O)}. Proposition 15 implies that there are no singlepoint Gleason parts of the algebra A(V~,r]) within the set M A(V~.rl) \ bV~,r] U {w} x Y, where w is the origin of the G Adisc ll}a/l' As an immediate consequence we obtain that A(V~,r]) is isomorphic to a Gdisc algebra if and only if the set Y consists of one point. In particular, in the above setting the algebra A(V~'1']) is isomorphic to a Gdisc algebra if and only if every Blaschke product Bn has a single singular point z6n ) in D~) such that B n (Z6 n») = z6n+1) for all n hig enough. 3. Gleason parts of inductive limits of disc algebras on Gdiscs. The celebrated theorem by Wermer [36] asserts that in every nonsinglepoint Gleason part of the maximal ideal space of a Dirichlet algebra can be embedded an analytic disc. Therefore it is of particular interest to locate singlepoint Gleason parts of an algebra, and especially those of them that do not belong to the Shilov boundary. While every point in the Shilov boundary is a separate Gleason part (e.g. Gamelin [14]), the opposite is not always true, i.e. there are singlepoint Gleason parts outside the Shilov boundary. For instance, if G is a solenoid group with a dense in lR. dual group, then the origin w = ({O} x G)/( {O} x G) E Ir»a of the Gdisc Ir»a is a singlepoint Gleason part for the Gdisc algebra Ar+. Of course w (j. 8 Ar+ = G.
Given a sequence of Blaschke products B = {Bn}~=1 on ll}, consider the Blaschke inductive limit algebra A(V B ) = [lim {A(ll}k) , Bn] on the compact > k+oo
set VB = lim {ll}k, Bkd. Recall that the Shilov boundary of A(VB ) is the group f
k+oo
TB = lim {1l'k' Bkd. Let Br be the set of all Blaschke products on ll} whose zeros fk+oo
are inside the disc Ir»[O,r] ones at O.
= {Izl
~ r}, and let B~
c Br
be the set of the vanishing
PROPOSITION 16 [21]. Let B be a finite Blaschke product with B(O) = O. Consider the sequence B = {B, B, ... }. If the Blaschke inductive limit algebra A(VB) = [lim {A(Ir»k) , Bd]' Ir»k = Ir», Bk = B is isometrically isomorphic to a > k+oo
Gdisc algebra, then necessarily B(z) = cz", where c E Ir»,
lei =
1, and n E N.
ANALYTIC FUNCTIONS ON COMPACT GROUPS
315
THEOREM 17 (Grigoryan, Tonev [21]). Let B be a finite Blaschke product on IDJ. The Blaschke inductive limit algebra A(D B ) is isometrically isomorphic to a Gdi8c algebra if and only if B(z) is conjugate to a power z'" of z, i.e. if and only if there is an mEN and a Mobius transformation 7 : IDJ t IDJ such that (7 1 OB 0 7)(Z) = zm.
m
THEOREM 18 [21J. Suppose that Bn E and ordBn > 1 for every Then tllere is only one singlepoint Gleason part in the set DB \ TB.
17,
E N.
In particular, if B E Sr, B(O) =f. 0, and Bk(z) = zd k BCk, dk > 1 then there is only one singlepoint Gleason part in the set DB \ TB. The proof of Theorem 18 involves a thorough study of onepoint Gleason parts of the algebras involved. 4. Inductive limits of Hoospaces on Gdiscs. Let I = {iZ+ 1}k'=1 be a sequence of homomorphisms iZ+l : HOO(IDJ) t HOO(IDJ). Consider the inductive sequence HOO(lDJd iL H OO (1DJ 2) 2L HOO(IDJ3) .fL ... of algebras HOO(lDJ k ) ~ HOO(IDJ). Every adjoint mapping (iZ+l)* : Mk f Mk+l maps the maximal ideal space Mk+l of HOO(IDJk+d into the maximal ideal space Mk of HOO(lDJ k ). The inverse limit
Ml
1;2)" ~1_
M2
1;3)" ~2_
M3
1;4)* ~3_
M4
1;5)" ~4_
••• f  
DI
is the maximal ideal space of the inductive limit algebra
HOO(DI) = [lim {HOO(lDJk),i~+I}J. + k+oo Recall that the open unit disc IDJ is a dense subset of every Mk. In general, the mappings (i~+l)* are not obliged to map IDJ k+1 onto itself. The most interesting situations, though, are when they do. Here we suppose that the mappings (iZ+l)* are inner nonconstant functions on IDJ. For instance, algebras of type H oo (D I ) are the algebras [lim {H OO (lDJ k), (zdk)*}dkEAJ = HOO(D A) c H OO (IDJ CA ), and also + k+oo the algebras of type HOO(DB) = [~{HOO(IDJd, Bk}], where B = {Bdk'=1 is k+oo a sequence of finite Blaschke products Bk : IDJ t IDJ. Note that HOO(DB) is a commutative Banach algebra of functions on DB. Let A = {dd~1 be the sequence of orders of Blaschke products {Bd~1 from the mentioned above example, and let rA C Q be the group generated by l/mk, k = 0,1,2, ... , where mk = I1~=1 d/, mo = 1. THEOREM 19 [21J. Let B = {B k }k'=1 be a sequence of finite Blaschke products on ~, each with at most one singular points zbk ), and such that Bk(zbk+ 1») = zbk ). Then the algebra HOO(D B ) is isometrically isomolphic to the algebra HOO(DA) for A = {ddk'=1 with dk = ordBk. For instance, if the Blaschke products Bk are of type Bk(Z) = zd k 'Pdz), where 'Pk are Mobius transformations and dk > 1, then the algebra HOO(D B ) is isometrically isomorphic to the algebra HOO(D A), where A = {1/dd~I' If every Blaschke
316
T. TONEV AND S. GRIGORYAN
product Bk in Theorem 19 is a Mobius transformation, then the algebra HOO(DB) is isometrically isomorphic to the algebra H oo . Indeed, the last theorem implies that HOO(D B ) ~ HOO(DA) with A = {I, I, ... }. Therefore rA = Z and GA = T. Let iP = {'P1, 'P2, ... ,'Pk, ... } be a sequence of nonconstant inner functions on ][)l. Consider the inverse sequence ][)l1 +'£l. ][)l2 where][)lk
~][)l.
~
][)l3
~
][)l4
~
...
Denote by Dq, its inverse limit. The inductive limit lim {HZ", 'PkH" t
k+oo
of the adjoint composition inductive sequence
Hf" 1.4 H2'
~
H'3
~
... 10 'Pk,
of algebras HZ" = HOO(][)lk) ~ HOO(][)l), where 'PkU) = is a subalgebra of BC(Dq,), the algebra of bounded continuous functions on the set Dq,. The closure HOO(Dq,) of lim {H OO , 'Pk} in BC(Dq,) is a commutative Banach subalgebra of t
k+oo
HOO(][)le). Its elements are referred to as iPhyperanalytic lunctions on Dq,. Recall that according to the classical corona theorem for the space HOO on the unit circle (Carleson [8]), given h, ... , /k, functions in Hoo with L~=lllil :::: a > 0 on ][)l, there exist functions gl, ... ,gk in Hoo such that L~=lligj = 1 on ][)l; If IIIi 1100 :::; 1, then 9j can be chosen to satisfy the estimates II 9j II :::; C (k, a) for some constant C(k, a) > O. Next theorem is the corona problem for the algebra HOO(Dq,). THEOREM 20 (Grigoryan, Tonev [21]). If h, 12, ... , In, IIIi II :::; 1, are iPhyperanalytic functions on Dq, for which Ih(x)1 + ... + I/n(x)1 :::: 8 > 0 for each x E Dq" then there is a constant K(n,8) and iPhyperanalytic functions gl, ... , gn on Dq, with Ilgj II :::; K(n, 8), such that the equality h (x )gl (x) + ... + In (x )gn (x) = 1 holds for every point x in the set Dq,. In the case when iP = {Z2, z3, ... , zn+1 ... } the corresponding set Dq, coincides with the open big disc ][)le over the compact abelian group G = ij, and the algebra HOO(Dq,) coincides with the set He of hyper analytic functions. In this case Theorem 20 reduces to the corona theorem for the algebra He of hyperanalytic functions on G with estimates (cf. Tonev [32]). 5. Hoospaces on solenoidal groups. Let G be a solenoidal abelian group, i.e. r = G c R Let HOO(][)le) be the algebra of bounded functions in the open Gdisc ][)le that can be approximated on compact subsets of][)le by functions 1, I E Ar+. For every I E HOO(][)le), the limits
f*(g) = lim I(r)(g), where I(r)(g) = r+1
f(r. g)
exist for almost all 9 E G, and f* E Hoo (G, a). The space of functions f*, I E HOO(][)le) we denote again by HOO(][)le). The algebra HOO(][)le) we interpret as a subspace of the set of functions in LOO(G,a) that are boundary values of continuous functions on ][)le, equipped with the norm 11/1100 = lim sup I/(r)(g)l. Denote
r+1 gEe
ANALYTIC FUNCTIONS ON COMPACT GROUPS
317
by 'HOC(lD>a) the weak*closure of Ar+ in LOC(G,a) (cf. Gamelin [14]). Clearly HOC(lD>a) is a closed subalgebra of 'HOC (lD>a). Let I be a directed set. We consider a family {rdiEI of subgroups of r indexed by I, such that ril C r i2 whenever il < i2. Let r = limri , and H~(lD>a) denotes + iEI the space of functions f E Hoc (lD>a) with sp (I) C ri . The family {Hr: (lD>a) hEI of subalgebras in HOC(lD>a) is ordered by inclusion. Denote by Hr(lD>a) the closure of the set U H~(lD>a) = limH~(lD>a) with respect to the norm II . lIoc. Hr(lD>a) iEI ~ is the set of Ihyperanalytic functions on lD>a. In a similar way we define the algebra 'Hf(lD>a) as the II . Ilocclosure of the inductive limit ~ 'H~(lD>a), where iEI 'H~(lD>a) = {f E 'HOC(lD>a) : sp (I) c rd· THEOREM 21 (Grigoryan, Tonev [22]). Let G be a solenoidal group such that its dual group r = G is the inductive limit of a family {rdiEI of subgroups of r, i.e. r = lim r • Let Hr='• (lD>a) and 'Hr', (lD>a) be the spaces offunctions in Hoc (lD>a) + i iEI [resp. in 'HOC(lD>a)] with spectra in ri , i E I. Then the following statements are equivalent. (a) HOO(lD>a) = Hr(lD>a) and 'HOC(lD>a) = 'Hf(lD>a). (b) HOC(lD>a) = U Hr:(lD>a) and 'HOC(lD>a) = U 'H~(lD>a). iEI iEI (c) Every countable subgroup ro in r is contained in some group from the family {rdiEI. Example 4. Let r = Q be the group of rational numbers with the discrete topology. Assume that {rdiEI is an inductive system of subgroups of Q such that Q = lim n. The last theorem implies that if Q itself is not one of the groups in the + iEI family {rdiEI, then Hr(lD>a) =I H""(lD>a). In the case when all subgroups ri , i E I are isomorphic to Z, the algebra Hr(lD>a) coincides with the algebra of hyperanalytic functions (e.g. [34]). As seen above, in this case the space Hr (lD>a) differs from HOC (lD>a). The properties of subalgebras of HOC (lD>a) on general compact groups G are less known. In particular it is not known if they possess a corona, and their maximal ideal spaces and Shilov boundaries lack a good description. Example 5. Let r = R and let A C R+ be a basis in R over the field Q of rational numbers. Then R = ~ r(y,n), where (y,n)EJ
Given an (y, n) E J, consider the set
T. TONEV AND S. GRIGORYAN
318
The closure HJ'(lJJJc) of the set
U
H0',n)(lJJJc) under the
II . lloonorm, i.e.
the
(y,n)EJ
inductive limit algebra
lim H(ooy,n )(lJJJc) is a subalgebra of HOC (lJJJc). There arises _ (y.n)EJ
the question of whether or not the algebra HJ'(lJJJ c ) coincides with HOO(lJJJ c ). THEOREM 22 (Grigoryan, Tonev [22]). The set HJ(lJJJ c ) =
lim H(ooy,n ) (lJJJ c ) _ (y,n)EJ
is a proper closed subalgebra of H OO (lJJJ c ).
c HOO(lJJJ c ) is easy (e.g. [12]). Assume that H0',n) (lJJJc). By the previous theorem, the countable
Proof. The inclusion HJ'(lJJJc)
HOO(lJJJ c ) = HJ(lJJJc) =
~ (y,n)EJ
subgroup Q c IR is a group in the family {rh,n) hy,n)EJ' which is impossible since r(y,n) is isomorphic to Z} for some kEN.
The algebra HOO(lJJJ c ) is isometrically isomorphic to the algebra HfPr+(IR)(IR) C
HOO (1R) of boundary values of almost periodic r+functions on IR that are analytic in the upper half plane. Similarly, the algebra HJ'(lJJJ c ) is isomorphic to a subalgebra HJ'(IR) of HZ,r+ (R) (1R). As the last theorem shows, these algebras are different. Algebras of type HI (lJJJ c ) were introduced in connection with the corona problem for algebras of analytic rfunctions (Tonev [32]). R. Curto, P. Muhly and J. Xia [12J have introduced similar algebras of this type in connection with their study of WienerHopf operators with almost periodic symbols.
6. Bourgain algebras and inductive limits of algebras. Bourgain algebras of a Banach space were introduced in 1987 by J. Cima and R. Timoney [9J as a natural extension of a construction due to J. Bourgain [5J. The concept of tightness of an algebra was introduced by Cole and Gamelin [lOJ. Let A c B be two commutative Banach algebras, and 11" A : B t B I A is the natural projection. The mapping Sf: A t (f A+A)IA c BIA; Sf: 9 ~ 1I"A(fg) is called the Hankel type operator. DEFINITION 7. An element fEB is said to be (a) a Bourgain element, (b) a wcelement, (c) a celement for A, if the Hankel type operator Sf : A t BIA is correspondingly (a) completely continuous, (b) weakly compact, (c) compact. The Bourgain algebra of A relative to B is the space A~ of all Bourgain elements for A in B [9J. PROPOSITION 17 [35J. If the range Sf(A) = 1I"A(f A) of the Hankel type operator Sf for an fEB is finite dimensional then f E A~ In particular, (As)f(C) = C(G) if As is a maximal algebra and xS \ X is a finite set for a character X E O\S. Indeed, X E (As)f(C) by the above proposition. Since X ~ S, then X ~ As, and consequently (As)f(C) = C(G) by the maximality of A.
ANALYTIC FUNCTIONS ON COMPACT GROUPS
319
Example 6. Let H be a finite group, G = (H E& Zr and S So' H E& Z+. Then (As)f(G) = C(G).
r r
r
Note that if = G and XS\ X is finite for every X E then every character X E G has finitely many predecessors in r. As it follows from Proposition 17, (Ar)f(G) = C(G), and therefore the corresponding big disc algebra Ar possesses the DunfordPettis property. THEOREM 23 (Yale, Tonev) [35]. Let G =,BJR be the Bohr compactification of lR. The Bourgain algebra (AIR+ )f(G) of the big disc algebra AIR+ is isomorphic to AIR+' Proof. Clearly, JR is a subset of (AIR+ )f(G). Since, as one can easily see, the
seqnence of real valued functions 'Pn(x) = as n
+
00 for every x
E
11 +2ei ';i 12n
converges pointwisely to
1
2n . JR, then the real valued functIOns 'l/Jn(g) = 11 + X2a(g)1
converge pointwisely to 1 as n + 00 for every 9 E G. Suppose that X3 E (AIR+ )f(G). Consider the sequence ~n(g) = 'l/Jn(g) 1, where 'l/Jn is as before. {X1~n}n is a weakly null sequence in AR+ since it converges pointwisely to 0 on G. By the Bourgain algebra property there are functions hn E AIR+ such that IIX3X1~n  hnll < l/n for every n, where II . II is the sup norm on G. By integrating over Ker(x~), if necessary, we can assume that hn
then (X',pn)(g)
1 • = qn(X") for some polynomIal qn'
~ (x«g))" ( 1 + ;~(g))
2n the jth Cesaro mean
af" =
n
= max I(X1~n)(g) gE G = ~Eas IPn (xa (g)) ( z)
Fa, j
~
= (1 + z)2n + 2n(1 + z)2n1 = (2n +
= max I(X 1'I/Jn)(g) gE G
 X1(g)  (xa (g)) 3n qn P (z)
~~(g))" ~ Pn(X" (g)).
= max 1(~X1~n)(g)  hn(g)1 gE G
(X3hn)(g)1
ZTO
= (1+Z)2n 2,
SO+S1+"'+S, j +1 J of Pn, where Sk is the kth
partial sum of Pn, we have 4n(2n + l)a~~(z) 1 + z)(l + z)2n1. Now II~X1~n  hnll
(1 +
If Pn(z)
zn
X1(g)  X3(g)h n (g)1
(x~ (g)) I = ~tf IPn(Z)_zn_ z3nqn (z)l·
z3n q (z)
P Note that a" " (z) because the Cesaro mean a2n de2n  (z) = a 2nn  pends only on the first 2n terms of the Taylor series. Since the inequality max la~(z)1
zE'lr
f E A(1l') we see that
:::; max If(z)1 holds for every zE'lr
maxlaPn ( zE'lr
2n
z)

zn
(z)1
p (z) = maxla n zE'lr 2n
zn

z3n q (z) n
(z)1
T. TONEV AND S. GRIGORYAN
320
However, O"~~(z)_zn (z) = O"~~(z) (z)  zn(n + 1)/(2n + 1) and thus O"~~(z)_zn (1)
1/2 as n > 00 for odd n. Hence AIR+ by the maximality of AIR+'
>
X3 f/. (AIR+)f(O) and consequently (AIR+)f(O) =
THEOREM 24 (Tonev [33]). Let {AO" }O"EL', {BO" }O"EE be two monotone increasing families of closed subspaces of a commutative Banach algebra B such that BO" are algebras, and AO" c BO" for every 0" E E. Let A = [ U AO"] be a (linear) subO"EE
space, and let B = [ U BO"] be a subalgebra of B. Suppose that for every
0"
E E
O"EE
tllere is a bounded linear mapping r 0" : B > BO", such that (i) rO"IB" = idB" (ii) rO"(fg) = frO"(g) for every f E BO", 9 E B (iii) sup Ilr0" II < 00. O"EE Then A~ c [ U (AO")~"]. O"EE Proof. Let fEB be a Bourgain element for A. Fix a 0" E E, and consider a weakly null sequence {
IlrO"(f)
>
O.
Consequently, r 0" (f) is a Bourgain element for AO", i.e. r 0" (f) E (AO")~" for every E E. Note that under the hypotheses every fEB is approximable by the elements rO"(f) in the norm of B. Indeed, let fO"n E BO"n be such that fO"n > f. Then IlfrO"n(f)II::; IlffO"nll+llrO"n(fO"JrO"..(f)II::; IlffO""II+supllrO"..IlllfO"n fll· Hence rO"n(f) > f and, consequently, f E [ U (AO")~"]. O"EE
0"
lim rO", let Hf? = {! E HOO(J]))o) : sp (f) c rO"}. Note that H't:.. is a + " O"EE closed sub algebra of HOO(J]))o), and H'f:. c Hr:. if and only if rO" err. Therefore, the family {H'f:. }O"EE of subalgebras of HOO (J]))o) is ordered by the inclusion. Denote by H~ the closure of the set U H'f:. with respect to the norm II . 1100' Theorem O"EE 24 implies that if r = lim rO" and G = f, then the Bourgain elements for H roo are + + O"EE approximable in the L 00  norm on G by Bourgain elements for H'f:., 0" E E. Note that H~, HOO(J]))o), and the weak* closure HOO(G, dO") of Ar+ in LOO(G, dO") are commutative Banach subalgebras of LOO(G,dO"), which are in principle different from each other, except in the case of G = 'll', when they coincide (Grigoryan [19]).
If r
=
The algebra HQ;/n = H OO 0 Xl/n = {! 0 Xl/n: f E HOO} is a closed subalgebra of HOO(J]))o). The closure HQ' of U HQ' with respect to the norm II . 1100 is the + nEN lin algebra of hyperanalytic junctions on G = f3Q (cf. Tonev [34]). By Theorem 24 the Bourgain algebra of is contained in the algebra + C(G).
Hift
Hift
ANALYTIC FUNCTIONS ON COMPACT GROUPS
321
THEOREM 25 (Tonev [33]). If the hypotheses of Theorem 24 are satisfied. then A~c c [ U (AO')~~]; A~ c [ U (AO')~O']; O'EE
(H~J~:(G) .
c [U
O'EE
(H~)~:(G)]; (HrJ~OO(G)
O'EE
In particular, the algebra H~
c [U
(H~)~OO(G)].
O'EE
+ C(,8Q)
contains the spaces (H~)~:({3Q) and
(H~ )~OO({3Q). A uniform algebra A C C(X) is said to be tight [strongly tight] if every f E C(X) is a weelement [resp. celement] for A, i.e. if (A)~~G) = C(X) [resp. (A)f(G) = C(X)] (cf. Cole, Gamelin [10], also Saccone [30]). Theorem 25 implies that the algebra H~ is neither tight nor strongly tight.
References
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DEPARTMENT OF MATHEMATICAL SCIENCES, UNIVERSITY OF MONTANA, MISSOULA, MONTANA 598121032 CHEBOTAREV INSTITUTE OF MATHEMATICS AND MECHANICS, KAZAN, SIA
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