Banach Function Algebras, Arens Regularity, and BSE Norms (CMS/CAIMS Books in Mathematics, 12) [1st ed. 2024] 3031445317, 9783031445316

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CMS/CAIMS Books in Mathematics

Harold Garth Dales Ali Ülger

Canadian Mathematical Society Société mathématique du Canada

Banach Function Algebras, Arens Regularity, and BSE Norms

CMS/CAIMS Books in Mathematics Volume 12

Series Editors Karl Dilcher Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, Canada Frithjof Lutscher Department of Mathematics, University of Ottawa, Ottawa, ON, Canada Nilima Nigam Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada Keith Taylor Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, Canada Associate Editors Ben Adcock Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada Martin Barlow University of British Columbia, Vancouver, BC, Canada Heinz H. Bauschke University of British Columbia, Kelowna, BC, Canada Matt Davison Department of Statistical and Actuarial Science, Western University, London, ON, Canada Leah Keshet Department of Mathematics, University of British Columbia, Vancouver, BC, Canada Niky Kamran Department of Mathematics and Statistics, McGill University, Montreal, QC, Canada Mikhail Kotchetov Memorial University of Newfoundland, St. John’s, Canada Raymond J. Spiteri Department of Computer Science, University of Saskatchewan, Saskatoon, SK, Canada

CMS/CAIMS Books in Mathematics is a collection of monographs and graduatelevel textbooks published in cooperation jointly with the Canadian Mathematical Society- Societé mathématique du Canada and the Canadian Applied and Industrial Mathematics Society-Societé Canadienne de Mathématiques Appliquées et Industrielles. This series offers authors the joint advantage of publishing with two major mathematical societies and with a leading academic publishing company. The series is edited by Karl Dilcher, Frithjof Lutscher, Nilima Nigam, and Keith Taylor. The series publishes high-impact works across the breadth of mathematics and its applications. Books in this series will appeal to all mathematicians, students and established researchers. The series replaces the CMS Books in Mathematics series that successfully published over 45 volumes in 20 years.

Harold Garth Dales · Ali Ülger

Banach Function Algebras, Arens Regularity, and BSE Norms

Harold Garth Dales Department of Mathematics and Statistics University of Lancaster Lancaster, UK

Ali Ülger Department of Mathematics Bo˘gaziçi University ˙Istanbul, Türkiye

ISSN 2730-650X ISSN 2730-6518 (electronic) CMS/CAIMS Books in Mathematics ISBN 978-3-031-44531-6 ISBN 978-3-031-44532-3 (eBook) https://doi.org/10.1007/978-3-031-44532-3 Mathematics Subject Classification: 46J, 46E, 46B

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

This book will discuss a certain number of mutually related properties of Banach function algebras, equivalently, of commutative, semisimple Banach algebras, on locally compact spaces and, in particular, on locally compact groups. Some of the theory will also apply to more general classes of Banach algebras, and we shall give several examples of (non-commutative) Banach algebras. Two of the main topics to be discussed are BSE algebras and the notion of Arens regularity. Both involve the bidual of a Banach algebra. In Chapter 5, on BSE norms and BSE algebras, we shall give many new results. The notion of Arens regularity and a closely related topic involving the topological centres of a Banach algebra are more than half a century old, and are much-studied subjects. The two subjects are closely related to the geometry of Banach space, and there is a substantial number of accumulated results; we shall present in Chapter 6, and earlier, a good many of these results, some new, concentrating on determining whether key examples are Arens regular or, at the other extreme, strongly Arens irregular: for example, all C ∗ -algebras are Arens regular, but all group algebras are strongly Arens irregular. Let (A, ||·||) be a natural Banach function algebra on a non-empty, locally compact space K. For each x ∈ K, let ε x be the evaluation functional at x, regarded as an element of the dual space A , and take L(A) to be the linear span of these functionals in A . A bounded, continuous function f on K is a BSE function if there is a constant β  0 such that   ⎞ ⎛ n n    ⎟⎟ ⎜⎜⎜ ⎜  αi f (xi )  β ||λ|| ⎜⎝λ = αi ε xi ∈ L(A)⎟⎟⎟⎠.  i=1  i=1 In this case, the BSE norm, denoted by || f ||BSE , of f is the infimum of these constants β. The space of BSE functions on K is denoted by C BS E (A). We shall show in Chapter 5 that, with respect to the pointwise product, (C BS E (A), || · ||BSE ) is itself a Banach function algebra on K such that A ⊂ C BS E (A) ⊂ C b (K), where C b (K) is the space of bounded, continuous functions on K; C BS E (A) is the BSE algebra of A; and A is a BSE algebra if C BS E (A) is equal to M(A), the multiplier algebra of A.

v

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We shall see that the BSE norm on A is such that (A, || · ||BSE ) is a normed algebra, and clearly | f |K  || f ||BSE  || f || ( f ∈ A), where | · |K is the uniform norm on K. In particular cases, the BSE norm might be equivalent to the given norm || · || (in which case ‘A has a BSE norm’) or to the uniform norm. This norm has arisen, perhaps implicitly, in a number of contexts, especially in connection with Banach function algebras defined on locally compact groups. The abstract notions of a BSE algebra and a BSE norm were introduced in 1990 by Takahasi and Hatori [297] as an abstraction of a classical theorem of harmonic analysis, the Bochner–Schoenberg–Eberlein theorem [276, Theorem 1.9.1]. Since their introduction, these notions have attracted a considerable amount of attention and have been widely studied, for example, in [63, 200, 297, 298, 299]. Two of the key notions that will be studied here are BSE algebras and BSE norms on Banach function algebras. Another key feature of Banach algebras is the existence of two ‘Arens products’, which we shall call  and , on the bidual A of a Banach algebra A; the Banach space A is a Banach algebra with respect to both of these products, and A is a closed subalgebra of (A , ) and (A , ). These products were first defined by Arens in 1951 [7, 8], and have been intensively studied in subsequent decades. The Banach algebra A is Arens regular if these two products coincide on A , and strongly Arens irregular if the opposite extreme occurs. This will be the topic of Chapter 6. The BSE norm on a natural Banach function algebra A on a locally compact space K is in fact the restriction to A of the quotient norm on the algebra Q(A) := (A , )/L(A)⊥ , which is also a Banach function algebra on its character space; the algebra C BS E (A) is the closed subalgebra Q(A) ∩ C b (K) of Q(A), and so C BS E (A) is a Banach function algebra on K. This shows the close connection between BSE norms and Arens products on the bidual space A . The problems that have been studied concerning the above notions, and those that we introduce here, include the following. Here A is a natural Banach function algebra on a non-empty, locally compact space K. 1.

Determine the algebras C BS E (A) and M(A), and decide when they are equal, i.e., whether or not A is a BSE algebra.

2.

Determine when A has a BSE norm (and when its given norm is equal to || · ||BSE ).

3. 4.

Characterize those A such that C BS E (A) = C b (K). Study the stability of BSE algebras under various Banach-algebra constructions.

Preface

vii

Explore the topology of pointwise convergence τ p on the character space ΦA , and in particular consider the closures of the closed unit balls A[1] and M(A)[1] in the space (C b (ΦA ), τ p ). 6. Define and study the properties of the Banach function algebra that is the quotient Q(A) = A /L(A)⊥ of A . 7. Determine when A is Arens regular or strongly Arens irregular or neither, and how many points are necessary to ‘determine the topological centre’. 8. Determine the answers to the above questions in the special cases where A is a Banach sequence algebra, where A is an ideal in its bidual, where A is Arens regular, or where A is a dual Banach algebra (and determine the relationship of these additional notions). Determine when the algebra C BS E (A) has any or all of these various properties.

5.

9.

In particular, determine the answers to the above questions for standard Banach function algebras such as group algebras, (weighted) semigroup algebras, Banach sequence algebras, Fourier algebras, Fourier–Stieltjes algebras, FigàTalamanca–Herz algebras, uniform algebras, and projective tensor products of Banach function algebras, including the Varopoulos algebra.

In this text, we shall present a considerable number of new results about these problems; we shall also include some known results, with proofs when the result is particularly relevant to our main aims or not in a standard text; some results previously noted for special algebras are proved in greater generality. Throughout we shall assume knowledge of standard functional analysis, algebra, measure theory, and a little complex analysis, although we shall often recall the basic definitions, mainly to fix notation, and some relevant theorems. In Chapter 1, we shall establish our notation and give some basic results, and in particular we shall recall basic properties of Banach spaces, operators on Banach spaces, and tensor products of Banach spaces, often without proofs. In Chapter 2, we shall introduce suitable background on general Banach algebras and especially C ∗ -algebras. Then, in §2.3, we shall give a detailed account of the two Banach algebras, (A , ) and (A , ), that are the bidual algebras of a Banach algebra A, and, in §2.4, we shall consider when a Banach algebra is a dual Banach algebra. In Chapter 3, we shall concentrate on Banach function algebras. In particular, in §3.2, we shall discuss Banach sequence algebras and, in §3.3, the projective tensor product of two Banach function algebras. In §3.4, we shall study variants of the separating ball property, and in §3.5 the concept of pointwise approximate identities. An important class of Banach function algebras is that of uniform algebras, which have been intensively studied; these will be discussed in §3.6, where we shall be particularly concerned with Gleason parts. In Chapter 4, we shall give background on Banach algebras that are based on a locally compact group G, including the group algebra L1 (G) and the measure algebra M(G). In §4.3, we shall discuss Fourier and Fourier–Stieltjes algebras, A(Γ) and B(Γ), that are both Banach function algebras on a locally compact group Γ. There is a

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generalization of these algebras, the Figà-Talamanca–Herz algebras A p (Γ), defined for 1 < p < ∞, and they will be introduced in §4.4. In the study of the above concepts, the notions of various approximate identities in a Banach function algebra and in its maximal ideals will be important. We shall define contractive and pointwise contractive Banach function algebras in §3.1 and §3.5, respectively. In §3.6, we shall show that a unital uniform algebra is contractive if and only if it is a Cole algebra, which means that every point of its character space is a strong boundary point for the algebra, and then we shall prove in Theorem 5.4.12(i) the stronger result that a contractive, unital Banach function algebra which has a BSE norm is equivalent to a Cole algebra. In Theorem 5.4.12(ii), we shall show that pointwise contractive unital Banach function algebras which have a BSE norm are equivalent to uniform algebras in which every Gleason part is a singleton. However, we shall show in Example 5.4.16 that there are contractive Banach function algebras whose BSE norm is equal to the uniform norm, and hence is not equivalent to the given norm. In our final chapter, Chapter 6, we shall consider whether various Banach algebras A are Arens regular or strongly Arens irregular, and consider when ‘small’ subsets of A determine the left topological centre. In particular, we shall consider these questions in §6.2 when A = A⊗ B is the projective tensor product of two Banach algebras, A and B, that are Arens regular, and in §6.3 when (weighted) semigroup algebras have these properties. We shall conclude by showing in §6.4 that each group algebra is strongly Arens irregular, with a two-point subset that determines the left topological centre.

Lancaster, UK ˙Istanbul, Türkiye July 2022

Harold Garth Dales Ali Ülger

Acknowledgements

The authors thank David Blecher, Yemen Choi, Matthew Daws, Haresh Dedania, Joel Feinstein, Mahmoud Filali, Jorge Galindo, Fereidoun Ghahramani, Alexender Izzo, Marcej de Jeu, Anthony To-Ming Lau, Matthias Neufang, Hung Le Pham, Ajit Iqbal Singh, Dona Strauss, and Vladimir Troitsky for enjoyable conversations, and valuable comments on the draft version of the text. The first author is very grateful to the London Mathematical Society for travel grants that helped him to visit Istanbul for discussions on this work and to the second author for generous hospitality in Istanbul on many occasions. We would like to thank sincerely the referees of the first and the second reviews of the book for their time and for their many valuable comments. The authors are very grateful to the Editor, Professor Keith Taylor, for handling the manuscript, and for his patience during this lengthy process. We also express our thanks to the personnel of Springer, especially to Donna Chernyk and Jayanthi Narayanaswamy, for their professional works. A couple of months after we have completed the book and submitted the manuscript to the editor, Garth Dales passed away in October 2022—an enormous loss to the mathematical community. We have been working together since 2012; that was an enormous loss for me too. I lost both a friend and a colleague. I am deeply grateful to Garth Dales for his many years of hard work in preparation of the book; without his diligence this book would not have been published.

ix

Contents

1

Banach spaces and operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Basic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Bounded linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 16 39 63

2

Banach algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.1 Algebras and Banach algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.2 C ∗ -algebras and von Neumann algebras . . . . . . . . . . . . . . . . . . . . . . 94 2.3 Biduals of Banach algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 2.4 Dual Banach algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

3

Banach function algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Banach function algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Banach sequence algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Projective tensor products of Banach function algebras . . . . . . . . . 3.4 The separating ball property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Pointwise approximate identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Uniform algebras and Gleason parts . . . . . . . . . . . . . . . . . . . . . . . . .

151 152 170 181 189 197 208

4

Banach algebras on locally compact groups . . . . . . . . . . . . . . . . . . . . . . 4.1 Group and measure algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Locally compact abelian groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Fourier and Fourier–Stieltjes algebras . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Figà-Talamanca–Herz algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

229 230 253 270 285

5

BSE norms and BSE algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The space L(A) and the algebra Q(A) . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Basic definitions and results about BSE norms . . . . . . . . . . . . . . . . 5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

293 294 309 326

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5.4 5.5 6

Pointwise contractive Banach function algebras and 1 -norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 Banach function algebras that are ideals in their bidual . . . . . . . . . 349

Arens regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Topological centres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Arens regularity of projective tensor products . . . . . . . . . . . . . . . . . 6.3 Biduals of semigroup algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Arens regularity for algebras on locally compact groups . . . . . . . .

359 359 374 383 406

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 Index of Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 Index of Theorems and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 Index of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

Chapter 1

Banach spaces and operators

In this first chapter, we shall recall some notation and preliminary results within the functional analysis and related subjects that we shall use. In §1.1, we shall establish some basic notation, touching upon lattices, semigroups, linear spaces and linear maps, standard topology, topological semigroups, ˇ measure spaces, Boolean algebras, continuous functions, and the Stone–Cech compactification of a completely regular topological space. Our main starting point will be the basic theory of Banach spaces, their dual spaces, and the relevant weak and weak-∗ topologies, to be given in §1.2; we shall briefly recall a number of classical theorems, and some related topics. We shall then, in §1.3, move to the Banach space of bounded linear operators between two Banach spaces, and consider various closed linear subspaces of this Banach space, especially those consisting of the compact and of the weakly compact operators. Finally, in §1.4, we shall recall the basic features of the projective and injective tensor products of two Banach spaces.

1.1 Basic notation In this section, we shall first recall some basic notation and general results that will be used throughout this work. In particular, we shall define the space C 0 pKq of continuous functions vanishing at infinity on a non-empty, locally compact space K; this will be the ambient setting for the Banach function algebras, to be defined in Chapter 3. Of course, C 0 pKq, with respect to the uniform norm | · |K , is the canonical commutative C ∗ -algebra. We set: N “ {1, 2, 3, . . . }, the set of natural numbers; we write Z for the set of integers, so that pZ, +q is an abelian group, and then set Z+ “ {0, 1, 2, . . . }, Z´ “ {0, ´1, ´2, . . . }, and Z• “ Z \ {0}. For n P N, we set Nn “ {1, . . . , n} and Z+n “ {0, 1, . . . , n}. We denote the real line by R, and the closed unit interval in R by I “ © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 H. G. Dales and A. Ülger, Banach Function Algebras, Arens Regularity, and BSE Norms, CMS/CAIMS Books in Mathematics 12, https://doi.org/10.1007/978-3-031-44532-3_1

1

2

1 Banach spaces and operators

r0, 1s; further, R+ “ {r P R : r ě 0} and R+• “ {r P R : r > 0}. The set of rational numbers is Q, with obvious definitions of Q+ , Q+• , and Q• ; when we write p/q for an element of Q• , we suppose that p P Z, that q P N, and that |p| and q are coprime integers. For s, t P R, we set s ∧ t “ min{s, t} ,

s ∨ t “ max{s, t} .

Take p with 1 < p < 8. Then the conjugate index to p is denoted by p , where 1/p + 1/p “ 1; we also regard 8 and 1 as the conjugates of 1 and 8, respectively. The complex plane is C, and C• “ C \ {0}; D “ {z P C : |z| < 1} is the open unit disc in C, and T “ {z P C : |z| “ 1} is the unit circle. For z P C, we shall frequently write z “ x + iy, where x, y P R. We set Π “ {z “ x + iy P C : x > 0} , the open right-hand half-plane, with closure Π. The coordinate functional on C is the function Z : z Ñ z, and the coordinate projections on Cn are denoted by Z1 , . . . , Zn .  Proposition 1.1.1 Let {α1 , . . . , αn } be a finite set in C such that | {αi : i P F}| ď 1 n for each non-empty subset F of Nn . Then i“1 |αi | ď 4. Proof First suppose that α1 , . . . , αn P R, and set F “ {i P Nn : αi < 0}. Then clearly n 

    |αi | “  {αi : i P F} +  {αi : i P Nn \ F} ď 2 .

i“1

In the general case, set αi “ βi + iγi for i P Nn , where βi , γi P R. For each subset F of Nn , we have           {βi : i P F} ∨  {γi : i P F} ď 1  {αi : i P F} +  {αi : i P F} ď 1 , 2 n n n and so i“1 |αi | ď i“1 |βi | + i“1 |γi | ď 4. Let S be a non-empty set. For s, t P S , we take δ s,t to be equal to 1 when s “ t and to 0 when s  t. The cardinality of S is denoted by |S |, and the characteristic function of a subset T of S is denoted by χT ; however we shall often write 1 or 1S for χS and write χ s or δ s for the characteristic function of {s} for s P S . The set of all subsets of S is the power set of S ; it is denoted by PpS q. Let U, V P PpS q. Then UΔV “ pU \ Vq ∪ pV \ Uq is the symmetric difference of U and V. Let S and T be non-empty sets. Then T S is the set of all functions from S into T . The first infinite ordinal is ω, and the first uncountable ordinal is ω1 ; the cardinals corresponding to ω and ω1 are ℵ0 and ℵ1 , respectively, and the cardinality of the continuum is denoted byc, so that c “ 2ℵ0 , and the Continuum Hypothesis, CH, is the statement that c “ ℵ1 . Results that depend on CH are denoted by (CH).

1.1 Basic notation

3

Proposition 1.1.2 Let S and T be non-empty sets, and let f : S × T Ñ C be a function. Suppose that psα q and ptβ q are nets in S and T , respectively, such that a “ limα limβ f psα , tβ q and b “ limβ limα f psα , tβ q both exist. Then there are subsequences psαm q and ptβn q of the nets psα q and ptβ q, respectively, such that a “ limm limn f psαm , tβn q and b “ limn limm f psαm , tβn q. Proof For each α and β, set aα “ limβ f psα , tβ q and bβ “ limα f psα , tβ q, respectively. We claim that there exist subsequences psαm q of psα q and ptβn q of ptβ q such that:     a ´ aαm  < 1/m ; b ´ bβn  < 1/n ;   bβn ´ f psαm , tβn q < 1/m pn P Nm´1 q ;   aαm ´ f psαm , tβn q < 1/n pm P Nn´1 q .     First choose α1 and β1 so that a ´ aα1  < 1 and b ´ bβ1  < 1. Now assume that α1 , . . . , αk and β1 , . . . , βk have been chosen appropriately. Choose αk+1 so that   a ´ aαk+1 
0, there   exists U P N x such that  f pyq ´ f pxq < ε whenever y P U and f P F ; the family F is equicontinuous on X if it is equicontinuous at each point of X. The following is a form of the Ascoli–Arzelà theorem; see [90, Theorem IV.6.7] or [279, p. 369], for example. Theorem 1.1.11 Let K be a non-empty, compact space. Suppose that F is a nonempty family in CpKq that is pointwise bounded and equicontinuous. Then F is relatively compact in pCpKq, | · |K q.

1.1 Basic notation

15

A Hausdorff topological space X is completely regular if, for each x P X and each U P N x , there exists f P CpXq with f pxq “ 1 and supp f Ă U. Thus X is completely regular if and only if the cozero sets are a basis for the topology. Each locally compact space is completely regular; the product of two completely regular spaces is completely regular; each subspace of a compact space is completely regular. Let X be a non-empty, completely regular topological space. A fundamental ˇ compactification theorem [139, Theorem 6.5] states that X has a Stone–Cech compactification β X with the following equivalent properties: (a) every continuous mapping from X into a compact space K has a continuous extension from β X into K; (b) each f P C b pXq has an extension to a function f β P Cpβ Xq; (c) disjoint zero sets in X have disjoint closures in β X. Further, β X is unique in the sense that any compactification with the stated properties is homeomorphic to β X by a homeomorphism that leaves each point of X fixed. Let U be a subset of X. Then the closure of U in β X is denoted by U, and we set U ∗ “ U ∩ pβ X \ Xq, so that U ∗ is the growth of U in β X. In particular, X ∗ “ β X \ X. For each completely regular topological space X, the space β X is a Stonean space. Note that, when X is not compact, the non-empty subset X ∗ of β X does not contain any isolated point, and so β X is not scattered. ˇ In particular, we shall consider the Stone–Cech compactification β S of a discrete set S . In this case, the family of sets of the form U for subsets U of S coincides with κ the family of clopen subsets of β S . Note that |β S | “ 22 when |S | “ κ, so that |β N| “ |N∗ | “ 2c . ˇ The Stone–Cech compactification of a discrete set S can also be characterized in terms of ultrafilters. Here the power set PpS q is a complete Boolean algebra with respect to the operations of union and intersection. An ultrafilter with respect to the Boolean algebra PpS q is exactly an ultrafilter on the set S , and so the Stone ˇ space, S tpPpS qq, is immediately identified with the Stone–Cech compactification β S . Indeed, for p P β S , set ! Up “ T Ă S : p P T , where the closure is taken in β S . Then, for each p P β S , the family U p is an ultrafilter on S which converges to p, and the map p Þ→ U p is a bijection from β S onto the family of all ultrafilters on S . Let F be the family of finite subsets of a set S , so that F is an ideal in the Boolean algebra PpS q; the quotient Boolean algebra is PpS q/F , and it is easy to see that the Stone space S tpPpS q/F q is identified with the growth S ∗ of S in β S . Let S 1 and S 2 be two non-empty sets. Then there are two projections π j : ps1 , s2 q Þ→ s j ,

S1 × S2 Ñ S j ,

for j “ 1, 2; these maps have continuous extensions to maps π j : β pS 1 ×S 2 q Ñ β S j , and so there is a continuous surjection

16

1 Banach spaces and operators

π : u Þ→ pπ1 puq, π2 puqq,

βpS 1 × S 2 q Ñ β S 1 × β S 2 .

(1.1.6)

This map is a homeomorphism if and only if at least one of the two sets S 1 and S 2 is finite. Finally in this section, we mention a basic theorem of complex analysis that we shall use. Let U be a non-empty, open subspace of C. Then OpUq denotes the algebra of all analytic functions on U. Then OpUq is a Fréchet algebra with respect to the compact-open topology; see [50, §2.2], for example. A family F in OpUq is a normal family if every sequence in F has a subsequence that converges in this compactopen topology. The following is Montel’s theorem; see [278, Theorem 14.6], for example. Theorem 1.1.12 Let U be a non-empty, open subspace of C, and let F be a family in OpUq such that sup{| f |K : f P F } < 8 for each compact subspace K of U. Then F is a normal family.

1.2 Banach spaces We shall now give some background on the theory of normed and Banach spaces, recalling the statement of a number of classic theorems of functional analysis that we shall use. A particularly central example to be considered will be the Banach space pC 0 pKq, | · |K q, described above, and its dual space of measures, denoted by MpKq. We shall also briefly mention Banach lattices. There is an enormous literature on Banach-space theory; standard texts include [2, 4, 51, 76, 81, 84, 90, 99, 118, 207, 208, 226, 227, 238, 279, 283]. Let E be a Banach space, with dual space E  and bidual space E  . The appropriate weak and weak-∗ topologies on E and E  , and the classic theorems on these spaces will be used throughout the text; a particularly useful result will be Grothendieck’s iterated-limit criterion, to be given in Theorem 1.2.34. In particular, in the present section, we shall define for Banach spaces the key properties of being weakly sequentially complete and the Schur property, prove that the space  1 pS q has the Schur property, define property pVq of Pełczy´nski, the Radon–Nikodým property, and the Grothendieck property; these notions will all be used later. Definition 1.2.1 Let E be a linear space over a field K. Then · : E Ñ R+ is a norm if: x + y ď x + y px, y P Eq; if αx “ |α| x pα P K, x P Eq; if

x “ 0 implies that x “ 0. In this case pE, · q is a normed space; pE, · q is a Banach space if the metric px, yq Þ→ x ´ y is complete. Let pE, · q be a normed space. The closed ball in E that is centred at 0 and of radius r ě 0 is Errs ; in particular Er1s is the closed unit ball in E. Let S be a subset

1.2 Banach spaces

17

of E. Then S rrs “ S ∩ Errs and S denotes the closure of S in the norm topology of E. A subspace is closed in E if it is a linear subspace that is closed in pE, · q; a closed linear subspace F of E is a linear subspace that is closed in E and such that the norm on F is the restriction of the norm on E. Let F be a closed linear subspace of a Banach space pE, · q, with quotient map q : E Ñ E/F. The quotient norm is defined to be

y “ inf{ x : qpxq “ y}

py P E/Fq .

Of course, pE/F, · q is a Banach space. The space F is complemented in E if there is a closed linear subspace G of E such that E “ F ‘ G, in which case F “ E/G. It is elementary that finite-dimensional linear subspaces and closed linear subspaces of finite codimension in a Banach space E are complemented in E. Let E be a Banach space, and let F and G be linear subspaces of E. When we write E “ F ‘ G, we imply that F and G are closed linear subspaces of E, and hence that each of F and G is complemented in E; in this case, the norm on E is equivalent to the norm given by

x + y “ x + y

px P F, y P Gq .

Let E be a Banach space. Then a null sequence in E is a sequence pxn q in E such that limnÑ8 xn “ 0; the space of null sequences in E is c0 pEq, and c0 pEq is itself a Banach space for the norm defined by " "" "pxn q"" “ sup{ xn : n P N} ppxn q P c0 pEqq . More generally, let ppEn , · n qq be a sequence of Banach spaces. Then c0 pEn q is the Banach space of sequences pxn q, where xn P En pn P Nq and limnÑ8 xn n “ 0. Let {pEα , · α q : α P A} be a family of normed spaces, defined for each α in a non-empty index set A (perhaps finite). Then we shall consider the following spaces. First, set # $ "" "" 8  pEα q “ pxα : α P Aq : "pxα q" “ sup xα α < 8 . α

Similarly, for p with 1 ď p < 8, we define ⎧ ⎫ ⎞1/p ⎛ ⎪ ⎪ "" "" ⎜⎜⎜ ⎟⎟⎟ ⎪ ⎪ ⎨ p p ⎜⎜⎝ xα α ⎟⎟⎠ < 8⎬ " "  pEα q “ ⎪ : α P Aq : q px “ px . ⎪ α α ⎪ ⎪ ⎩ ⎭ α

Clearly  8 pEα q and  p pEα q are normed spaces; they are Banach spaces if each of the spaces Eα is a Banach space. We write E ‘8 F

and

E ‘p F

18

1 Banach spaces and operators

for the sum of two normed spaces E and F with the appropriate norms, etc., and, for n P N, we shall write np pEq for E n with the norm given by ⎞1/p ⎛ n "" ⎜⎜⎜ "" ⎟⎟ p ⎜ "px1 , . . . , xn q" “ ⎜⎝ xi ⎟⎟⎟⎠

px1 , . . . , xn P Eq .

i“1

We also write  p pEq and  8 pEq for  p pEn : n P Nq and  8 pEn : n P Nq, respectively, when each En is equal to E. Let pΩ, Σ, μq be a measure space. For each p with 1 ď p < 8, we define # $ % L p pΩ, μq “ L p pμq “ f : Ω Ñ C : f is Σ-measurable, | f | p dμ < 8 K

and

%

f p “

Ω

1/p | f | p dμ

p f P L p pΩ, μqq .

As usual, we identify equivalent functions f and g in L p pμq, that is, those with

f ´ g p “ 0. Then pL p pΩ, μq, · p q is a Banach space. The family of essentially bounded, Σ-measurable functions on Ω is the standard Banach space that is denoted by pL8 pΩ, μq, · 8 q “ L8 pμq , where we identify functions in L8 pΩ, μq if they are equal locally almost everywhere (with respect to μ), and we set  

λ 8 “ ess sup{λpxq : x P Ω} pλ P L8 pΩ, μqq . Suppose that μ is a σ-finite, positive measure on a second countable topological space X and that 1 ď p < 8. Then the Banach space L p pX, μq is separable [42, Proposition 3.4.5]. Let X be a non-empty topological space. We have mentioned the canonical Banach space pC b pXq, | · |X q. In particular, we shall frequently discuss the Banach space pC 0 pKq, | · |K q for a locally compact space K. Two monographs that concentrate on these Banach spaces are [51] and [293]. Definition 1.2.2 Let pE, · , ďq be such that pE, · q is a normed space and such that pE, ď q is a vector lattice. Then · is a lattice norm, and pE, · , ďq is a normed lattice, provided that x ď y whenever x, y P E and |x| ď |y|. In the case where pE, · q is a lattice norm and a Banach space, the structure pE, · , ďq is a real Banach lattice. For the theory of Banach lattices, see [1], [3], [51], [61], [227], [239], [290], [293], [305], [324], and [331], for example. Let pE, · , ďq be a normed lattice. Then the map x Þ→ |x| is continuous on pE, · q, and so all the lattice operations are continuous on pE, · q. It follows that

1.2 Banach spaces

19

E + is closed in pE, · q. Indeed, each order-closed subset of E is closed in pE, · q; in particular, every band in E is closed in pE, · q. Let pΩ, μq be a measure space, and take p with 1 ď p ď 8. The real-valued functions in the space L p pΩ, μq form the real-linear Banach space L pR pΩ, μq; we set f ď g in L pR pΩ, μq if f ptq ď gptq for almost all t P Ω to form the real Banach lattice pL pR pΩ, μq, ď q. Again, for each locally compact space K, the Banach space C0,R pKq is a real Banach lattice. In the case where K is compact, the function 1K is a strong unit for the real Banach lattice C R pKq. Suppose that E is a linear space such that E “ E R ‘ iE R for a real Banach lattice pE R , · q. Then we make the following definitions. First, set E + “ E +R . Now take z P E, say z “ x + iy, where x, y P E R , and define the modulus |z| P E + of z by  1/2 (1.2.1) |z| “ |x|2 + |y|2 (the right-hand side of (1.2.1) is well defined in E R by the ‘Yudin–Krivine functional calculus’). Alternatively, we can set |z| “ |x + iy| “ sup{x cos θ + y sin θ : 0 ď θ ď 2π} ;

(1.2.2)

the supremum always exists in E + and the two definitions of |z| coincide. We then define (1.2.3)

z “ |z| pz P Eq . We see that · is a norm on E and that pE, · q is a Banach space. This complexification of a real Banach lattice is defined to be a (complex) Banach lattice. For example,  8 , c, and  p for 1 ď p < 8 are Banach lattices. Let X be a nonempty topological space. Then pC b pXq, | · |X q is a Banach lattice, etc. Let pΩ, Σ, μq be a measure space, and take p with 1 ď p ď 8. Then L p pΩ, μq is a Banach lattice. Let E be a Banach lattice. A subset B of E is order-bounded if there exists x P E + such that |y| ď x py P Bq. A linear subspace F of E is an order-ideal in E if x P F whenever x, y P E with |x| ď |y| and y P F, so that F ∩E R is an order-ideal in E R and F is the complexification of F ∩ E R . An order-ideal F is a band or a projection band if F ∩ E R is a band or a projection band in E R , respectively. Thus an order-ideal F in E is a projection band if and only if E “ F + F d , where F d “ {x P E : |x| K |z| pz P Fq} . Definition 1.2.3 A Banach lattice E is Dedekind complete if E R is Dedekind complete as a lattice. Clearly, to show that a Banach lattice E is Dedekind complete, it suffices to show that each increasing net in E + that is bounded above has a supremum. Let pΩ, μq be a measure space, and take p with 1 ď p < 8. Then it is standard that the Banach lattice L p pΩ, μq is Dedekind complete [305, Theorem 2.1.15], and, in the case where the measure μ is σ-finite, the Banach lattice L8 pΩ, μq is Dedekind complete. By [305, Corollary 3.3.63], every band is a projection band in a Dedekind

20

1 Banach spaces and operators

complete Banach lattice, and so this holds for each Banach lattice L p pΩ, μq when 1 ď p < 8. The following well-known theorem is proved in [51, Theorem 2.3.3] and [305, Theorem 2.1.24]. Theorem 1.2.4 Let K be a non-empty, compact space. Then the Banach lattice CpKq is Dedekind complete if and only if K is Stonean. Let pE, · , ďq be a Banach lattice. Then E is order-continuous if xα Ñ x in o pE, · q whenever xα ´ → x. For example, the Banach lattices L p pΩ, μq for each measure space pΩ, μq and p with 1 ď p < 8 and the space c 0 are order-continuous. Every order-continuous Banach lattice E is Dedekind complete [305, Theorem 2.6.11], and every closed order-ideal in E is a band [305, Proposition 3.3.43], and hence a projection band. Let K be a non-empty, locally compact space. Then MpKq and M R pKq are the Banach spaces of all complex-valued and real-valued, regular Borel measures on K, respectively; here

μ “ |μ| pKq pμ P MpKqq , where |μ| is the total variation measure of μ, so that pMpKq, · q is a Banach space. A measure μ P MpKq is supported on a Borel subset B of K if |μ| pK \ Bq “ 0. The support of a measure μ P MpKq is denoted by supp μ : it is the complement of the union of the open sets U in K such that |μ| pUq “ 0, and so is a closed subspace of K. For μ, ν P M R pKq, set μ ď ν if μpBq ď νpBq pB P BK q and # pμ ∨ νqpBq “ sup{μpCq + νpB \ Cq : C P BK , C Ă B} , pB P BK q , pμ ∧ νqpBq “ inf{μpCq + νpB \ Cq : C P BK , C Ă B} , essentially following the Riesz–Kantorovich formulae of Theorem 1.1.5. Then pM R pKq, · , ďq is a real Banach lattice, and the operations ∨ and ∧ coincide with those given earlier. The space MpKq is a Banach lattice, and it is Dedekind complete. The cone of measures in MpKq taking values in R+ is denoted by MpKq+ , so that lin MpKq+ “ MpKq, and the measures μ P MpKq+ with μpKq “ 1 are the probability measures on K, forming the set PrpKq. Again, for μ P M R pKq, we define μ+ “ μ ∨ 0, μ´ “ p´μq ∨ 0, and |μ| “ μ+ + μ´ “ μ ∨ p´μq , so that μ+ , μ´ P MpKq+ , μ “ μ+ ´ μ´ , and |μ| coincides with the total variation measure. Proposition 1.2.5 Let K be a non-empty, locally compact space. Suppose that μ P M R pKq and μ “ ν1 ´ν2 , where ν1 , ν2 P MpKq+ and μ “ ν1 + ν2 . Then ν1 “ μ+ and ν2 “ μ´ .

1.2 Banach spaces

21

Proof Since μ “ ν1 ´ ν2 , it follows that |μ| ď ν1 + ν2 , and so |μ| + ν “ ν1 + ν2 for some ν P MpKq+ . Thus

ν1 + ν2 + ν “ μ + ν “ ν1 + ν2 . It follows that ν “ 0, and hence that 2ν1 “ μ + |μ| “ 2μ+ . Thus ν1 “ μ+ , and then ν2 “ μ´ . The measures μ P MpKq for which every set A with |μ| pAq > 0 contains a point x with |μ| p{x}q > 0 are the discrete measures, and the measures μ P MpKq such that μp{x}q “ 0 for each x P K are the continuous measures. The sets of discrete and continuous measures on K are denoted by Md pKq and Mc pKq, respectively; they are closed linear subspaces of MpKq, and MpKq “ Md pKq ‘1 Mc pKq .

(1.2.4)

We identify the space Md pKq with  1 pKq and with ⎫ ⎧ ⎪ ⎪   ⎪ ⎪ ⎬ ⎨ α s δ s : μ “ , μ“ |α s | < 8⎪ ⎪ ⎪ ⎪ ⎭ ⎩ sPK

sPK

where we are now regarding δ s as the measure that is the ‘point mass’ at s. Take μ P MpKq. Then μ is normal if x fα , μy Ñ 0 for each net p fα : α P Aq in pC 0 pKq+ , ďq such that fα ↓ 0 in the lattice; the space NpKq of normal measures on K is a closed sublattice of MpKq. A measure μ P MpKq is normal if and only if |μ| pLq “ 0 for each compact subspace L of K with intK L “ ∅. See [51, §4.7] for a discussion of NpKq and several examples. Definition 1.2.6 Let K be a non-empty, compact space. Then K is hyper-Stonean if K is Stonean and if the union of the supports of the normal measures on K is dense in K. For the theory of hyper-Stonean spaces, see [51, Chapter 5]; for a topological characterization of these spaces by van der Walt, see [320]. For example, let K be a Stonean space such that the set DK of isolated points of K is dense in K. Then K is certainly hyper-Stonean; it is homeomorphic to β DK . In particular, β N is a hyper-Stonean space. Let E be a locally convex space, so that E is a linear space equipped with a family P of seminorms on E that together separate the points of E, in the sense that, for each x P E • , there exists p P P with ppxq  0. We shall denote by E  the dual space of E, so that E  is the space of all continuous linear functionals on E. The action of λ P E  on x P E gives the complex number λpxq that we shall usually denote by xx, λy, and so the duality is specified by px, λq Þ→ xx, λy ,

E × E Ñ C ;

22

1 Banach spaces and operators

occasionally, we shall write xx, λyE,E  to indicate the spaces with respect to which the duality is calculated. In the case where E is a normed space, the dual space E  is itself a Banach space for the norm specified by  

λ “ sup{xx, λy : x P Er1s } pλ P E  q . We shall sometimes write ·  for the norm on the dual space E  of a normed space pE, · q. Let pE, · E q and pF, · F q be Banach spaces such that F is a"linear"subspace of E and x F ě x E px P Fq. Take λ P E  . Then λ | F P F  and ""λ | F ""F  ď λ E  ; when F is dense in E, the map λ Þ→ λ | F ,

E Ñ F ,

is an injection, and so we can regard E  as a linear subspace of F  . Let {pEn , · n q : n P N} be a sequence of normed spaces, and take p such that 1 ď p < 8. Then it is easy to see that we can identify the dual space of the normed space c 0 pEn q with  1 pEn q and the dual space of the normed space  p pEn q with  q pEn q, where q is the conjugate index to p ; both dualities are given by xx, λy “

8 

xxn , λn y px “ pxn q P  p pEn q, λ “ pλn q P  q pEn qq ,

(1.2.5)

n“1

and similarly with c 0 pEn q. Definition 1.2.7 Let E be a normed space. Then the bidual of E is pE  q “ E  . Thus E  is a Banach space. We regard E as a linear subspace of E  via the canonical embedding κE : E Ñ E  , where xκE pxq, λyE  ,E  “ xx, λyE,E 

px P E, λ P E  q ;

the space E is reflexive if κE pEq “ E  . We shall usually identify E with its image κE pEq in E  .  For example, for p with 1 ď p < 8, we have p p q “  p , and  p is reflexive  1  1  8 when 1 < p < 8. Also, c 0 “  and c 0 “ p q “  . It is a theorem of Nakano [3, Theorem 4.9], [305, Theorem 11.1.7] that a Banach lattice E is order-continuous if and only if E is an order-ideal in E  . Example 1.2.8 We give an example of a particular Banach space that will arise later. Let S be a non-empty set, and let ω : S Ñ R+• be a function. Then ⎧ ⎫ ⎪ ⎪    ⎪ ⎪ ⎨ ⎬  1  f psq ωpsq < 8⎪  pS , ωq “ ⎪ f psqδ s : f ω “ f “ . ⎪ ⎪ ⎩ ⎭ sPS

sPS

Clearly p 1 pS , ωq, · ω q is a Banach space. The dual of this Banach space is

1.2 Banach spaces

23

   1 pS , ωq “  8 pS , 1/ωq “ {λ P C S : sup{λpsq /ωpsq : s P S } < 8} , with the norm denoted by · ω , so that  

λ ω “ sup{λpsq /ωpsq : s P S } pλ P  8 pS , 1/ωqq . The duality x · , · yω is defined as follows: for f “  λ “ sPS λpsqδ s P  8 pS , 1/ωq, we set  f psqλpsq . x f, λyω “

 sPS

f psqδ s P  1 pS , ωq and

sPS

The space

  c 0 pS , 1/ωq “ λ P CS : lim λpsq/ωpsq “ 0 sÑ8

is a closed linear subspace of  8 pS , 1/ωq, and  1 pS , ωq is the dual space of c0 pS , 1/ωq, with the same duality. In the case where ω “ 1S , we obtain the space  1 pS q. Clearly  1 pS , ωq is a linear subspace of  1 pS q when ωpsq ě 1 ps P S q.

Example 1.2.9 Let pΩ, Σ, μq be a measure space, and take p with 1 ď p < 8. In the case where 1 < p < 8, we identify the dual space L p pΩ, μq with the space Lq pΩ, μq, where q “ p ; by Hölder’s inequality, f g P L1 pΩ, μq and f g 1 ď f p g q when f P L p pΩ, μq and g P Lq pΩ, μq, and the duality is given by % x f, gy “ f ptqgptq dμptq . Ω

The Banach spaces L p pΩ, μq are reflexive when 1 < p < 8. Similarly, we have L1 pΩ, μq “ L8 pΩ, μq at least when μ is σ-finite.

Definition 1.2.10 Let E and F be linear spaces. A pairing pE, Fq of E and F is a bilinear functional px, yq Þ→ xx, yy , E × F Ñ C , such that, for each x P E • , there is y P F such that xx, yy  0 and, for each y P F • , there is x P E such that xx, yy  0. Again, we shall sometimes write xx, yyE,F for xx, yy. Let pE, Fq be such a pairing, and define   py pxq “ xx, yy px P Eq

24

1 Banach spaces and operators

for y P F. Then each py is a seminorm on E, and the family {py : y P F} defines a topology on E with respect to which E is a (Hausdorff) locally convex space; this topology is called the weak topology associated with the pairing, and it is denoted by σpE, Fq. It is easy to see that σpE, Fq is the weakest topology such that each seminorm py is continuous on E; a linear functional λ on E is continuous with respect to σpE, Fq if and only if there exists y P F with λpxq “ xx, yy px P Eq, and so pE, σpE, Fqq “ F . Now let E be a normed space, so that pE, E  q is a pairing for the functional px, λq Þ→ xx, λy, E × E  Ñ C. Then the topology σpE, E  q is the weak topology on E. Let pxγ q be a net in E, and take x P E. Then limγ xγ “ x weakly (i.e., with respect to the weak topology) if and only if limγ xxγ , λy “ xx, λy pλ P E  q. The closure of a set S in E with respect to the weak topology is called the weak closure  of S , and a subset B of E is weakly bounded if sup{xx, λy : x P B} < 8 for each λ P E. Again let E be a normed space, and now consider the pairing pE  , Eq for the functional pλ, xq Þ→ xx, λy, E  × E Ñ C. The topology σpE  , Eq is the weak-∗ topology on E  . Clearly σpE  , Eq Ă σpE  , E  q; every weakly convergent net in E  is weak-∗ convergent. We have pE, σpE, E  qq “ E 

and

pE  , σpE  , Eqq “ E .

The closure of a subset B of E  in the topology σpE  , Eq will often be denoted by σ B , and we shall sometimes write wk∗ – lim λα “ λ α

to show that the net pλα q converges to λ in pE  , σpE  , Eqq. The weak-∗ topology on E  is σpE  , E  q. Proposition 1.2.11 Let E be a normed space, and let S be a relatively weakly compact subset of E  . Then the weak and weak-∗ closures of S in E  coincide. Proof Let S 1 and S 2 be the weak and weak-∗ closures of S , respectively. Then S 1 Ă S 2 . Take λ P S 2 . There is a net pλα q in S such that λα Ñ λ weak-∗. Since S is relatively weakly compact, we may suppose that λα Ñ μ weakly for some μ P S 1 . Clearly μ “ λ, and so λ P S 1 . Hence S 1 “ S 2 . Let E be a normed space, and suppose that wk∗ – limα λα “ λ in E  . Then

λ ď lim inf λα . α

(1.2.6)

  xλ, xy > λ ´ ε, and then there with Indeed, take ε > 0. There exists x P E r1s     exists an index αε such that xλα , xy > xλ, xy ´ ε pα ě αε q. This implies that lim inf α λα ě λ ´ 2ε. This holds for each ε > 0, and so (1.2.6) follows.

1.2 Banach spaces

25

The dual space of the Banach space pC 0 pKq, | · |K q is identified with MpKq by the Riesz representation theorem. For details of this theorem, see [23, §§7.10, 7.11] and the classic texts of Halmos [160] and Rudin [278, Theorem 6.19], for example. Theorem 1.2.12 Let K be a non-empty, locally compact space. Then MpKq is the dual space of C 0 pKq, with the duality % f psq dμpsq p f P C 0 pKq, μ P MpKqq , x f, μy “ K 

so that C 0 pKq “ MpKq. Further, C 0 pKq acts on MpKq: we define f · μ P MpKq for f P C 0 pKq and μ P MpKq by setting % f psqgpsq dμpsq “ x f g, μy pg P C 0 pKqq , xg, f · μy “ K

so that f · μ ď | f |K μ . For each λ P B b pKq, define κpλq on MpKq by % λpsq dμpsq pμ P MpKqq . xκpλq, μy “

(1.2.7)

K

We see immediately that κpλq P MpKq “ C 0 pKq . We shall frequently use the following forms of the Hahn–Banach theorem; see [4, Theorem 3.6 and Corollaries 3.4, 3.14, 3.15, and 3.27], etc. Theorem 1.2.13 Let F be a closed linear subspace of a normed space E, and take λ P F  . Then there exists μ P E  with μ | F “ λ and μ “ λ . Suppose that x0 P E with dpx0 , Fq > 0. Then there exists λ P E  with λ | F “ 0, with λ “ 1, and with xx0 , λy “ dpx0 , Fq. Let E be a normed space, and take x P E. Then it follows from Theorem 1.2.13  with x “ xx, λy, and so that there exists λ P Er1s    };

x “ max{ xx, λy  : λ P Er1s

 we shall use this implicitly many times. A subset S of Er1s is a norming subset if

 

x “ sup{ xx, λy  : λ P S } .

 ; in this case, A linear subspace F of E  is norming if Fr1s is a norming subset of Er1s   E embeds isometrically into F . Suppose that E “ G for a Banach space G, and regard G as a closed linear subspace of E  . Then G is a norming subspace of E  .

26

1 Banach spaces and operators

Corollary 1.2.14 Let E be a Banach space, and let F be a linear subspace of E  . Then F is a norming subspace of E  if and only if the set Fr1s is weak-∗ dense in  . Er1s Theorem 1.2.15 Let E be a locally convex space. (i) Let F be a linear subspace of E, and take λ P F  . Then there exists μ P E  with μ | F “ λ. In particular, for each x P E • , there exists μ P E  with xx, μy  0. (ii) Let C be an absolutely convex, closed subset of E, and take x0 P E \ C. Then   there exists λ P E  with xx, λy ď 1 px P Cq and xx0 , λy > 1. Corollary 1.2.16 Let E be a locally convex space. Then a linear functional λ on E is continuous if and only if ker λ is closed in E. We shall also use many times the following basic Banach–Alaoglu theorem [4, Theorem 3.21] and Goldstine’s theorem, a corollary of the Hahn–Banach theorem [4, Corollary 3.30]. Theorem 1.2.17 Let E be a normed space.  is weak-∗ compact in E  for each r ě 0. (i) The closed ball Errs

(ii) For each m ě 0 and M P E  with M “ m, there is a net in Erms that converges to M in the weak-∗ topology. For the following result, see [90, V.5.6]. Proposition 1.2.18 Let E be a normed space. Then the following conditions on a linear functional M on E  are equivalent: (a) M P κE pEq; (b) M is weak-∗ continuous on E  ;  . (c) M is weak-∗ continuous on Er1s Definition 1.2.19 Let pE, Fq be a pairing of linear spaces. Let V be a linear subspace of E. The annihilator V K of V in F is V K “ {y P F : xx, yy “ 0 px P Vq} Let V be a linear subspace of F. Then the pre-annihilator V J of V in E is V J “ {x P E : xx, yy “ 0 py P Vq} . Proposition 1.2.20 Let pE, Fq be a pairing of linear spaces, and let V be a closed linear subspace of E. Then V “ V KJ .

1.2 Banach spaces

27

Proof Certainly V Ă V KJ . Take x P E \ V. By Theorem 1.2.15(i), there is an element y P F with y P V K and xx, yy  0, and so V KJ Ă V. Hence V “ V KJ . Corollary 1.2.21 Let E be a Banach space. (i) Suppose that F is a closed linear subspace of E. Then F “ F KJ . (ii) Suppose that F is a linear subspace of E  . Then F JK is the weak-∗ closure of F in E  . Proof This follows by applying Proposition 1.2.20 to the pairings pE, E  q and pE  , Eq, respectively. Let F be a closed linear subspace of a Banach space E, and regard F as a subspace of E  . Then F  is equal to the weak-∗ closure of F in E  . Proposition 1.2.22 Let F be a closed linear subspace of a Banach space E. Then F  is identified with the weak-∗ closed linear subspace F KK of E  . Proposition 1.2.23 Let E, F, and G be three Banach spaces such that F is a closed linear subspace of E and G is a closed linear subspace of F  with G “ E. Then Fr1s is σpE, Gq-dense in Er1s . Proof Note that xx, yyF,F  “ xy, xyG,E px P F, y P Gq. Write C for the σpE, Gq-closure of Fr1s in E, so that C Ă Er1s . Assume to the contrary that C  Er1s , and take  x0 in Er1s \ C. By Theorem 1.2.15(ii), there exists  y P pE, σpE, Gqq “ G with xy, xyG,E  ď 1 px P Cq, but such that xy, x0 yG,E > 1. Since y P G Ă F  , we have    

y “ sup{xx, yyF,F   : x P Fr1s } “ sup{xy, xyG,E  : x P Fr1s } ď 1 ,   and so xy, x0 yG,E  ď 1, a contradiction. Thus C “ Er1s . For an example of the above situation, take E “ MpRq and F “ L1 pRq, so that F is a closed linear subspace of E and F  “ L8 pRq. Also take G “ C 0 pRq Ă L8 pRq, so that G is a closed linear subspace of F  , and G “ E by Theorem 1.2.12. Take f P L1 pRq and λ P C 0 pRq. Then % f ptqλptq dt x f, λyF,F  “ xλ, f yG,E “ R

and L pRqr1s is weak-∗ dense in MpRqr1s . We shall use the following classical theorems; see [2, Theorem 1.6.3 and Appendix F] or [90, V.5.7, V.2.6, V.6.1, and V.6.4], or [238, Chapter 2], for example. 1

28

1 Banach spaces and operators

Theorem 1.2.24 Let E be a normed space. (i) (Banach–Dieudonné) A convex set C in E  is weak-∗ closed if and only if  is weak-∗ closed for each r > 0. C ∩ Errs (ii) (Mazur) For each convex set C in E, the norm and weak closures of C in E are equal. (iii) (Mazur) Let K be a compact subset of E. Then aco K is also compact. (iv) (Krein–Šmulian) Let K be a weakly compact subset of E. Then aco K is also weakly compact. (v) (Eberlein–Šmulian) A subset of E is relatively weakly compact if and only if it is relatively weakly sequentially compact. In particular, it follows from (ii) that, for a Banach space E, the closed unit ball Er1s is weakly closed. We note that it follows from Theorem 1.2.24(v) that a Banach space is reflexive if and only if each bounded sequence has a weakly convergent subsequence. Example 1.2.25 Consider a sequence pFn q of Banach spaces, each with the weak & topology, σpFn , Fn q, and set F “ nPN Fn , the product space, taken with the product topology. Then F is a locally convex space. Further, the weak topology σpF, F  q is the same as the above product topology. Let E be a closed linear subspace of pF, σpF, F  qq. Then it follows from the Hahn–Banach theorem that the weak topology σpE, E  q on E is equal to the restriction of the weak topology σpF, F  q to E. In particular, set E “  8 pFn q, regarded as a linear subspace of F. Then E is a closed subspace of pF, σpF, F  qq. For n P N, take Ln to be a weakly compact subset & Ln , so that L Ă Er1s . Then it follows from Tychonoff’s of pFn qr1s , and set L “ product theorem that L is weakly compact in E.

Let E be a real Banach lattice, with dual Banach space E  . Then E  is ordered by the requirement that λ P E  belongs to pE  q+ if and only if xx, λy ě 0 px P E + q. One checks easily that this ordering gives a lattice ordering, and so E  becomes a real Banach lattice. The equations that define the lattice operations are the Riesz– Kantorovich formulae, as in Theorem 1.1.5. Indeed, take λ, μ P E  . Then xx, λ ∨ μy and xx, λ ∧ μy are defined for x P E + by # xx, λ ∨ μy “ sup{xy, λy + xz, μy : y, z P E + , y + z “ x} , (1.2.8) xx, λ ∧ μy “ inf{xy, λy + xz, μy : y, z P E + , y + z “ x} , and then λ ∨ μ and λ ∧ μ are extended linearly to all of E  . The dual of a real Banach lattice E is also a real Banach lattice for these operations. To see this, take λ, μ P E  with |λ| ď |μ|. Then, for each x P E, we have     λpxq ď |λ| p|x|q ď |μ| p|x|q “ sup{μpyq : y P E, |y| ď |x|} ď μ x ,

1.2 Banach spaces

29

and so λ ď μ . This proves that E  is a real Banach lattice. This is the dual Banach lattice of E. It is standard that a dual Banach lattice is Dedekind complete. Indeed, let E be a Banach lattice and take F to be a non-empty, bounded subset of pE  q+ . Consider the net G of finite subsets of F , and, for each x P E + , set A ' E  λpxq “ lim x, S :S PG . Then λ is additive, homogeneous, and positive on E + , and thus λ extends uniquely ( to an element, also λ, of E  . Clearly λ “ {F : F P F }, and so E  is Dedekind complete. Let F be a real Banach lattice, and set E “ F ‘ iF, its complexification. Let λ be a continuous, real-linear functional on F. Then λ extends to a continuous, complexlinear functional, also denoted by λ, on E: indeed, we define λpx + iyq “ λpxq + iλpyq px, y P Fq , and so we may regard F  as a real-linear subspace of E  . For each λ in E  , there exist λ1 and λ2 in F  such that λpxq “ λ1 pxq + iλ2 pxq px P Fq, and so E  is isomorphic as a complex Banach space to the complexification F  ‘ iF  . In fact, this identification is isometric; the details of this are given in [239, Proposition 2.2.6], for example. Since F  ‘ iF  is a Banach lattice, we can now conclude that E  is a Banach lattice. Hence the dual of a Banach lattice is a Banach lattice. Thus we obtain the dual Banach lattice of a Banach lattice. Note that positive measures on a locally compact space K correspond to positive linear functionals on C 0 pKq, in the sense that, for μ P MpKq, we have μ P MpKq+ if and only if x f, μy ě 0 p f P C 0 pKq+ q. We also note that, in the case where K is compact and μ P MpKq, we have μ P MpKq+

if and only if

x1K , μy “ μ .

(1.2.9)

Definition 1.2.26 A (real or complex) Banach lattice pE, · q is an AL-space (or abstract L-space) if

x + y “ x + y

whenever

x, y P E

with

x ∧ y “ 0,

and an AM-space (or abstract M-space) if

x ∨ y “ max{ x , y } whenever

x, y P E

with

x ∧ y “ 0.

An AM-space with a unit is an AM-space with a strong unit e such that

x “ inf{α P R+ : |x| ď αe}

px P Eq .

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1 Banach spaces and operators

For example, each space of the form L1 pΩ, μq, where pΩ, μq is a measure space, is an AL-space, and each space C 0 pKq, where K is a non-empty, locally compact space, is an AM-space. Let E be a Banach lattice. Then it is standard that E is an AL-space if and only if

x + y “ x + y

px, y P E, x K yq ,

and an AM-space if and only if

x ∨ y “ max{ x , y } px, y P E, x K yq . Every band in an AL-space is a projection band. For the following duality result, see [3, Theorem 4.23], [239, Proposition 1.4.7], or [305, §6.1], for example. Theorem 1.2.27 Let E be a Banach lattice, with dual Banach lattice E  . Suppose that E is an AL-space. Then E  is an AM-space with a unit e such that epxq “ x px P E + q. Suppose that E is an AM-space. Then E  is an AL-space. In particular, the bidual space E  is an AL-space whenever E is an AL-space. It follows that MpKq and MpKq are AL-spaces for each non-empty, locally compact space K. An identification of AL-spaces and AM-spaces will be given in Theorems 1.3.77 and 1.3.79. The following result is easily seen; the implication (b) ⇒ (a) follows from Lebesgue’s dominated convergence theorem. Proposition 1.2.28 Let K be a non-empty, locally compact space. Take a sequence p fn q in C 0 pKq and f P C 0 pKq. Then the following are equivalent: (a) limnÑ8 fn “ f weakly; (b) p fn q is bounded and limnÑ8 fn “ f pointwise on K. Proposition 1.2.29 Let K be a non-empty, compact space, and take μ P M R pKq. Suppose that pμα q is a net in M R pKq such that μα ď μ for each α and such that wk∗ – limα μα “ μ in M R pKq. Then wk∗ – limβ μ+α “ μ+ in M R pKq. Proof Let pμβ q be a subnet of pμα q such that wk∗ – lim μ+β “ ν1 β

and

wk∗ – lim μ´ β “ ν2 β

in M R pKq, say, so that ν1 , ν2 P MpKq+ and μ “ ν1 ´ ν2 . Then, using (1.2.9), we have " " "" "" ""μ+ "" + "μ´ " yq “ lim x1K , ν1 y + x1K , ν2 y “ limpx1K , μ+β y + x1K , μ´ " β" β β β β "" "" “ lim "μβ " ď μ ď ν1 + ν2 “ x1K , ν1 y + x1K , ν2 y , β

1.2 Banach spaces

31

and hence μ “ ν1 + ν2 . By Proposition 1.2.5, this implies that ν1 “ μ+ and ν2 “ μ´ . The result follows. Now suppose that S is a semigroup, that s P S , and that λ P  8 pS q. We define ps · λqptq “ λptsq ,

pλ · sqptq “ λpstq pt P S q ,

(1.2.10)

so that s · λ “ λ ◦ R s and λ · s “ λ ◦ L s . Note that we have |s · λ|S ď |λ|S and |λ · s|S ď |λ|S . In the case where S is a compact, right topological semigroup, the ‘right-translation’ is transferred to the Banach space MpS q as follows: take f P CpS q, so that s · f P CpS q ps P S q, and then define x f, R s μy “ xs · f, μy

p f P CpS qq .

(1.2.11)

Clearly R s μ P MpS q+ for each μ P MpS q+ . Proposition 1.2.30 Let S be a compact, right topological semigroup. (i) Let μ P MpS q+ and s P S . Then R s μ “ μ . (ii) Let μ P M R pS q and s P S with s right cancellable. Then R s μ “ μ . Proof (i) We have R s μ “ x1S , R s μy “ xs · 1S , μy “ x1S , μy “ μ . " " (ii) We have μ “ μ+ ´ μ´ with μ “ μ+ + ""μ´ "". Take ε > 0. Then there + + are two disjoint, "" "" compact subspaces K and L of S such that μ pKq > μ ´ ε and ´ ´ μ pLq > "μ " ´ ε. The sets R s pKq and R s pLq are disjoint because R s is injective on S , and so " "

μ ě R s μ “ ""R s μ+ ´ R s μ´ "" ě pR s μ+ qpR s pKqq + pR s μ´ qpR s pLqq " " " " “ μ+ pKq + μ´ pLq > ""μ+ "" + ""μ´ "" ´ 2ε “ μ ´ 2ε . This holds for each ε > 0, and so R s μ “ μ . We now give a (special case of) a famous result of Grothendieck [157]. Definition 1.2.31 Let X and Y be non-empty sets, and let f : X × Y → C be a function. Then: (i) f clusters on X × Y if lim lim f pxm , yn q “ lim lim f pxm , yn q

mÑ8 nÑ8

nÑ8 mÑ8

whenever pxm q and pyn q are sequences in X and Y, respectively, each consisting of distinct points, and both repeated limits exist; (ii) f 0-clusters on X × Y if lim lim f pxm , yn q “ lim lim f pxm , yn q “ 0

mÑ8 nÑ8

nÑ8 mÑ8

whenever pxm q and pyn q are sequences in X and Y, respectively, each consisting of distinct points, and both repeated limits exist.

32

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Suppose that h : X × Y → C is bounded and that pxm q and pyn q are two sequences of distinct points in X and Y, respectively. Then there are subsequences pxm j q and pynk q of pxm q and pyn q, respectively, such that the two repeated limits of phpxm j , ynk q : j, k P Nq both exist. This shows that h fails to 0-cluster on X × Y if and only if there exist sequences pxm q and pyn q of distinct points in X and Y, respectively, such that one of the two repeated limits of phpxm , yn q : m, n P Nq exists and is non-zero. Let K and L be infinite compact spaces with respective dense subspaces K0 and L0 , and let f : K × L Ñ C be a bounded function. We suppose that the functions x Þ→ f px, yq, K Ñ C, and y Þ→ f px, yq, L Ñ C, are continuous for each y P L0 and each x P K0 , respectively. For y P L0 , we set fy : x Þ→ f px, yq,

K Ñ C,

so that fy P CpKq. In the next theorem, we set F “ { fy : y P L0 }. Theorem 1.2.32 Let K and L be infinite compact spaces with respective dense subspaces K 0 and L 0 , and let f : K × L Ñ C and F be as above. Then the following conditions are equivalent: (a) F is relatively weakly compact in CpKq; (b) F is relatively weakly sequentially compact in CpKq; (c) aco F is relatively weakly sequentially compact in CpKq; (d) the function f clusters on K 0 × L 0 . Proof We may suppose that | f |K×L “ 1. (a) ⇔ (b) This is the Eberlein–Šmulian theorem. (a) ⇔ (c) This follows from the Krein–Šmulian theorem. (a) ⇒ (d) Take pxm q and pyn q to be sequences in K 0 and L 0 , respectively, such that the two repeated limits of p f pxm , yn q : m, n P Nq both exist. Let h P CpKq be a weak accumulation point of { fyn : n P N}, and let x P K be an accumulation point of {xm : m P N}. Then lim lim f pxm , yn q “ lim fyn pxq “ hpxq

nÑ8 mÑ8

nÑ8

and lim lim f pxm , yn q “ lim hpxm q “ hpxq ,

mÑ8 nÑ8

mÑ8

where we are using the fact that various limits exist, and so f clusters on K 0 × L 0 . (d) ⇒ (b) By Proposition 1.2.28, it suffices to show that h P CpKq whenever h is the pointwise limit of a sequence in F . Assume that this is not the case, so that there exist x0 P K and  δ > 0 such  that each neighbourhood of x0 in K contains a point x P K 0 with hpxq ´ hpx0 q ě δ. Choose any y1 P L 0 , and then choose x1 P K 0 such that      f px0 , y1 q ´ f px1 , y1 q < 1 and hpx1 q ´ hpx0 q ě δ ;

1.2 Banach spaces

33

this is possible because fy1 P CpKq. Having specified x1 , . . . , xn in K 0 and y1 , . . . , yn  in L 0 , choose yn+1 P L 0 with  f pxi , yn+1 q ´ hpxi q < 1/n pi P Z+n+1 q, and then choose xn+1 P K 0 \ {x1 , . . . , xn } with      f px0 , yi q ´ f pxn+1 , yi q < 1/n pi P Nn+1 q and hpx0 q ´ hpxn+1 q ě δ ; this is possible because fy1 , . . . , fyn P CpKq. The sequence pxm q consists of distinct points of K 0 . Assume first that yn “ y for infinitely-many n P N. Then f pxm , yq “ hpxm q pm P Nq and limm f pxm , yq “ hpx0 q, a contradiction. Hence we may suppose that the sequence pyn q consists of distinct points of L 0 . Clearly lim lim f pxm , yn q “ lim f px0 , yn q “ hpx0 q

nÑ8 mÑ8

nÑ8

of pxm q, we and limnÑ8 f pxm , yn q “ hpxm q pm P Nq. By passing to a subsequence  may suppose that limmÑ8 hpxm q “ α for some α P D; we have hpx0 q ´ α ě δ, and so the two repeated limits of the double sequence p f pxm , yn q : m, n P Nq both exist, but are unequal, a contradiction of the hypothesis that f clusters on K 0 × L 0 . Hence h P CpKq, as claimed. Corollary 1.2.33 Let X and Y be non-empty, completely regular spaces, and let f : X × Y Ñ C be a bounded, separately continuous function that 0-clusters on ∗ X  × Y.  Take x P X and ε > 0. Then there is a finite subset F of Y such that  f px, yq < ε py P Y \ Fq. Proof We may suppose that f is a bounded function on β X × βY such that the functions x Þ→ f px, yq, β X Ñ C, and y Þ→ f px, yq, βY Ñ C, are continuous for each y P Y and each x P X, respectively. Assume towards a contradiction that there is no   such set F. Then there is a sequence pyn q of distinct points of Y with  f px, yn q ě ε pn P Nq. Since f clusters on X × Y, it follows from Theorem 1.2.32 that F is relatively weakly sequentially compact in Cpβ Xq, and so, passing to a subsequence of pyn q, there exists h P Cpβ Xq such that fyn Ñ h weakly  in Cpβ Xq. Inductively choose a sequence pxm q of distinct points in X such that  f pxm , yn q > ε/2 pn P Nm q for each m P N. We may suppose that the two repeated limits of p f pxm , yn q : m, n P Nq both exist, say α “ limnÑ8 limmÑ8 f pxm , yn q. Then |α| ě ε/2, a contradiction of the fact that f 0-clusters on X × Y. Thus the required set F exists. Let E be a non-zero Banach space. We shall apply the above Theorem 1.2.32   , σpE  , Eqq, with L “ pEr1s , σpE  , E  qq, with L 0 “ with K “ K 0 “ pEr1s pEr1s , σpE, E  qq, and with f pλ, Mq “ xM, λy pλ P K, M P Lq , so that the hypotheses of Theorem 1.2.32 are satisfied. This gives us the following Grothendieck’s iterated-limit criterion.

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1 Banach spaces and operators

Theorem 1.2.34 Let E be a Banach space, and let L be a non-empty, bounded set in E  . Then L is relatively weakly compact in E  if and only if, given any sequence pxm q in Er1s and any sequence pλn q in L, the two iterated limits lim lim xxm , λn y and

mÑ8 nÑ8

lim lim xxm , λn y

nÑ8 mÑ8

are equal whenever both exist. The following is the Krein–Milman theorem [4, Therorem 3.31]. Theorem 1.2.35 Let L be a non-empty, compact, convex subset of a locally convex space. Then L “ copex Lq. In particular, suppose that E is a normed space, and set  . Then copex Lq is dense in pL, σpE  , Eqq. L “ Er1s The next result is an immediate corollary of the Krein–Milman theorem; see [51, Corollary 4.4.6]. Corollary 1.2.36 Let K be a non-empty, locally compact space. Then the space Md pKqr1s “  1 pKqr1s is weak-∗ dense in MpKqr1s . Recall that a compact space pK, τq is completely metrizable whenever there is a countable set { fn : n P N} in CpKq • that separates the points of K. Indeed, given such a set, define d : K × K Ñ R+ by   8   fn pxq ´ fn pyq dpx, yq “ px, y P Kq . 2n | fn |K n“1 Then d is a metric on K and the identity map from pK, τq to pK, dq is clearly continuous, and so this map is a homeomorphism because pK, τq is compact. Thus pK, τq is completely metrizable. For example, let E be a separable Banach space, and set  , σpE  , Eqq. Then K is completely metrizable. K “ pEr1s Let K be a compact space. Then we note that pCpKq, | · |K q is separable if and only if K is metrizable [51, Theorem 2.1.7(i)]. We recall that a sequence pxn q in a Banach space E is weakly Cauchy if the sequence pxxn , λyq is convergent for each λ P E  . Suppose that F is a closed linear subspace of E and that pxn q is weakly Cauchy in F. Then pxn q is also weakly Cauchy as a sequence in E. The following property of Banach spaces will be used rather often. Definition 1.2.37 A Banach space E is weakly sequentially complete if every weakly Cauchy sequence in E is weakly convergent in E. A closed subspace of a weakly sequentially complete Banach space is itself weakly sequentially complete, and every reflexive space is weakly sequentially complete. An example of a non-reflexive Banach space E such that both E and E  are weakly sequentially complete is given by Bourgain and Pisier in [26]. The following result is also proved in [26].

1.2 Banach spaces

35

Proposition 1.2.38 Let E be a Banach space, with a closed subspace F. Suppose that both F and E/F are weakly sequentially complete. Then E is weakly sequentially complete. It follows easily from Proposition 1.2.28 that C 0 pKq is not weakly sequentially complete whenever K is an infinite, locally compact space. The following classical theorem is given in [81, Theorems VII.11 and VIII.12] and [90, IV.8.3 and IV.8.6], for example. Theorem 1.2.39 Let K be a non-empty, locally compact space. (i) A bounded sequence pμn q in MpKq converges weakly to μ P MpKq if and only if limnÑ8 μn pBq “ μpBq pB P BK q, and so MpKq is weakly sequentially complete. (ii) For each positive Borel measure μ on K and each p with 1 ď p < 8, the space L p pK, μq is weakly sequentially complete. The following notion will be important for us. Definition 1.2.40 A Banach space E has the Schur property if weak and norm sequential convergence coincide on E. To show that E has the Schur property, it suffices to show that every weakly null sequence in E is a null sequence. Proposition 1.2.41 Each Banach space with the Schur property is weakly sequentially complete. Proof Let E be a Banach space with the Schur property, and take a weakly Cauchy sequence pxn q in E. For each two strictly increasing sequences pmk q and pnk q in N, the " sequence " pxmk ´ xnk q converges weakly to 0. By the Schur property, limkÑ8 "" xmk ´ xnk "" “ 0, and so pxn q is a Cauchy sequence in pE, · q. Hence there exists x P E such that limnÑ8 xn “ x in pE, · q, and so limnÑ8 xn “ x weakly in E. Thus E is weakly sequentially complete. Clearly, each finite-dimensional Banach space and each closed subspace of a Banach space with the Schur property also have the Schur property, and a Banach space has the Schur property whenever every closed and separable subspace has the Schur property. It follows from Proposition 1.2.41 that C 0 pKq does not have the Schur property whenever K is an infinite, locally compact space. The following instructive proof is based on one given in [83]; see also [51, Corollary 4.5.8] and [238, Example 2.5.24]. The result is due to Tanbay [301]. Theorem 1.2.42 Let pEk , · k q be a sequence of Banach spaces such that each Ek has the Schur property. Then the Banach space  1 pEk : k P Nq has the Schur property.

36

1 Banach spaces and operators

 Proof Set E “  1 pEk : k P Nq, so that E  “  8 pEk : k P Nq, and set B “ Er1s , with the weak-∗ topology. We may suppose that E, and hence each Ek , is separable, so that pEk qr1s is completely metrizable, with metric dk , say. Take λ 0 “ pλk0 q P B. It is a consequence of earlier remarks that we may suppose that λ 0 has a base of weak-∗ neighbourhoods of the form

U “ {λ “ pλk q P B : dk pλk , λk0 q < δ

pk P NN q} ,

(1.2.12)

where δ > 0 and N P N. pnq Take a weakly null sequence pxpnq q in E. The sequence pxk : n P Nq is weakly convergent to 0 for each k P N, and hence · k -convergent to 0 because Ek has the Schur property, and so N "  "" pnq """ " xk "k “ 0 pN P Nq . nÑ8

lim

(1.2.13)

k“1

Fix ε > 0, and define Bm “

) něm

 ε  λ P B : xxpnq , λy ď 3

pm P Nq .

 Then each Bm is closed in B, Bm Ă Bm+1 pm P Nq, and {Bm : m P N} “ B, and so, by the Baire category theorem, Theorem 1.1.7, there exist m0 P N, δ > 0, and λ 0 “ pλk0 q P Bm0 such that Bm0 contains a weak-∗-open set U of the form specified in equation (1.2.12). By equation (1.2.13), we may suppose by increasing m0 that N "  "" pnq """ ε " xk "k ď 3 k“1

pn ě m0 q .

Now take n ě m0 , and define μ “ pμk q P U Ă Bm0 as follows. First, we set  μk “ λk0 pk P NN q."Then, "" for each k > N, we take an element μk P pEk qr1s " pnq pnq such that xxk , μk y “ "" xk "" ; such a μk exists by the Hahn–Banach theorem. Then k   xxpnq , μy ď ε/3, and so N " 8   "" pnq "" "" pnq """ pnq "x " ď xxk , μk y " xk "k + k“1

k“N+1

N " N     "" pnq """  pnq  pnq 0  ε ε ε   + , μy ď xx + x " k "k xxk , λk y ď + + “ ε . 3 3 3 k“1 k“1

" " Thus limnÑ8 "" xpnq "" “ 0, as required.

1.2 Banach spaces

37

Corollary 1.2.43 Let S be a non-empty set. Then the Banach space  1 pS q has the Schur property. On the other hand, it is clear that the Banach spaces  p do not have the Schur property when 1 < p < 8: in this case, the sequence pδn q is weakly convergent to the zero sequence, but the sequence is not norm convergent in  p . Similarly, the Banach space L1 pIq does not have the Schur property: the sequence pZ n | Iq is weakly convergent to the zero function, but the sequence is not norm convergent. 8 Definition 1.2.44 A series n“1 xn in a Banach space E is weakly unconditionally  8  xx , λy < 8 pλ P E  q. Cauchy if n“1

n

 Equivalently [81, Chapter V, Theorem 6], 8 n“1 xn is weakly unconditionally Cauchy if and only if there exists m > 0 such that "" "" n "" " sup "" αk xk """ ď m " nPN " k“1 8 . for each pαk q in r1s The following property was introduced by Pełczy´nski [258]. For the definitions and subsequent remarks, see [161, §§III.3.3, IV.2] and [325, §III.D.33], where more results are given.

Definition 1.2.45 Let E be a Banach space. Then E has property pVq if each nonempty, bounded subset S of E  with the property that limnÑ8 supλPS xxn , λy “ 0  for each weakly unconditionally Cauchy series 8 n“1 xn in E is relatively weakly compact. Let E be a Banach space, with a closed linear subspace F. Suppose that E has property pVq. Then E/F has property pVq, and so F also has property pVq when it is complemented in E. However, in general, F does not necessarily have property pVq. The following key result is essentially a consequence of Grothendieck’s characterization of relatively weakly compact subsets of MpKq [258, Theorem 1]. The theorem is proved in [325, Theorem III.C.31] in greater generality; the disc algebra ApDq and general uniform algebras will be defined in §3.6. Theorem 1.2.46 Let K be a non-empty, locally compact space. Then C 0 pKq has property pVq. Further, the disc algebra has property pVq. Indeed, many uniform algebras and other subalgebras of CpKq have property pVq, as proved by Saccone in [284]. (However, not all uniform algebras have property pVq; in fact, it is a result of Milne [241] that every Banach space is isometrically isomorphic to a complemented subspace of a uniform algebra.) It is a theorem of Pfitzner [260] that every C ∗ -algebra has property pVq; see Theorem 2.2.12. The following proposition and other related results are stated in [161, §3.3].

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Proposition 1.2.47 Let E be a Banach space such that E has property pVq. Then E  is weakly sequentially complete. Let E be a Banach space. Then a bounded subset B of E is dentable if, for each ε > 0, there exists x P B such that x  co pB\Bε pxqq. Following [27, Theorem 2.3.6], we say that a closed, bounded subset B of E has the Radon–Nikodým property (RNP) if every closed subset of B is dentable. Then [27, Theorem 3.6.1 and Corollary 4.2.15] show the following. Proposition 1.2.48 (i) Let E be a Banach space. Then each weakly compact subset of E has the RNP. (ii) Let E be a separable Banach space. Then each weak-∗-compact subset of E  with the RNP is norm-separable in E  , and so each weakly compact subset of E  is norm-separable in E  . Definition 1.2.49 A Banach space E has the Radon–Nikodým property (RNP) if every closed, bounded subset of E is dentable. The RNP delineates when there is an E-valued version of the standard Radon– Nikodým theorem. The texts [27] and [84] contain many different characterizations of the Radon–Nikodým property; for example, see Chapters III, IV, and VII of [84]. In particular, pages 217/218 summarize many equivalent formulations of this property, and pages 218/219 specify many Banach spaces that do and do not have this property. For example, we have the following. Proposition 1.2.50 All reflexive spaces and all separable dual spaces have the Radon–Nikodým property. For a locally compact space K, the Banach space MpKq has the Radon–Nikodým property if and only if K is scattered. The Banach spaces L1 pIq, c 0 , and  8 do not have the Radon–Nikodým property. Proof See [84], and also [283, Examples 4.2 and 4.5, and Corollaries 5.42 and 5.45], together with remarks in [283, p. 80]. The following result is [283, Proposition 5.49]. Proposition 1.2.51 Let E be a Banach space with the Radon–Nikodým property. Then every closed linear subspace of E has the Radon–Nikodým property. Thus a Banach space that contains c 0 as a closed linear subspace is not weakly sequentially complete and does not have the Radon–Nikodým property. Definition 1.2.52 Let E be a Banach space. Then E is a Grothendieck space if every weak-∗-convergent sequence in E  is weakly convergent.

1.3 Bounded linear operators

39

It is proved in [51, Theorem 4.5.6] that a space CpKq is a Grothendieck space whenever K is a Stonean space. The following result is given in [239, Proposition 5.3.11]. Proposition 1.2.53 Let E be a Banach space, and suppose that E  has property pVq. Then E  is a Grothendieck space. For a fine account by González and Kania of Grothendieck spaces and current open questions, see [141]. For example, it is not known whether every Grothendieck space has property pVq.

1.3 Bounded linear operators In this section, we shall introduce the basic definitions and the theorems that we shall use concerning bounded linear operators between normed, and especially Banach, spaces. In particular, we shall define the notions of linearly homeomorphic and linearly isometric pairs of normed spaces; we shall then state a collection of the foundational theorems of functional analysis involving bounded linear operators. We shall discuss quotients and biduals of Banach spaces, when a Banach space is a dual space and some forms of its predual, and we shall also discuss when a closed linear subspace of a Banach space is complemented in the ambient space, and, in particular, when a Banach space is complemented in its bidual. It will be relevant when we discuss the Arens regularity of Banach algebras to know when a Banach space contains a complemented copy of the spaces c 0 or of  1 . We shall then define some special subspaces of the space of bounded linear operators between two Banach spaces. Indeed, we shall discuss the spaces of finite rank, approximable, compact, weakly compact, and p-summing operators, and mention completely continuous operators; a theme will be to present some results that show that every operator between two specified Banach spaces is weakly compact, or even compact. We shall also recall the definitions of the approximation property, the bounded approximation property, and the Dunford–Pettis property for Banach spaces, and we shall include three classical fixed-point theorems that will be used. We shall conclude the section with some discussion of Banach-lattice isomorphisms and the Banach space Br pE, Fq of regular operators from a Banach lattice E to a Banach lattice F, and we shall define KB-spaces within the class of Banach lattices. Definition 1.3.1 Let E, F, and G be normed spaces. The normed space of all (bounded linear) operators from E to F is denoted by pBpE, Fq, · op q , where

40

1 Banach spaces and operators

T op “ sup{ T x : x P Er1s }

pT P BpE, Fqq ;

the operator T P BpE, Fq is a contraction if T op ď 1. We write BpEq for BpE, Eq. The normed space of all (bounded bilinear) operators from the product space E × F to G is pBpE × F, Gq, · op q. Of course, E  “ BpE, Cq, and pBpE, Fq, · op q and pBpE × F, Gq, · op q are Banach spaces whenever F and G, respectively, are Banach spaces. We shall note in §2.1 that BpEq is always a Banach algebra. Definition 1.3.2 Let E and F be two normed spaces. A bijection T : E Ñ F is a linear homeomorphism if T and T ´1 are both bounded linear operators. The spaces E and F are linearly homeomorphic, written E ∼ F, if there is a linear homeomorphism from E onto F; a homeomorphic embedding from E into F is a linear homeomorphism from E onto a linear subspace of F, and then the subspace of F is a copy of E. The two normed spaces E and F are linearly isometric, written E – F, if there is a linear isometry from E onto F; an isometric embedding from E into F is a linear isometry from E onto a linear subspace of F. Let pE, · E q and pF, · F q be two Banach spaces that are linearly isometric, with a specified identification of E and F. Then we shall sometimes write pE, · E q “ pF, · F q . For example, let X be a completely regular topological space. Then the map f Þ→ f β ,

C b pXq Ñ Cpβ Xq ,

is a linear isometry, and so pC b pXq, | · |X q “ pCpβ Xq, | · |β X q. Example 1.3.3 Suppose that E and F are Banach spaces with E ∼ F, and suppose that E is weakly sequentially complete or has the Schur property. Then F also has the corresponding property. For example, let S be a non-empty set, and let ω : S Ñ R+• be a function, as in Example 1.2.8. Then  1 pS , ωq –  1 pS q, and so  1 pS , ωq has the Schur property.

Definition 1.3.4 Let E and F be Banach spaces, and take T P BpE, Fq. The operator T fixes a Banach space G if there is a copy of G in E and the map T | G : G Ñ F is a homeomorphic embedding.

1.3 Bounded linear operators

41

Suppose that · 1 and · 2 are two norms on a linear space E. Then we say that

· 1  · 2 if there is a constant C > such that x 1 ď C x 2 px P Eq, and

· 1 ∼ · 2 when the two norms are equivalent, so that · 1  · 2 and · 2  · 1 . In this case, the identity map from pE, · 1 q to pE, · 2 q is a linear homeomorphism. We now state some famous theorems of functional analysis. The first result is the open mapping lemma [4, Lemma 3.38] and the open mapping theorem [4, Theorem 3.40]. Theorem 1.3.5 Let E and F be Banach spaces, and take T P BpE, Fq. (i) Suppose that there exists m > 0 such that T pErms q ⊃ Fr1s . Then T is a surjection. (ii) Suppose that the operator T is a surjection. Then there exists m > 0 such that T pErms q ⊃ Fr1s , so that T is an open mapping. In the case where the map T in clause (ii) is a bijection, it follows that T is a linear homeomorphism; this is Banach’s isomorphism theorem. The following is the closed graph theorem [4, Theorem 3.45]. Theorem 1.3.6 Let E and F be Banach spaces, and take T P LpE, Fq. Suppose that y “ 0 whenever limnÑ8 T xn “ y for some null sequence pxn q in E. Then T P BpE, Fq. The following is the uniform boundedness theorem [4, Theorem 3.34]. Theorem 1.3.7 Let E be a Banach space, let {Eα : α P A} be a family of normed spaces, and let T α : E Ñ Eα be a bounded operator for each α P A. Suppose that sup{ T α x : α P A} < 8 for each x P E. Then sup{ T α op : α P A} < 8. Corollary 1.3.8 Let E be a normed space. Then a subset of E is bounded if and only if it is weakly bounded. For example, each weakly Cauchy sequence is bounded. The following corollary of Theorem 1.3.7 is the Banach–Steinhaus theorem [4, Theorem 3.36]. Theorem 1.3.9 Let E and F be Banach spaces, and take a sequence pT n q in BpE, Fq. Suppose that T : E Ñ F is such that pT n xq converges to T x in F for each x P E. Then T P BpE, Fq and T op ď lim inf nÑ8 T n op .

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1 Banach spaces and operators

Let E and F be normed spaces. The dual of an operator T P BpE, Fq is denoted by T  P BpF  , E  q, so that xx, T  λy “ xT x, λy px P E, λ P E  q . Note that T  is injective if and only if T pEq is dense in F; that T is injective if and only if T  pF  q is σpE  , Eq-dense in E  ; that T pEq is closed in F if and only if T  pF  q is σpE  , Eq-closed in E  if and only if T  pF  q is closed in E  . Further, T is a surjection if and only if T  P BpF  , E  q is a homeomorphic embedding, and so E  contains a copy of F  , and T  is a surjection if and only if T is a homeomorphic embedding. The bidual of the operator T is T  “ pT  q P BpE  , F  q; of course, T  | E “ T and " " ""  "" "T "op “ ""T  ""op “ T op , and so the map

T Þ→ T  ,

BpE, Fq Ñ BpE  , F  q ,

is an isometric embedding. Note that T  is surjective if and only if T is surjective; further, T  is injective with closed range whenever T is injective with closed range, and T  : E  Ñ F  is a homeomorphic embedding whenever T : E Ñ F is a homeomorphic embedding. Let E and F be normed spaces. Then a linear map from E  to F  is weak-∗ continuous if it is continuous as a map from pE  , σpE  , Eqq to pF  , σpF  , Fqq. It is easy to see that T  is weak-∗ continuous in BpF  , E  q for each T P BpE, Fq; further, for each weak-∗-continuous operator S in BpF  , E  q, there is a unique operator T P BpE, Fq with T  “ S . In particular, T  is weak-∗ continuous in BpE  , F  q. The following result is given by Bermúdez and Kalton in [18, Proposition 2.3]. Proposition 1.3.10 Let E and F be Banach spaces, and let T P BpE, Fq be an injection. Then the following are equivalent: (a) xn Ñ 0 weakly for each sequence pxn q in Er1s such that limnÑ8 T xn “ 0; (b) the map T  : E  Ñ F  is an injection. We now make some remarks about quotient maps. Proposition 1.3.11 Let E be a Banach space, and take F to be a closed linear subspace of E  , with quotient map q : E  Ñ E  /F. Then, for each y P E  /F, there exists λ P E  with qpλq “ y and λ “ y . Proof There is a sequence pλn q in E  such that qpλn q “ y pn P Nq and such that limnÑ8 λn “ y . Then pλn q is bounded in E  , and so has a subnet pλα q that converges in pE  , σpE  , Eqq, say to λ. By equation (1.2.6), λ ď lim inf α λα “

y , and so λ has the required properties.

1.3 Bounded linear operators

43

Proposition 1.3.12 Let E be a Banach space, and take F to be a closed linear subspace of E, with quotient map q : E Ñ E/F. Then q : E  Ñ E  /F  is the quotient map. Proposition 1.3.13 Let E be a Banach space such that E  has the Schur property, and take p with 1 < p < 8. Then  p is not a quotient of E. 

Proof Assume that q : E Ñ  p is a quotient map. Then q :  p Ñ E  is a homeo morphic embedding, and so  p has the Schur property, a contradiction. Definition 1.3.14 Let E and F be normed spaces, and take T P BpE, Fq. Then T is a quotient operator if T maps the open unit ball in E onto the open unit ball in F. Let F be a closed linear subspace of a Banach space E. Then clearly the quotient map q : E Ñ E/F is a quotient operator. Proposition 1.3.15 Let E be a separable Banach space. Then: (i) there is an isometric embedding of E into  8 ; (ii) there is a quotient operator from  1 onto E . Proof (i) Let S “ {xn : n P N} be a dense subset of {x P E : x “ 1}. For each n P N, there exists λn P E  with xxn , λn y “ λn “ 1. Then the map T : x Þ→ pxx, λn yq ,

E Ñ 8 ,

is an isometric embedding. (ii) Let S “ {xn : n P N} be a dense subset of Er1s . We define T : pαn q Þ→

8 

αn xn ,

1 Ñ E .

n“1

Then clearly T is a linear contraction with T δn “ xn pn P Nq.  j Now take x P E with x 0 such that x + 8 j“1 ε < 1. First choose n1 P N such that "" x ´ xn1 "" < ε, and then inductively choose a strictly increasing sequence pnk q in N such that "" ⎞"" ⎛ k "" ⎟" ⎜⎜⎜ "" x ´ ⎜⎜⎜ ε j´1 xn ⎟⎟⎟⎟⎟""" < εk pk P Nq . j ⎠" ⎝ "" " j“1

Set α “

8

j“1

ε j´1 δn j , so that α P  1 with α 1 < 1 and x“

8 

ε j´1 xn j “ T α P T p 1 q .

j“1

This shows that T :  1 Ñ E is a quotient operator.

44

1 Banach spaces and operators

Proposition 1.3.16 Let pE, · q be a Banach space with closed linear subspaces F and G. Then the following are equivalent: (a) F + G is closed in E ; (b) pF + Gq/G is closed in E/G ; (c) there is a constant m > 0 such that, for each element x P F + G, there exist y P F and z P G with x “ y + z and y ď m x . Proof (a) ⇒ (b) This is immediate. (b) ⇒ (c) Consider the map T : y + F ∩ G Þ→ y + G ,

F/pF ∩ Gq Ñ pF + Gq/G .

This map is a linear bijection, and y + G ď y + F ∩ G py P Fq, so that T is a ´1 contraction." Thus, "" by Banach’s isomorphism theorem, the map T is also bounded. " ´1 Take m “ "T " + 1. For x P F + G with x  0, choose y1 P F and z1 P G with x “ y1 + z1 . Then " " " "

y1 + F ∩ G ď ""T ´1 "" y1 + G “ ""T ´1 "" x + G < m x , and so there exists w P F ∩ G with y1 + w < m x . Set y “ y1 + w and z “ z1 ´ w. Then y P F and z P G with x “ y + z and y ď m x , as required. (c) ⇒ (a) Let pxk q be a sequence in F + G that converges to x P E. We may suppose that xk+1 ´ xk < 1/pm · 2k q pk P Nq. Then there exist sequences pyk q in  F and pzk q in G with xk+1 ´ xk “ yk + zk and yk < 1/2k for k P N. Thus 8 k“1 yk 8 and k“1 zk converge in F and G, respectively, say to y P F and z P G. We have x “ x1 +

8 8 8    pxk+1 ´ xk q “ x1 + yk + zk “ x1 + y + z P F + G , k“1

k“1

k“1

whence F + G is closed in E, giving (a). In the next result, q : E Ñ E/G is the quotient map, and we identify F, a linear subspace of E, with a linear subspace of E/G, so that · F , as defined, is a norm on F. Corollary 1.3.17 Let pE, · q be a Banach space, and suppose that " F"and G are closed linear subspaces of E such that F ∩ G “ {0}. Set x F “ ""qpxq"" px P Fq. Then pF, · F q is closed as a subspace of E if and only if F + G is closed as a subspace of E. Proof Suppose that pF, · F q is closed as a linear subspace of E. Then clearly q´1 pFq “ F + G is closed as a subspace of E. The converse follows from Proposition 1.3.16, (a) ⇒ (c). The following result is standard [4, §3.16].

1.3 Bounded linear operators

45

Theorem 1.3.18 Let E be a Banach space, and let F be a closed linear subspace of E, with quotient map q : E Ñ E/F. (i) The map λ Þ→ λ ◦ q, pE/Fq Ñ F K , is the canonical linear isometry. (ii) Set ιpλ + F K qpyq “ xy, λy pλ P E  , y P Fq . Then ι : E  /F K Ñ F  is the canonical linear isometry, and so !  " ""  "λ + F K "" “ inf{ λ + μ : μ P F K } “ sup xζ, λy : ζ P Fr1s

(1.3.1)

for each λ P E  . In fact, by Proposition 1.3.11, given λ P E  , there exists μ P F K such that " "" "λ + F K "" “ λ + μ . (1.3.2) As an immediate corollary to Theorem 1.3.18, we see the following; it will be used frequently. Here we identify x P E with κE pxq P E  . Let E be a Banach space, and take F to be a closed linear subspace of E  , so that F K Ă E  . Then " ""   " x + F K "" “ sup {xx, λy : λ P Fr1s } px P Eq . (1.3.3) Let F be a closed linear subspace of a Banach space E. Then it follows easily that we can identify the bidual space pE/Fq with the quotient space E  /F  “ E  /F KK . Further, the canonical embedding κE/F of E/F into E  /F  is an isometric embedding, and also dpx, Fq “ dpκE pxq, F  q px P Eq, where d is the metric in E  . In particular, (1.3.4) κE pEq ∩ F  “ κE pFq . Corollary 1.3.19 Let E be a Banach space, and let F be a norming subspace of E  . Then the map x Þ→ x + F K is a linear embedding from E into F  “ E  /F K . Proposition 1.3.20 Let F be a closed linear subspace of a Banach space E. Then the linear subspace κE pEq + F  of E  is closed in E  . Proof Take x P E \ F and Λ P F  . Then x + Λ ě dpκE pxq, F  q “ dpx, Fq, and so there exists y P F with x + y ď 2 x + Λ , and this is trivial if x P F. Thus x + Λ “ px + yq + pΛ ´ yq, where x + y P κE pEq and Λ ´ y P F  , and so the condition in clause (c) of Proposition 1.3.16 is satisfied (with m “ 2). By that proposition, κE pEq + F  is closed in E  . Proposition 1.3.21 Let E and F be Banach spaces, and suppose that T : E Ñ F is a surjective operator. Then, for each μ P pker T qK , there exists λ P F  with μ “ λ ◦ T . Proof Let q : E Ñ E/ ker T be the quotient map, and take Tr : E/ ker T Ñ F to be the bijection induced by T , so that T “ q ◦ Tr. Then Tr : F  Ñ pker T qK is

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1 Banach spaces and operators

also a bijection, and so, given μ P pker T qK , there exists λ P F  with Tr pλq “ μ. Since Tr ◦ q “ T  and since q is the identity map on pker T qK , it follows that T  pλq “ Tr pλq “ μ, and so μ “ λ ◦ T . Corollary 1.3.22 Let E be a Banach space, and suppose that G is a weak-∗-closed linear subspace of E  . Then there exists a Banach space H such that G – H  and such that the topologies σpE  , Eq and σpG, Hq agree on G. Proof Consider the space GJ “ {x P E : xx, λy “ 0 pλ P Gq}, and then set H “ E/GJ . By Theorem 1.3.18(i), there is a canonical linear isometry from H  onto GJK . Since G is weak-∗ closed in E  , it follows from Corollary 1.2.21(ii) that GJK “ G, and so G – H  . It is clear that the topologies σpE  , Eq and σpG, Hq agree on G. Proposition 1.3.23 Let E and F be Banach spaces, and suppose that T P BpE  , Fq is surjective and such that T : pE  , σpE  , E  qq Ñ pF, σpF, F  qq is continuous. Then T pEq “ F.  q ⊃ Fr1s . Proof By the open mapping theorem, there exists m > 0 such that T pErms  Take y P Fr1s . Then there exists an element M P Erms with T pMq “ y. Let pxν q be a net in Erms such that limν xν “ M in pE  , σpE  , E  qq. Since the map T : pE  , σpE  , E  qq Ñ pF, σpF, F  qq is continuous, limν T xν “ y in pF, σpF, F  qq, and so the weak closure of the convex set T pErms q contains Fr1s . By Mazur’s theorem, Theorem 1.2.24(ii), T pErms q ⊃ Fr1s . By the open mapping lemma, Theorem 1.3.5(i), T pEq “ F.

Definition 1.3.24 Let E be a Banach space. Then a predual of E is a pair pF, T q consisting of a Banach space F and a linear homeomorphism T : E Ñ F  ; the predual is isometric if T is an isometry. The space E is a dual space if it has a predual. Two preduals pF1 , T 1 q and pF2 , T 2 q of E are (isometrically) equivalent if there is an (isometric) linear homeomorphism S : F2 Ñ F1 such that T 1 “ S  ◦ T 2 . A concrete predual of E is a closed linear subspace F of E  such that the map T F : E Ñ F  defined by pT F xqpλq “ xx, λyE,E 

px P E, λ P Fq

(1.3.5)

is a linear homeomorphism. Preduals and concrete preduals are discussed in [71] and [126], where the following results are given. Clearly each concrete predual of E is a predual of E. Suppose that F is a concrete predual of E. Then the above linear map T F is always a contraction, and xλ, T F xyF,F  “ xx, λyE,E 

px P E, λ P Fq .

(1.3.6)

1.3 Bounded linear operators

47

" " Further, T F x F  ď x ď c T F x F  px P Eq, where c “ ""T F´1 ""; the predual F is isometric if and only if c “ 1. The map T F : F  Ñ E  is also a linear homeomorphism such that T F | F is the identity on F. Almost always in this work, a specified ‘predual’ of a Banach space will obviously be a concrete predual, and so we shall just call it a ‘predual’. Note that, if we start with a Banach space E and form its dual E  , regarded as a closed linear subspace of E  , then E  is a concrete predual of E  . Recall that a Banach space E may be linearly isometric to E  , but not reflexive. This is the case for the James space J, to be defined in Example 3.2.10; here J – J  , but the canonical image of J in J  has codimension 1. Thus J  is an isometric predual of J, but it is not a concrete predual. Example 1.3.25 (i) Let S be a non-empty set, and let ω : S Ñ R+• be a function. We noted in Example 1.2.8 that c 0 pS , 1/ωq is a predual of the Banach space  1 pS , ωq; clearly, it is a concrete predual when considered as a closed linear subspace of the dual Banach space  8 pS , 1/ωq. In fact, the Banach space  1 has many isometric concrete preduals, no two of which are even mutually linearly homeomorphic as Banach spaces; see [51, §6.3] and [71] for a discussion of the preduals of  1 . (ii) Let K be a non-empty, locally compact space. Then it follows from the Riesz representation theorem, Theorem 1.2.12, that C 0 pKq “ MpKq, and it is clear that C 0 pKq is an isometric concrete predual of MpKq. Proposition 1.3.26 Let E be a Banach space, and suppose that pF, T q is a predual of E. Take G to be the image of the map T  ◦ κF : F Ñ E  . Then G is a concrete predual of E and pG, TG q is equivalent to pF, T q; if pF, T q is an isometric predual of E, then so is G. Further, σpE, Gq “ σpE, Fq. Proof Clearly G is a closed linear subspace of E  . Set S “ T  ◦ κF : F Ñ E  , so that S is a homeomorphic embedding and  S : E  Ñ F  is a surjection. Then xλ, pS  ◦ κE qpxqyF,F  “ xx, S λyE,E  “ xλ, T xyF,F 

px P E, λ P Fq ,

and so T “ S  ◦ κE . For x P E and λ P F, we have xλ, pS  ◦ TG qpxqyF,F  “ pTG xqpS λq “ xx, S λyE,E  “ xλ, T xyF,F  , and so S  ◦ TG “ T . It follows that TG “ pS  q´1 ◦ T : E Ñ G is a linear homeomorphism, and hence pG, TG q is a concrete predual of E that is equivalent to pF, T q. Clearly G is an isometric predual when pF, T q is an isometric predual of E. Now suppose that pxα q is a net in E. Then limα xα “ 0 with respect to the topology σpE, Fq if and only if limα xλ, T xα yF,F  “ 0 pλ P Fq, and this holds if and only if limα xxα , pT  ◦ κF qpλqyE,G “ 0 pλ P Fq, i.e., if and only if limα xα “ 0 with respect to the topology σpE, Gq. Thus σpE, Gq “ σpE, Fq.

48

1 Banach spaces and operators

Thus concrete preduals correspond to equivalence classes of preduals. Note that, when pF, T F q is a concrete predual, and we then consider the corresponding space G defined in Proposition 1.3.26, necessarily G “ F. Indeed, take λ P F Ă E  and x P E. Then xx, pT F ◦ κF qpλqyE,E  “ xλ, T F xyF,F  “ xx, λyE,E  , and hence T F ◦ κF : F Ñ E  is the inclusion map. Let E be a dual Banach space. Then E has a unique predual if any two preduals are equivalent in the sense of Definition 1.3.24. Proposition 1.3.27 Let E be a Banach space, and suppose that pF1 , T 1 q and pF2 , T 2 q are preduals of E, with corresponding concrete preduals G1 and G2 , respectively. Then σpE, F1 q “ σpE, F2 q if and only if pF1 , T 1 q and pF2 , T 2 q are equivalent if and only if G1 “ G2 . Proof Certainly pF1 , T 1 q and pF2 , T 2 q are equivalent if and only if G1 “ G2 , and σpE, F1 q “ σpE, F2 q when G1 “ G2 . Now suppose that G1  G2 , say there exists λ P G1 \ G2 . Then there exists M P E  such that xM, λy “ 1, but M P GK 2 . There is a bounded net pxα q in E such that limα xα “ M in pE  , σpE  , E  qq. But now limα xα “ 0 in pE, σpE, G2 qq and limα xxα , λy “ 1, and so it is not true that limα xα “ 0 in pE, σpE, G1 qq. Thus σpE, G1 q  σpE, G2 q, and hence σpE, F1 q  σpE, F2 q. The following result shows that there is always a small element of uniqueness about concrete preduals. Proposition 1.3.28 Let E be a Banach space, and suppose that F and G are concrete preduals of E with F Ă G. Then F “ G. Proof Assume towards a contradiction that F  G. Then there exists M P E  such that M | F “ 0 and M | G  0. Since G is a concrete predual of E, there exists x P E with xx, λy “ xM, λy pλ P Gq; in particular, x  0. However xx, μy “ 0 pμ P Fq, and so the canonical map E Ñ F  is not an isomorphism, a contradiction of the fact that F is a concrete predual of A. Hence F “ G. Let E be a Banach space. An element P P BpEq is a projection if P2 “ P, so that PpEq is a closed linear subspace of E. Suppose that F is a closed linear subspace of E. Then F is complemented in E if and only if there is a projection P P BpEq with PpEq “ F; we say that F is c-complemented (for c ě 1) if there is such a projection P in BpEqrcs . In this case, P P BpE  q, and P is a projection with range F  , so that F  is complemented in E  . For example, let H be an infinite-dimensional Hilbert space. Then we can regard  2 as a closed, 1-complemented subspace of H. Proposition 1.3.29 Let E be a Banach space such that  2 is a complemented subspace of E  . Then  2 is a quotient of E.

1.3 Bounded linear operators

49

Proof There is a projection P P BpE  q such that PpE  q “  2 . Then the operator T :“ IE  ´ P P BpE  q is also a projection, with range p 2 q “  2 . The map T : pE  , σpE  , E  qq Ñ pF, σpF, F  qq is continuous, with F “  2 , and so T pEq “  2 by Proposition 1.3.23. The following theorem originates with Sobczyk, and is given in [51, Theorem 2.4.14]. Theorem 1.3.30 Let E be a separable Banach space containing c 0 as a closed linear subspace. Then c 0 is 2-complemented in E. We shall later consider Banach spaces that are complemented in their biduals; the class of such spaces is rather large. For example, suppose that E is a dual Banach space, so that E ∼ F  for a Banach space F; as in Proposition 1.3.26, we regard F as a concrete predual of E, so that F is a closed linear subspace of E  . Then the dual of the canonical embedding of F into F  is a bounded projection P : Λ Þ→ Λ | κF pFq,

E  Ñ E ,

such that P : F  Ñ F  op “ 1 (this is the Dixmier projection), and so we can write E  “ κE pEq ‘ F K “ E ‘ F K

(1.3.7)

as a Banach space. This shows that E is complemented in its bidual. Proposition 1.3.31 Let E be a Banach space that is 1-complemented in E  and is a dual space. Then E has an isometric concrete predual. Proof Let F be a concrete predual of E. By Theorem 1.3.18(ii), ι : E  /F K Ñ F  is an isometry (when E  /F K has the quotient norm). Thus

T F x F  “ inf{ x + y : y P F K } ď x

px P Eq .

Let P : E  Ñ with P “ 1. Then, for each x P E and y P F K , we "" E be a projection "" have x “ "Ppx + yq" ď x + y , and so x ď T F x F  . Hence the map T F : E Ñ F  is an isometry, and so F is an isometric concrete predual of E. However we note the following famous theorem of Phillips [51, Theorem 2.4.11]; see also page 55. Theorem 1.3.32 The space c 0 is not complemented in its bidual  8 . More generally, the spaces C 0 pKq for a locally compact space K are not complemented in their biduals whenever K contains a convergent sequence of distinct points [51, Corollary 2.4.17]; this is the case whenever K is an infinite, compact, metrizable space. See [51] for more examples.

50

1 Banach spaces and operators

Let E be any Banach space, and regard E as a linear subspace of E  , so that E K is a linear subspace of E  , and E  as a linear subspace of E  . Then we have E  “ E  ‘ E K

(1.3.8)

as a Banach space. Let pxα q be a net in E that converges weakly to x P E. Then it follows that, also, pxα q converges weakly to x P E  . By Proposition 1.2.47, E  is weakly sequentially complete whenever E has property pVq; the following result, from [161, Proposition III.3.6], is slightly stronger. Proposition 1.3.33 Let E be a Banach space with property pVq. Then the Dixmier projection P : E  Ñ E  maps weak-∗-convergent sequences in E  to weakly convergent sequences in E  . It will be important for us later, perhaps surprisingly, to know whether a Banach space contains a copy of c 0 or of  1 , and we state a few such results here. Theorem 1.3.34 Let E be a Banach space. Then the following are equivalent: (a) E contains a complemented copy of  1 ; (b) E  contains a copy of c 0 ; (c) there is a continuous linear surjection from E onto  1 . In this case, E  contains a complemented copy of  8 . Proof (a) ⇒ (b) Certainly E  contains a complemented copy of  8 , and hence a copy of c 0 , whenever E contains a complemented copy of  1 . (b) ⇒ (a) This is given in [81, Theorem 10, p. 48] and [226, Proposition 2.e.8]. (a) ⇒ (c) This is immediate from the definitions. (c) ⇒ (a) Let Q : E Ñ  1 be a continuous linear surjection. By the open mapping theorem, Theorem 1.3.5(ii), there is a bounded sequence pxn q in E such that Qpxn q “ δn pn P Nq. The closed linear subspace F :“ lin {xn : n P N} of E is a copy of  1 , and so there is R P Bp 1 , Eq with Rpδn q “ xn pn P Nq. Then R ◦ Q : E Ñ E is a projection onto F, and so  1 is complemented in E. The next result is given in [84, p. 219]. Proposition 1.3.35 A Banach space that is the dual of a Banach space that contains a copy of  1 does not have the Radon–Nikodým property. The following Banach–Mazur theorem is proved in [2, Theorem 1.4.3]. Theorem 1.3.36 The Banach space CpIq contains an isometric copy of every separable Banach space, and, in particular, of  1 . The most important result in this area is Rosenthal’s  1 -theorem [274]; see [2, Corollary 10.2.2] or [81, Chapter XI], for example.

1.3 Bounded linear operators

51

Theorem 1.3.37 Let E be a Banach space. Then every bounded sequence in E has a weakly Cauchy subsequence if and only if E does not contain a copy of  1 . Corollary 1.3.38 Let E be a Banach space that is weakly sequentially complete. Then either E is reflexive or E contains a copy of  1 . The following theorem concerning Banach spaces of the form C 0 pKq is given in [259]; in fact, many other equivalent conditions are given in this source. We shall give one more equivalence in Corollary 1.3.71. More general results are contained in [207, §§8, 13]. Theorem 1.3.39 Let K be a non-empty, locally compact space. Then the following conditions on K are equivalent: (a) the topological space K is scattered; (b) the Banach space C 0 pKq does not contain a copy of  1 ; (c) Mc pKq “ {0} and C 0 pKq “  1 pKq. Let E and F be Banach spaces. We now turn to consideration of particular classes of operators in the Banach space BpE, Fq. First, for λ0 P E  and y0 P F, we define y0 ⊗ λ0 to be the bounded operator y0 ⊗ λ0 : x Þ→ xx, λ0 y y0 ,

E Ñ F;

(1.3.9)

clearly y0 ⊗ λ0 op “ y0 λ0 . The space that is the linear span of these continuous rank-one operators consists of the continuous, finite-rank operators; it is denoted by F pE, Fq, The dual of the above operator T “ y0 ⊗ λ0 P F pE, Fq is the operator T  : μ Þ→ xy0 , μy λ0 ,

F Ñ E ,

so that T  P F pF  , E  q. In the special case that F “ E, we note that px1 ⊗ λ1 q ◦ px2 ⊗ λ2 q “ xx2 , λ1 y x1 ⊗ λ2

px1 , x2 P E, λ1 , λ2 P E  q .

(1.3.10)

Take x0 P E, λ0 P E  , and T P BpEq. Then we also have T ◦ px0 ⊗ λ0 q “ T x0 ⊗ λ0 ,

px0 ⊗ λ0 q ◦ T “ x0 ⊗ T  pλ0 q .

(1.3.11)

Definition 1.3.40 Let E and F be Banach spaces, and take T P BpE, Fq. Then: T is compact if the closure of T pEr1s q in F is compact in the norm topology of F; T is weakly compact if the weak closure of T pEr1s q in F is compact with respect to the weak topology on F; T is weakly precompact if each sequence in T pEr1s q has a weakly Cauchy subsequence; T is completely continuous if pT xn q is convergent in F whenever pxn q is weakly convergent in E.

52

1 Banach spaces and operators

The subsets of BpE, Fq consisting of the compact operators and of the weakly compact operators are denoted by KpE, Fq and WpE, Fq, respectively, and the closure of F pE, Fq in BpE, Fq is ApE, Fq, so that ApE, Fq consists of the approximable operators; these three spaces are closed linear subspaces of pBpE, Fq, · op q, and clearly F pE, Fq Ă ApE, Fq Ă KpE, Fq Ă WpE, Fq Ă BpE, Fq . We write F pEq, ApEq, KpEq, and WpEq for F pE, Eq, ApE, Eq, KpE, Eq, and WpE, Eq, respectively. Let E be a Banach space, and suppose that pxn q is a normalized basic sequence to the unit basis of  1 if there is β > 0 in E. Then the sequence n q is equivalent ""px "" n n " " such that β k“1 |ck | ď k“1 ck xk whenever n P N and c1 , . . . , cn P C. (In this case the sequence pxn q generates in E a closed linear subspace isomorphic to  1 .) Take T P BpE, Fq. Then T is weakly precompact if and only if T pEr1s q does not contain a normalized basic sequence that is equivalent to the unit basis of  1 . Clearly each weakly compact operator is weakly precompact. Each compact operator is completely continuous, and the set of completely continuous operators is a closed linear subspace of BpE, Fq. We shall several times use the following basic result; the case of weakly compact operators follows from the Eberlein–Šmulian theorem, Theorem 1.2.24(v). Theorem 1.3.41 Let E and F be Banach spaces. Then T P BpE, Fq is [weakly] compact if and only if pT xn q has a [weakly] convergent subsequence in F for each bounded sequence pxn q in E. Suppose that E, F, G, H are Banach spaces, that R P BpE, Fq, that S P KpF, Gq, and that T P BpG, Hq. Then T ◦ S ◦ R P KpE, Hq. Similarly, T ◦ S ◦ R P WpE, Hq whenever S P WpF, Gq. The following result is given in [81, Chapter XIII, Lemma 2]. Proposition 1.3.42 Let E be a Banach space, and let K be a weakly closed subset of E. Suppose that, for each ε > 0, there is a weakly compact set Kε in E such that K Ă Kε + Erεs . Then K is weakly compact. Let E and F be Banach spaces, and take T P BpE, Fq. It is a standard fact that the operator T  is compact (respectively, weakly compact) if and only if T is compact (respectively, weakly compact); it is also standard that T is weakly compact if and only if T  pE  q Ă F and that a Banach space E is reflexive if and only if WpE, Fq “ BpE, Fq for each Banach space F. This implies the following remark. Proposition 1.3.43 Let E and F be Banach spaces. (i) Suppose that either E or F is reflexive. Then BpE, Fq “ WpE, Fq. (ii) Suppose that there is an operator in WpE, Fq that is an epimorphism. Then F is reflexive.

1.3 Bounded linear operators

53

Proposition 1.3.44 Let E and F be Banach spaces, with E  {0}. Suppose that F contains a copy of c 0 . Then KpE, Fq contains a copy of c 0 . Proof Let T : c 0 Ñ F be a homeomorphic embedding, and take λ P E  such that λ “ 1. Then the map α Þ→ T α ⊗ λ, c 0 Ñ KpE, Fq, is a homeomorphic embedding. The following characterization of compact operators is given in [90, Theorem VI.5.6]. Theorem 1.3.45 Let E and F be Banach spaces, and take T P BpE, Fq. Then T is compact if and only if limα T  pμα q “ T  pμq in pE  , · q whenever pμα q is a bounded net in F  such that limα μα “ μ in pF  , σpF  , Fqq. We shall use the following theorem of Kalton [192, Theorem 1].   and Fr1s are both weak-∗-compact Let E and F be Banach spaces, so that Er1s   spaces; we set K “ Er1s × Fr1s , a compact space. For T P KpE, Fq, define   fT pM, λq “ xM, T  pλqy “ xT  pMq, λy pM P Er1s , λ P Fr1s q.

(1.3.12)

Then it follows from Theorem 1.3.45 that fT P CpKq, and it is clear that the map θ : T Þ→ fT , KpE, Fq Ñ CpKq, is an isometric embedding. Thus a subset H of KpE, Fq is relatively (weakly) compact if and only if θpHq is relatively (weakly) compact in CpKq. Theorem 1.3.46 Let E and F be Banach spaces. Suppose that pT n q is a sequence in KpE, Fq and that T P KpE, Fq. Then T n Ñ T weakly in KpE, Fq if and only if T n pλq Ñ T  pλq weakly in E  for each λ P F  . Proof Suppose that T n Ñ T weakly in KpE, Fq, and take λ P F  . For each M P E  , the map S Þ→ xM, S  pλqy is a continuous linear functional on KpE, Fq, and so xM, T n pλqy Ñ xM, T  pλqy as n Ñ 8. Thus T n pλq Ñ T  pλq weakly in E  . Now suppose that T n pλq Ñ T  pλq weakly in E  for each λ P F  . By two applications of the uniform boundedness theorem, Theorem 1.3.7, pT n q is bounded in KpF  , E  q, and so pT n q is bounded in KpE, Fq, whence p fTn q is a bounded sequence in CpKq. By the hypothesis, p fTn q is pointwise convergent to fT on K, and so it follows from Proposition 1.2.28 that fTn Ñ fT weakly in CpKq. Thus T n Ñ T weakly in KpE, Fq. Definition 1.3.47 A Banach space F is injective if, for every Banach space G, every closed linear subspace E of G, and every T P BpE, Fq, there is a Tr P BpG, Fq such that Tr | E “ T . The "space " F is c-injective, where c > 0, if, further, there exists such a Tr P BpG, Fq with ""Tr""op ď c T op .

54

1 Banach spaces and operators

Proposition 1.3.48 The space  8 pS q “ Cpβ S q is 1-injective for each non-empty set S . Proof Take a Banach space G, a closed linear subspace E of G, and an operator T P BpE,  8 pS qq. For each s P S , the functional λ s : x Þ→ pT xqpsq on E is continuous with λ s ď T . By the Hahn–Banach theorem, each λ s has a norm-preserving extension λ˜ s to G. Set pTr xqpsq “ xx, λ˜ s y ps P S , x P Gq . " " Then Tr P BpG,  8 pS qq is an extension of T with ""Tr""op “ T op . Let K be a non-empty, compact space. Then CpKq is 1-injective if and only if K is Stonean if and only if CpKq is 1-complemented in CpKq ; see [51, Theorem 6.8.3], where other equivalent conditions are also given.  Recall that a series 8 n“1 xn in a Banach space converges unconditionally if 8 x converges (in norm) for every permutation π of N. n“1 πpnq Definition 1.3.49 Let E and F be Banach spaces, and take T P BpE, Fq. Then T is  unconditionally converging if the series 8 n“1 T xn converges unconditionally in F 8 for each weakly Cauchy series n“1 xn in E. We now give various theorems that show some circumstances in which either BpE, Fq “ KpE, Fq or BpE, Fq “ WpE, Fq for Banach spaces E and F. For the following two theorems, which are due to Pełczy´nski [258], see [81, Exercise 12, p. 116], [141], and [325, Proposition III.C.35(b)]. Theorem 1.3.50 Let E be a Banach space. Then E has property pVq if and only if, for each Banach space F, every unconditionally converging operator T P BpE, Fq is weakly compact. Theorem 1.3.51 Let E and F be Banach spaces, and take T P BpE, Fq. (i) Either T is unconditionally converging or T fixes a copy of c 0 . (ii) Suppose that E has property pVq and that F does not contain a copy of c 0 . Then BpE, Fq “ WpE, Fq. Corollary 1.3.52 Let E and F be Banach spaces which have property pVq. Then BpE, F  q “ WpE, F  q. Proof Since F has property pVq, the space F  is weakly sequentially complete by Proposition 1.2.47, and so F  does not contain a copy of c 0 . Thus the result follows from Theorem 1.3.51(ii).

1.3 Bounded linear operators

55

Corollary 1.3.53 Let K be a non-empty, locally compact space. Then each operator from C 0 pKq to MpKq is weakly compact. Proof This follows since C 0 pKq has property pVq by Theorem 1.2.46 and the dual of C 0 pKq is MpKq. The following theorem of Rosenthal is from [273, Proposition 1.2]. Theorem 1.3.54 Let E and F be Banach spaces such that E is complemented in E  , and take T P BpE, Fq. Suppose that T fixes a copy of c 0 . Then T also fixes a copy of  8 . Corollary 1.3.55 Let E and F be Banach spaces. Suppose that E  has the property pVq and that F does not contain a copy of  8 . Then BpE  , Fq “ WpE  , Fq. Proof By Theorem 1.3.50, it suffices to show that each T P BpE  , Fq is unconditionally converging. Assume towards a contradiction that T P BpE  , Fq is not unconditionally converging. By Theorem 1.3.51(i), T fixes a copy of c 0 , and so T | c 0 : c 0 Ñ F is a homeomorphic embedding. Since E  is complemented in E  , Theorem 1.3.54 applies to show that E  has a closed linear subspace G such that G ∼  8 and T | G : G Ñ F is a homeomorphic embedding, and so F contain a copy of  8 , a contradiction. Thus BpE  , Fq “ WpE  , Fq. The above corollary gives a proof of a slightly stronger version of Phillips’ theorem, Theorem 1.3.32. Indeed, we apply the corollary with E “  1 and F “ c 0 to see that Bp 8 , c 0 q “ Wp 8 , c 0 q. Assume that there is an epimorphism in Bp 8 , c 0 q. Then, by Proposition 1.3.43(ii), the space c 0 is reflexive, a contradiction. In particular, c 0 is not complemented in  8 . The following is an immediate consequence of the definition of the Schur property for Banach spaces. Proposition 1.3.56 Let E and F be Banach spaces such that F or E  has the Schur property. Then WpE, Fq “ KpE, Fq. Proof We consider the case where E  has the Schur property. Indeed, let T : E Ñ F be a weakly compact operator. Then T  P BpF  , E  q is also weakly compact. As E  has the Schur property, T  is compact, so T is compact. Theorem 1.3.57 Let E and F be Banach spaces. Suppose that E has property pVq and that F has the Schur property. Then BpE, Fq “ KpE, Fq, and this space is weakly sequentially complete.

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1 Banach spaces and operators

Proof The space F does not contain a copy of c 0 , and so BpE, Fq “ WpE, Fq by Theorem 1.3.51(ii). By Proposition 1.3.56, WpE, Fq “ KpE, Fq. It follows that BpE, Fq “ KpE, Fq. Take T P BpE, Fq. Then T  pE  q Ă F, and the map T Þ→ T  ,

KpE, Fq Ñ KpE  , Fq ,

is an isometric embedding. Let P : E  Ñ E  be the Dixmier projection, so that PpT  pλqq “ T  pλq pλ P F  q. Now suppose that pT n q is a weakly Cauchy sequence in KpE, Fq. Then pT n q is a weakly Cauchy sequence in KpE  , Fq, and so, for each M P E  , the sequence pT n pMqq is weakly Cauchy in F. Since F has the Schur property, T n pMq Ñ S pMq, say, in F. By Theorem 1.3.9, S P BpE  , Fq. Set T “ S | E, so that T P KpE, Fq. Take λ P F  , so that pT n pλqq is a sequence in E  . For each M P E  and n P N, we have xM, T n pλqy “ xT n pMq, λy, and so wk∗ – limnÑ8 T n pλq “ S  pλq in E  . Also, PpT n pλqq “ T n pλq, and so, by Proposition 1.3.33, limnÑ8 T n pλqq “ PpS  pλqq weakly. For each x P E, we have xx, PpS  pλqqy “ xS x, λy “ xT x, λy “ xx, T  pλqy , and so PpS  pλqq “ T  pλq and limnÑ8 T n pλq “ T  pλq weakly. It follows from Theorem 1.3.46 that T n Ñ T weakly in KpE, Fq, and so KpE, Fq is weakly sequentially complete. The following theorem is proved in [239, Proposition 5.3.10]; other equivalent properties are listed in [84, p. 179]. Theorem 1.3.58 Let E be a Banach space. Then the following are equivalent; (a) E is a Grothendieck space; (b) BpE, c 0 q “ WpE, c 0 q; (c) BpE, Fq “ WpE, Fq for every separable Banach space F. We shall also use some results on p-summing operators. The main text on these operators is [83]; see also [19, 76, 283]. Let E be a Banach space, and take p with 1 ď p < 8 and n P N. Then we define μ p,n pxq for x “ px1 , . . . , xn q P E n by ⎫ ⎧⎛ n ⎞ ⎪ ⎪  p ⎟⎟⎟1/p ⎪ ⎪ ⎬ ⎨⎜⎜⎜⎜   μ p,n pxq “ sup ⎪ . ⎜⎝ xxi , λy ⎟⎟⎠ : λ P Er1s ⎪ ⎪ ⎪ ⎭ ⎩ i“1

Then μ p,n is the weak p -summing norm (at dimension n) on E n . Define T x : pζ1 , . . . , ζn q Þ→

n  j“1

ζjxj ,

Cn Ñ E .

1.3 Bounded linear operators

57

" " Set q “ p . Then μ p,n pxq “ ""T x : nq Ñ E "", and the map x Þ→ T x ,

pE n , μ p,n q Ñ Bpnq , Eq ,

is a linear isometry. Definition 1.3.59 Let E and F be Banach spaces, and suppose that 1 ď p ď q < 8. Then an operator T P BpE, Fq is pq, pq´summing if there is a constant C ě 0 such that ⎞1/q ⎛ n ⎟ ⎜⎜⎜ ⎜⎜⎝ T xi q ⎟⎟⎟⎟⎠ ď C μ p,n px1 , . . . , xn q px1 , . . . , xn P E, n P Nq . i“1

The smallest such constant C is denoted by πq,p pT q, and the linear space of these pq, pq-summing operators is Πq,p pE, Fq, with Π p pE, Fq for Π p,p pE, Fq and Πq,p pEq for Πq,p pE, Eq, and with π p pT q for π p,p pT q. The space Πq,p pE, Fq is a linear subspace of BpE, Fq, and pΠq,p pE, Fq, πq,p q is a Banach space. The space Π p pE, Fq of all p -summing operators has been widely studied. Let E, F, G, H be Banach spaces, and take R P BpE, Fq, S P Πq,p pF, Gq, and T P BpG, Hq. Then T ◦ S ◦ R P Πq,p pE, Hq. There is an inclusion theorem [83, Theorem 2.8]. Theorem 1.3.60 Let E and F be Banach spaces, and suppose that 1 ď p ď q < 8. Then Π p pE, Fq Ă Πq pE, Fq and πq pT q ď π p pT q pT P Π p pE, Fqq. In particular, every 1-summing operator is a 2-summing operator. There are compact operators that are not p -summing for any p, and there are p-summing operators that are not compact. However, every p-summing operator is weakly compact and is completely continuous [83, Theorem 2.17]. Key examples of a 1-summing operator are operators from  1 to  2 [83, Theorem 1.13]: Bp 1 ,  2 q “ Π1 p 1 ,  2 q. This is a special case of the following theorem of Grothendieck [83, Theorem 3.4]; here KG is Grothendieck’s constant. Theorem 1.3.61 Let pΩ, Σ, μq be a measure space, and let H be a Hilbert space. Then BpL1 pμq, Hq “ Π1 pL1 pμq, Hq, and π1 pT q ď KG T pT P BpL1 pμq, Hqq. A companion theorem to the above is the following [83, Theorem 3.5]. Theorem 1.3.62 Let K be a non-empty, compact space, and let H be a Hilbert space. Then BpCpKq, Hq “ Π2 pCpKq, Hq, and π2 pT q ď KG T pT P BpCpKq, Hqq. The next Π2 -extension theorem will be used several times; it is proved in [83, Theorem 4.15].

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Theorem 1.3.63 Let E and F be Banach spaces, and suppose that G is a Banach space that contains E as a closed linear subspace. Take T P Π2 pE, Fq. Then there exists Tr P Π2 pG, Fq such that Tr | E “ T and π2 pTrq “ π2 pT q. Corollary 1.3.64 Let E be a Banach space that contains a copy F of  1 . (i) Take T P BpF,  2 q. Then there exists Tr P Π2 pE,  2 q with Tr | F “ T . (ii) The space  2 is a quotient of E, and E  contains a copy of  2 . Proof (i) By Theorem 1.3.61, Bp 1 ,  2 q “ Π1 p 1 ,  2 q, and so T P Π2 pF,  2 q by Theorem 1.3.60. Thus the required operator Tr exists by Theorem 1.3.63. (ii) By Proposition 1.3.15(ii), there is a surjection T P BpF,  2 q. By (i), there is r T P Π2 pE,  2 q with Tr | F “ T , and R :“ Tr P BpE,  2 q is a surjection, so that  2 is a quotient of E. Thus the map R :  2 Ñ E  is a homeomorphic embedding. We now consider certain results connected with the approximation property and Dunford–Pettis property of Banach spaces. Definition 1.3.65 A Banach space E has the approximation property (AP) if, for every ε > 0 and every compact subset L of E, there exists an operator T P F pEq such that T x ´ x < ε px P Lq; E has the bounded approximation property (BAP) if there is a constant C > 0 (independent of ε and L) such that the operator T can be chosen with T op ď C. All Banach spaces with a basis have the BAP; however various examples, including the space BpHq for an infinite-dimensional Hilbert space H, lack the approximation property. Let F be a Banach space. Then F has the AP if and only if ApE, Fq “ KpE, Fq for every Banach space E; further, F has the AP whenever F  has the AP, so that a reflexive space F that has the AP is such that F  has AP. Proposition 1.3.66 The Banach spaces C 0 pKq and L p pK, μq for each non-empty, locally compact space K, for each p with 1 ď p ď 8, and each positive measure μ on K, have the bounded approximation property. See [226] for more on the approximation property. Definition 1.3.67 Let E be a Banach space. Then E has the Dunford–Pettis property if every weakly compact operator from E into any Banach space F is completely continuous. Thus E has the Dunford–Pettis property if and only if, for any two sequences pxn q in E and pλn q in E  that both converge weakly to 0, also limnÑ8 xxn , λn y “ 0. Clearly E has this property whenever E  does. See [238, Theorem 3.5.18]. Every Banach space with the Schur property has the Dunford–Pettis property. For much information about the Dunford–Pettis property of Banach spaces, including the above remarks, see the survey articles [80] and [141].

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Proposition 1.3.68 Let K be a non-empty, locally compact space. The Banach spaces C 0 pKq, the dual space C 0 pKq “ MpKq, and the spaces L1 pK, μq and the dual space L1 pK, μq “ L8 pK, μq for each σ-finite, positive measure μ on K have the Dunford–Pettis property. Proposition 1.3.69 Let E, F, and G be three Banach spaces. Then F has the Dunford–Pettis property if and only if T ◦ S P KpE, Gq whenever S P WpE, Fq and T P WpF, Gq for two Banach spaces E and G. Proof Suppose that F has the Dunford–Pettis property. The composition of a weakly compact operator followed by a completely continuous operator is a compact operator by the Eberlein–Šmulian theorem, Theorem 1.2.24(v), and hence T ◦ S P KpE, Gq because T is completely continuous. The converse is immediate For the following proposition, see [80, Theorem 3 and the following remark]; we also use Theorem 1.3.37. Proposition 1.3.70 Let E be a Banach space. Then E  has the Schur property if and only if E has the Dunford–Pettis property and E does not contain a copy of  1 . Corollary 1.3.71 Let K be a non-empty, locally compact space. Then MpKq has the Schur property if and only if K is scattered. Proof We apply Proposition 1.3.70 with E “ C 0 pKq. By Proposition 1.3.68, C 0 pKq has the Dunford–Pettis property, and so MpKq has the Schur property if and only if C 0 pKq does not contain a copy of  1 . By Theorem 1.3.39, this occurs if and only if K is scattered. Proposition 1.3.72 Let E be a Banach space such that E  has the Schur property. Then BpE, E  q “ KpE, E  q. Proof Take T P BpE, E  q, and let pxn q be a bounded sequence in E. By Proposition 1.3.70, E does not contain a copy of  1 , and so, by Rosenthal’s  1 -theorem, Theorem 1.3.37, pxn q has a weakly Cauchy subsequence, say pxnk q. Now take M P E  . Then xM, T xnk y “ xxnk , T  pMqy for each k P N, and so pT xnk q is a weakly Cauchy sequence in E  . Since E  is weakly sequentially complete, pT xnk q is weakly convergent, and so norm convergent, again because E  has the Schur property. Hence T P KpE, E  q. We shall now quote three of the large variety of fixed-point theorems that arise in functional analysis. The first is the Schauder–Tychonoff fixed-point theorem [90, V.10.5].

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Theorem 1.3.73 Let K be a non-empty, compact, convex set in a locally convex space. Then every continuous function from K to K has a fixed point. For the following Markov–Kakutani fixed-point theorem, see [90, V.10.6] or [256, Proposition (0.14)] for a classical proof. Theorem 1.3.74 Let K be a non-empty, compact, convex set in a locally convex space. Suppose that F is a commuting family of continuous, affine maps from K to K. Then the operators in F have a common fixed point in K. We also mention a related fixed-point theorem; this is the Ryll-Nardzewski fixedpoint theorem; see [87, §5, Theorem (12.4)]. Many other fixed-point theorems are proved in [87]. Theorem 1.3.75 Let K be a non-empty, weakly compact, convex set in a locally convex space pE, τq. Suppose that F is a semigroup of weakly continuous, affine maps from K to K such that 0 does not belong to the τ-closure of {S x ´ S y : S P F } whenever x, y P K with x  y. Then the operators in F have a common fixed point in K. We shall conclude this section with some further discussion of Banach lattices. Definition 1.3.76 Let E and F be Banach lattices that are the complexifications of the real Banach lattices E R and F R , respectively. An operator T P BpE, Fq is a Banach-lattice homomorphism or a Banach-lattice isomorphism if the map T | E R : E R Ñ F R is a lattice homomorphism or a lattice isomorphism, respectively; it is a Banach-lattice isometry if, further, T is a linear isometry. The Banach lattices E and F are Banach-lattice isomorphic or Banach-lattice isometric if there is a Banachlattice isomorphism or isometry, respectively, between them; an isometric lattice embedding is an isometric embedding that is a lattice homomorphism. The following central representation theorems are proved in [3, Theorems 4.27 and 4.29], [227, II. §1.b], [239, Theorems 2.7.1 and 2.1.3], and [305, Theorem 6.5.5 and Proposition 6.1.3], for example. Theorem 1.3.77 A Banach lattice is an AL-space if and only if it is Banach-lattice isometric to a Banach lattice L1 pΩ, μq for some measure space. Corollary 1.3.78 Each AL-space is Dedekind complete and weakly sequentially complete, every band is a projection band, and its norm is order-continuous. Theorem 1.3.79 Every AM-space with a unit is Banach-lattice isometric to a Banach lattice CpKq for some compact space K.

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We now give the definitions of two Banach spaces B r pE, Fq and B b pE, Fq. Let E and F be real Banach lattices. Then we have defined the spaces L r pE, Fq and L b pE, Fq of regular and order-bounded operators from E to F. Now suppose that E and F are Banach lattices. Then T P LpE, Fq is positive if T pE +R q Ă F +R . For a positive operator, we have |T x| ď T p|x|q px P Eq. Each operator in LpE, Fq has a unique expression in the form S + iT , where S and T belong to LpE R , F R q and pS + iT qpx + iyq “ S x ´ T y + ipS y + T xq

px, y P E R q ;

such an operator is regular or order-bounded if both S and T are regular or orderbounded, respectively. We have the following theorem [305, Theorem 4.1.8]. Theorem 1.3.80 Let E and F be Banach lattices. Then every order-bounded operator from E to F is bounded. Thus we denote the spaces of all positive, all regular, and all order-bounded operators from E to F by BpE, Fq+ , by B r pE, Fq, and by B b pE, Fq, respectively, and so BpE, Fq+ Ă B r pE, Fq Ă B b pE, Fq Ă BpE, Fq . We write B r pEq and B b pEq for B r pE, Eq and B b pE, Eq, respectively. In particular, we identify E ∼ with the dual space E  (cf. [3, Corollary 4.5]). We have already noted that E  is a Dedekind complete Banach lattice. The book [3] is devoted to positive operators on real Banach lattices (and more general spaces). See also [53]. Take T P BpE, Fq+ . Then + }.

T “ sup{ T x : x P Er1s

(1.3.13)

+ In particular, take λ P pE  q+ . Then λ “ sup{xx, λy : x P Er1s }. Let E and F be Banach lattices such that F is Dedekind complete. The regular norm · r is defined on B r pE, Fq by setting

T r “ |T |

pT P B r pE, Fqq .

Then pB r pE, Fq, · r q is a Dedekind complete Banach lattice [3, Theorem 4.74]. We shall use the following definition and theorem. The term ‘KB’ stands for Kantorovich–Banach. Definition 1.3.81 Let pE, · q be a Banach lattice. Then E is a KB-space if every + is convergent in pE, · q. increasing sequence in Er1s The following theorem is given in [3, Theorem 4.60], [239, Theorem 2.4.12], and [305, Theorems 9.4.3, 9.4.6, and 19.4.9].

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Theorem 1.3.82 Let pE, · q be a Banach lattice. Then the following conditions are equivalent: (a) E is a KB-space; (b) the space E R contains no closed sublattice F such that there is a Banachlattice isomorphism from c 0,R onto F; (c) the Banach space E is weakly sequentially complete; (d) E is a band in E  . Corollary 1.3.83 Let pE, · q be a Banach lattice that is an AL-space, and take M P E  . Then there are unique elements x P E and N P E d such that M “ x + N. Moreover, M “ x + N . Proof By Theorem 1.3.77, E is weakly sequentially complete, and so a KB-space. By Theorem 1.3.82, (c) ⇒ (d), E is a band in E  . By Theorem 1.2.27, E  is an AL-space, and so E is a projection band in E  . The result follows. Example 1.3.84 Let S be a non-empty set, and take E “ c 0 pS q. Then E  “  1 pS q and E  “  8 pS q “ Cpβ S q, as in Example 1.2.8. By the Riesz representation theorem, E  “ Mpβ S q, and so  1 pS q “ Mpβ S q “  1 pS q ‘1 MpS ∗ q “  1 pS q ‘1 c 0 pS qK .

(1.3.14)

The Banach space  1 pS q is a Banach lattice (for the pointwise ordering) that is an AL-space, as in Definition 1.2.26. By Theorem 1.3.82,  1 pS q is a projection band in the AL-space  1 pS q ; this is also immediate from equation (1.3.14).

Example 1.3.85 Let pΩ, μq be a measure space, and take p with 1 ď p < 8. Then L p pΩ, μq is a KB-space. Since L1 pΩ, μq is also an AL-space, L1 pΩ, μq is a projection band in L1 pΩ, μq . It follows that for each M P L1 pΩ, μq , there are unique elements f P L1 pΩ, μq and N P L1 pΩ, μqd such that M “ f + N and M “ f + N . Let K be a non-empty, locally compact space. As noted, MpKq is an AL-space. As in Theorem 1.2.39(i), the Banach lattice MpKq is weakly sequentially complete, and so MpKq is a projection band in MpKq , and so there is a similar unique decomposition of elements in MpKq .

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63

1.4 Tensor products We shall recall some definitions and results concerning tensor products of normed spaces. For the theory of such products, see [76, 82–84, 283]; in fact, we shall consider just the projective and injective tensor products of two Banach spaces. Later, we shall be concerned with the projective tensor product of two Banach algebras. Let E and F be Banach spaces. A key fact will be the identification of the dual p F with the space BpE, F  q of bounded space of the projective tensor product E ⊗ operators from E to the dual space of F. Let E and F be linear spaces. Then the tensor product of E and F is denoted by  E ⊗ F; each element z P E ⊗ F has a representation as z “ nj“1 x j ⊗ y j , where n P N, x1 , . . . , xn P E, and y1 , . . . , yn P F; in the case where z  0, we may suppose that the sets {x1 , . . . , xn } and {y1 , . . . , yn } are linearly independent. Now let G be another linear space, and take S : E × F Ñ G to be a bilinear map. Then there is a unique linear map T S : E ⊗ F Ñ G such that T S px ⊗ yq “ S px, yq

px P E, y P Fq .

It follows immediately that there is a linear isomorphism T : E ⊗ F Ñ F ⊗ E such that T px⊗yq “ y⊗ x px P E, y P Fq, and so E ⊗F and F ⊗E are linearly isomorphic. Definition 1.4.1 Let E and F be normed spaces. The projective tensor norm · π on E ⊗ F is defined by ⎧ ⎫ ⎪ ⎪ n " "" " n ⎪ ⎪   ⎪ ⎪ ⎨ " "" " ⎬ " " " " x ⊗ y , n P N x : z “ y pz P E ⊗ Fq ,

z π “ inf ⎪ ⎪ j j j j ⎪ ⎪ ⎪ ⎪ ⎩ j“1 ⎭ j“1 where the infimum is taken over all representations of z as an element of E ⊗ F. Then pE ⊗ F, · π q is a normed space; it is complete if either E or F is finite dimensional and the other is a Banach space, but it is not complete if both E and F are infinite-dimensional spaces; the Banach space which is its completion is denoted by p F, · π q . pE ⊗ p F is the following: for Banach spaces E, F, and G and The basic property of E ⊗ each bounded bilinear operator S P BpE × F, Gq, there is a unique bounded linear p F, Gq such that T S px ⊗ yq “ S px, yq px P E, y P Fq and such operator T S P BpE ⊗ that T S op “ S op . In particular, take λ P E  and μ P F  . Then there is an element, p Fq such that called λ ⊗ μ, in pE ⊗ pλ ⊗ μqpx ⊗ yq “ xx, λy xy, μy

px P E, y P Fq .

(1.4.1)

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p F and ε > 0. Then there are Let E and F be Banach spaces, and take z P E ⊗  bounded sequences pxn q in E and pyn q in F such that z “ 8 n“1 xn ⊗ yn and 8 

xn yn < z π + ε .

n“1

It follows that

p Fqr1s “ co{x ⊗ y : x P Er1s , y P Fr1s } . pE ⊗

(1.4.2)

pF Ñ F⊗ p E such that We see that there is a linear isometry T : E ⊗ T px ⊗ yq “ y ⊗ x

px P E, y P Fq ,

pF – F⊗ p E. We also note that E is linearly isometric to a complemented so that E ⊗ p F whenever F  {0}. linear subspace in E ⊗ Examples (i) Let E be a Banach space. Then there is a canonical linear isometry p E Ñ  1 pEq such that T p f ⊗ xq “ p f pnqxq for f “ p f pnqq P  1 and x P E. T : 1 ⊗ For details, see [283, Example 2.6]. (ii) Let S and T be non-empty sets. Then we identify the projective tensor product p  1 pT q with  1 pS × T q by setting  1 pS q ⊗ p f ⊗ gqps, tq “ f psqgptq

ps P S , t P T q  for f P  1 pS q and g P  1 pT q. Note that, for an element F “ αi j δpsi ,t j q in the space p  1 pT q, where {psi , t j q : i, j P N} is a set of distinct points in S × T , we have  1 pS q ⊗

F π “ F 1 “

8   α  , ij

(1.4.3)

i, j“1

p  1 pT q, · π q “ p 1 pS × T q, · 1 q. and so p 1 pS q ⊗ Let E and F be Banach spaces, and take G to be a closed linear subspace of p F is not necessarily a closed linear subspace of E ⊗ p F. E. Then, in general, G ⊗ p F is a closed linear subspace of E ⊗ p F in the special case that G is However, G ⊗ complemented in E. Further, we have the following [283, Corollary 2.14]. p F is a closed linear Proposition 1.4.3 Let E and F be Banach spaces. Then E ⊗  p  subspace of E ⊗ F . It is often not easy to give a simple description of a projective tensor product p F, even when E “ F “  2 . space E ⊗ p  2 , · π q. The tensor diagonal of  2 ⊗ p 2 Example 1.4.4 Consider the space p 2 ⊗ is the closed linear subspace D :“ lin{δn ⊗ δn : n P N}

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65

p  2 . It is shown in [283, Example 2.10] that D is linearly isometric to  1 and of  2 ⊗ p  2 by a projection of norm 1. This implies that the that D is complemented in  2 ⊗ 2p 2 space  ⊗  is not reflexive.

Example 1.4.5 In [26], Bourgain and Pisier give an example of a Banach space E that is weakly sequentially complete and has the Radon–Nikodým property, but is p E contains a copy of c 0 , and so has neither of these two properties. such that E ⊗

Let E and F be Banach spaces. Then there is a contraction p E  Ñ BpE, Fq T : F⊗

(1.4.4)

such that T py ⊗ λq is the operator y ⊗ λ for y P F and λ P E  that was defined in equation (1.3.9). The image of this map (with the quotient norm) is the space pNpE, Fq, · ν q of nuclear operators from E to F. Clearly F pE, Fq Ă NpE, Fq Ă ApE, Fq . p Fq induces an element of For λ P E  and μ P F  , the above element λ ⊗ μ in pE ⊗  p E. NpE, Fq . We shall write NpEq for NpE, Eq, the image of E ⊗ p Fq – BpE, F  q, where We recall the standard fact [283, Theorem 2.9] that pE ⊗ the linear isometry p Fq Ñ BpE, F  q , (1.4.5) Λ Þ→ T Λ , pE ⊗ satisfies the condition that p Fq q . xy, T Λ xy “ xx ⊗ y, Λy px P E, y P F, Λ P pE ⊗

(1.4.6)

This duality prescribes a weak-∗ topology on BpE, F  q. To be specific, to check that a subset S of BpE, F  qr1s is weak-∗ dense in BpE, F  qr1s , we must show that, given T P BpE, F  qr1s , and also given ε > 0, x1 , . . . , xk P Er1s , and y1 , . . . , yk P Fr1s , there exists S P S with    xyi , pS ´ T qpx j qy  < ε pi, j P Nk q . (1.4.7) p E  q – BpE, E  q, and, further, In particular, for each Banach space E, we have pE ⊗   p E q – BpEq when E is reflexive. pE ⊗ Definition 1.4.6 Let E be a non-zero Banach space, and let B be a closed linear  . Then subspace of BpEq. Take x P Er1s and λ P Er1s Br1s pxq “ {T x : T P Br1s }

and

rr1s pλq “ {T  λ : T P Br1s } . B

rr1s pλq are convex subsets of Er1s and E  , respectively. Clearly Br1s pxq and B r1s

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Proposition 1.4.7 Let E be a non-zero, reflexive Banach space, and let B be a norming subspace of BpEq. Take x0 P E and λ0 P E  with x0 “ λ0 “ 1. Then Br1s px0 q rr1s pλ0 q are dense in Er1s and E  , respectively. and B r1s p E  q , it follows that Proof Since B is a norming subspace of BpEq “ pE ⊗   sup{T ◦ px0 ⊗ λq : T P Br1s } “ x0 ⊗ λ op “ λ pλ P E  q .   Thus sup{xT x0 , λy : T P Br1s } “ λ pλ P E  q. By the Hahn–Banach theorem, Theorem 1.2.13, Br1s px0 q is dense in Er1s . rr1s pλ0 q is dense in E  . Similarly, B r1s Proposition 1.4.8 Let E and F be Banach spaces. Suppose that E has property pVq p Fq is weakly sequentially complete. and that F  has the Schur property. Then pE ⊗ p Fq “ BpE, F  q. By Theorem 1.3.57, BpE, F  q “ KpE, F  q is Proof We have pE ⊗ p Fq has the same property. weakly sequentially complete, and so pE ⊗ The following result of Emmanuele and Hensgen is the main theorem of [94]. Theorem 1.4.9 Let E and F be two Banach spaces with property pVq and such that p F has property pVq. WpE, F  q “ KpE, F  q. Then E ⊗ Definition 1.4.10 Let E and F be two Banach spaces, and take K and L to be nonempty subsets of E and F, respectively. Then K ⊗ a L is the subset K ⊗ a L “ {x ⊗ y : x P K, y P L} of E ⊗ F, where the subscript ‘a’ stands for ‘atomic’. Proposition 1.4.11 Let E and F be Banach spaces, and take non-empty subsets K and L of E and F, respectively. p F. (i) Suppose that K and L are compact. Then K ⊗ a L is a compact subset of E ⊗ (ii) Suppose that K is compact and L is weakly compact. Then K ⊗ a L is a weakly p F. compact subset of E ⊗ Proof Consider the continuous bilinear operator S : px, yq Þ→ x ⊗ y ,

pF. E×F Ñ E⊗

(i) The set K × L is compact in E × F, and so K ⊗ a L “ S pK × Lq is compact in p F. E⊗

1.4 Tensor products

67

(ii) By the Eberlein–Šmulian theorem, Theorem 1.2.24(v), we must show that, for every sequences pxn q in K and pyn q in L, the sequence pxn ⊗ yn q in K ⊗ a L has a weakly convergent subsequence. By passing to subsequences, we may suppose that pxn q and pyn q converge weakly in K and L, respectively, say to x P K and y P L. For p Fq , we have each operator T “ T Λ in the space BpE, F  q “ pE ⊗ xS pxn , yn q, T y “ xxn ⊗ yn , Λy “ xyn , T xn y Ñ xy, T xy “ xx ⊗ y, Λy “ xS px, yq, T y because K is compact, and so T xn Ñ T x with respect to the norm on F  . Hence p F. S pxn , yn q Ñ S px, yq weakly, and so K ⊗ a L is weakly compact in E ⊗ Since the projective tensor product of two reflexive Banach spaces is not necespF sarily reflexive, it is not always true that K ⊗ a L is a weakly compact set in E ⊗ whenever K and L are weakly compact sets in E and F, respectively. Let E, F, G, and H be Banach spaces, and take R P BpE, Fq and S P BpG, Hq. Then the map pH, px, yq Þ→ Rx ⊗ S y , E × G Ñ F ⊗ is a continuous bilinear operator, and so, by the basic property of tensor products, p G, F ⊗ p Hq such that there is a unique operator R ⊗ S P BpE ⊗ pR ⊗ S qpx ⊗ yq “ Rx ⊗ S y px P E, y P Gq . Further, R ⊗ S op “ R op S op . Thus there is a bilinear map Ψ : pR, S q Þ→ R ⊗ S ,

p G, F ⊗ p Hq ; BpE, Fq × BpG, Hq Ñ BpE ⊗

clearly Ψ “ 1. It follows that there is a contractive operator p : BpE, Fq ⊗ p BpG, Hq Ñ BpE ⊗ p G, F ⊗ p Hq Ψ

(1.4.8)

p pR ⊗ S q “ R ⊗ S for R P BpE, Fq and S P BpG, Hq. For details, see such that Ψ [283, Proposition 2.3]. It is easy to see, using the open mapping theorem, that R ⊗ S is a surjection whenever R and S are surjections. Proposition 1.4.12 Let E and F be Banach spaces, and take operators R P BpEq and S P BpFq. (i) Suppose that R and S are compact. Then R ⊗ S is compact. (ii) Suppose that R is compact and S is weakly compact. Then R ⊗ S is weakly compact. Proof (i) It follows from (1.4.2) and Theorem 1.2.24(iii) that it suffices to show that p F. This follows from Proposition RpEr1s q ⊗ a S pFr1s q is relatively compact in E ⊗ 1.4.11(i). (ii) It follows from (1.4.2) and Theorem 1.2.24(iv) that it suffices to show that p F. This follows from PropRpEr1s q ⊗ a S pFr1s q is relatively weakly compact in E ⊗ osition 1.4.11(ii).

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Proposition 1.4.13 Let E and F be Banach spaces, each containing a copy of  1 . p F contains a complemented copy of  1 , and BpE, F  q contains a compThen E ⊗ lemented copy of  8 . Proof By Corollary 1.3.64(ii), there are continuous linear surjections R : E Ñ p F Ñ 2 ⊗ p  2 is a continuous  2 and S : F Ñ  2 . Thus the map R ⊗ S : E ⊗ p  2 contains a complemented linear surjection. By Example 1.4.4, the space  2 ⊗ p  2 Ñ  1 . The map copy of  1 , and so there is a continuous linear surjection Q :  2 ⊗ p F Ñ  1 is a continuous linear surjection, and so, by Theorem Q ◦ pR ⊗ S q : E ⊗ p F contains a complemented copy of  1 . 1.3.34, (c) ⇒ (a), E ⊗  p Fq “ BpE, F  q, it follows from Theorem 1.3.34 that BpE, F  q conSince pE ⊗ tains a complemented copy of  8 . Corollary 1.4.14 Let E and F be non-zero Banach spaces. Suppose that E has property pVq and that WpE, F  q does not contain a copy of c 0 . Then E or F does not contain a copy of  1 . Proof By Proposition 1.3.44, F  does not contain a copy of c 0 . By Theorem p Fq “ WpE, F  q, and so pE ⊗ p Fq 1.3.51(ii), BpE, F  q “ WpE, F  q. Thus pE ⊗ p does not contain a copy of c 0 . By Theorem 1.3.34, (a) ⇒ (b), E ⊗ F does not contain a complemented copy of  1 , and so, by Proposition 1.4.13, E or F does not contain a copy of  1 . Definition 1.4.15 Let E and F be normed spaces. The injective tensor norm · ε on E ⊗ F is defined by  ⎫ ⎧ ⎪ ⎪   m ⎪ ⎪ ⎪ ⎪ ⎨  : λ P E  , μ P F  ⎬ xx , λy xy , μy ,

z ε “ sup ⎪  j j r1s r1s ⎪ ⎪ ⎪   ⎪ ⎪ ⎭ ⎩ j“1  where z “

m j“1

x j ⊗ y j is any representation of z in E ⊗ F.

Then pE ⊗ F, · ε q is a normed space; the Banach space which is its completion is denoted by q F, · ε q . pE ⊗ Note that

x ⊗ y π “ x ⊗ y ε “ x y

px P E, y P Fq

(1.4.9)

and that z ε ď z π pz P E ⊗ Fq. We may also view the elements of E ⊗ F as operators from E  into F, so that  z “ mj“1 x j ⊗ y j corresponds to the operator θpzq : λ Þ→

m  xx j , λy y j , j“1

E Ñ F ,

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69

q F, · ε q is the closure of θpE ⊗ Fq in BpE  , Fq. In and then the Banach space pE ⊗  q E  when E q E [50, Theorem 2.5.3(ii)], and so KpEq – E ⊗ particular, ApEq – E ⊗ pF Ñ has the approximation property. It is also clear that θ extends to a map θ : E ⊗ p Fq is contained in the linear subspace E ⊗ q F of BpE  , Fq and that the image θpE ⊗ p F into E ⊗ q F is not an injection, but BpE  , Fq. In general, the embedding of E ⊗ this is the case whenever either E or F has the approximation property; see [283, Chapter 4]. In fact, we have the following result. Proposition 1.4.16 Let E be a Banach space. Then the following conditions on E are equivalent: (a) E has the approximation property; p E  Ñ NpEq is injective; (b) the canonical map T : E ⊗ p F, · π q into pE ⊗ q F, · ε q is injective for every Banach (c) the embedding of pE ⊗ space F. The following result is [283, Exercise 4.5]. Proposition 1.4.17 Let E and F be Banach spaces each having the bounded approxq F and E ⊗ p F have the bounded approximation propimation property. Then both E ⊗ erty. We shall use the following results on tensor products; see [76] and [283, Corollary 4.13 and Theorems 4.21 and 5.33], for example. Clause (ii) is [31, Proposition 1]; somewhat more general results are given in this source. See also [32]. Recall that E “ c 0 is an example of a Banach space such that E  “  1 has the bounded approximation property and the Radon–Nikodým property. Theorem 1.4.18 Let E and F be Banach spaces. q F “ KpE, Fq. (i) Suppose that E  has the approximation property. Then E  ⊗ (ii) Suppose that E  has the approximation property and the Radon–Nikodým q F, considered naturally as a closed linear subspace of property. Then the space E ⊗ p   p F  , so that KpE, Fq “ BpE  , F  q. pE ⊗ F q , is a concrete predual of E  ⊗   (iii) Suppose that either E or F has the bounded approximation property. Then p F into pE ⊗ p Fq extends to a homeomorphic embedthe natural embedding of E ⊗  p  p Fq . ding of E ⊗ F onto a closed linear subspace of pE ⊗ (iv) Suppose that E and F are both reflexive and E has the approximation propp F is reflexive if and only if pE ⊗ p Fq “ E  ⊗ q F  if and only if E  ⊗ q F erty. Then E ⊗   is reflexive if and only if KpE, F q “ BpE, F q.

Chapter 2

Banach algebras

In this chapter, we shall introduce Banach algebras and recall their basic properties. General Banach algebras are considered in §2.1, and then we shall turn to a special case, that of C ∗ -algebras and von Neumann algebras, in §2.2. Our major theme, to be commenced in §2.3, will be consideration of the bidual space A of a Banach algebra A and of the two Arens products,  and , that are defined on the Banach space A , each making A into a Banach algebra that contains A as a closed subalgebra. The Banach algebra A is ‘Arens regular’ if these two products coincide on A . In §2.3, we shall give various examples of Arens regular Banach algebras and of Banach algebras that are not Arens regular; for example, every C ∗ -algebra A is Arens regular, and (A ,  ) is itself a von Neumann algebra, called the ‘enveloping von Neumann algebra’. However, the group algebra ( 1 (G),  ) of a group G is not Arens regular whenever G is infinite. These ideas will be substantially developed in Chapter 6. We shall also consider when a Banach algebra is an ideal in its bidual. Finally, in §2.4, we shall define and discuss ‘dual Banach algebras’. These are Banach algebras that are not just dual Banach spaces, but have a ‘Banach-algebra predual’. For example, a C ∗ -algebra is a dual Banach algebra if and only if it is a von Neumann algebra; a semigroup algebra  1 (S ) is a dual Banach algebra predual c 0 (S ) if and only if the semigroup S is weakly cancellative.

2.1 Algebras and Banach algebras In this section, we shall recall some properties of Banach algebras that we shall use. See [50] for an extensive account of the theory of Banach algebras; the basic properties of these algebras are given in [4]; classic texts include [25, 272, 332]; an account of commutative Banach algebras is given in [196]. Key examples will be the commutative C ∗ -algebra C 0 (K) of all continuous functions that vanish at infinity on a non-empty, locally compact space K and the algebra B(E) of all bounded linear operators on a Banach space E. In particular, we shall use the language of Banach © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 H. G. Dales and A. Ülger, Banach Function Algebras, Arens Regularity, and BSE Norms, CMS/CAIMS Books in Mathematics 12, https://doi.org/10.1007/978-3-031-44532-3_2

71

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2 Banach algebras

bimodules over a Banach algebra. We shall define the Banach algebra that is the projective tensor product of two Banach algebras, and we shall discuss Banach lattice algebras. The notion of (bounded and multiplier-bounded) approximate identities that will be introduced in this section will be very important for us. Let A be an algebra (always taken to be linear and associative and over the complex field C, unless stated otherwise). A non-zero element e of A is a left identity, respectively, right identity, if ea = a, respectively, ae = a, for all a ∈ A, and an identity if ea = ae = a (a ∈ A), in which case e is unique and is denoted by eA ; an algebra with an identity is a unital algebra. An idempotent in A is an element p ∈ A such that p2 = p, so that each left and right identity is an idempotent; the set of idempotents of A is denoted by I(A). The algebra formed by adjoining an identity, eA , to a non-unital algebra is A , so that A = CeA ⊕ A with the product given by (zeA + a)(weA + b) = zweA + wa + zb + ab

(z, w ∈ C, a, b ∈ A) ;

we take A = A when A is unital. For example, the linear space C S is a unital algebra for each non-empty set S , where we define f g for f, g ∈ C S by ( f g)(s) = f (s)g(s) (s ∈ S ). The opposite algebra to an algebra A is Aop . For example, let E be a linear space. Then L(E) and L(E) × L(E)op are unital algebras for the obvious products. Let S be a subset of an algebra A. Then the commutant of S is S c = {b ∈ A : ab = ba (a ∈ S )} , so that S c is a subalgebra of A; we set S cc = (S c ) c , the double commutant of S . Clearly S ⊂ S cc . The centre of A is denoted by Z(A), so that Z(A) = A c = {b ∈ A : ab = ba (a ∈ A)} . An element a ∈ A commutes with b ∈ A if ab = ba. For subsets S and T of an algebra A, set S · T = {ab : a ∈ S , b ∈ T } ,

S T = lin S · T .

We set S [2] = {ab : a, b ∈ S } and S 2 = lin S [2] ; the algebra A factors if A = A[2] and factors weakly if A = A2 . Further, S [n] = {a1 · · · an : a1 , . . . , an ∈ S } and S n = lin S [n] for n ∈ N. We write aS for {a}S , etc. A linear subspace I of an algebra A is a left ideal if AI ⊂ I, a right ideal if IA ⊂ I, and an ideal if it is both a left and right ideal. Let I be an ideal in an algebra A. The quotient space A/I is an algebra for the product specified by (a + I)(b + I) = ab + I

(a, b ∈ A) .

A left ideal M in A is a maximal left ideal if M  A and if I = A or I = M for each left ideal I in A with I ⊃ M. A left ideal I in A is modular if there exists u ∈ A such that a − au ∈ I (a ∈ A), so that u is a right modular identity for I. Similarly, we

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73

define maximal right ideals, maximal ideals, maximal modular left and right ideals, etc., in A. Let I be a left ideal in an algebra A such that I has a right modular identity u. It is immediate from Zorn’s lemma that the family of left ideals J in A such that J ⊃ I and u  J (when the family is ordered by inclusion) has a maximal member, say M. Clearly M is a maximal modular left ideal in A, and hence a maximal left ideal in A. An algebra A is a semi-direct product of a subalgebra B and an ideal I if A = B+ I and B ∩ I = {0}; in this case, we write A = BI. It follows that A/I = B. Let e be a right identity for an algebra A. Then eA is a right ideal in A for which e is the identity. Set J = {a − ea : a ∈ A} = {a ∈ A : ea = 0} .

(2.1.1)

Then J is an ideal in A such that AJ = {0}, and A = eA  J. Now take a ∈ Z(A). Then ea = a, and so a ∈ Z(eA). It follows that  Z(A) ⊂ {Z(eA) : e is a right identity for A} . (2.1.2) Proposition 2.1.1 Let A be an algebra. (i) Suppose that e1 and e2 are two right identities for A. Take a ∈ A. Then e1 a ∈ Z(e1 A) if and only if e2 a ∈ Z(e2 A). (ii) Suppose that e is a right identity for A and that p ∈ I(A). Then e − ep ∈ I(A). (iii) Suppose that e is a right identity for A and that B is a subalgebra of A such that B factors weakly and B ⊂ Z(A). Then BZ(A) = BZ(eA). Proof (i) Suppose that e1 a ∈ Z(e1 A), and take b ∈ A. Then (e2 a)(e2 b) = e2 ab = e2 (e1 a)(e1 b) = e2 (e1 b)(e1 a) = e2 be2 e1 a = (e2 b)(e2 a) , and so e2 a ∈ Z(e2 A). The result follows. (ii) It is immediate that (e − ep)2 = e − p. (iii) By Equation (2.1.2), Z(A) ⊂ Z(eA), and so BZ(A) ⊂ BZ(eA). Now take a ∈ A such that ea ∈ Z(eA) and take b ∈ B. For each c ∈ A, we have (ea)(ec) = (ec)(ea), and so bac = bca = cba. Thus ba = bea ∈ Z(A), and so BZ(eA) ⊂ Z(A). But now BZ(eA) = B 2 Z(eA) ⊂ BZ(A). We have shown that BZ(A) = BZ(eA). The Jacobson radical of an algebra A is defined to be the intersection of the maximal modular left ideals of A; it is denoted by rad A, with rad A = A when A has no maximal modular left ideals. In fact, rad A = rad Aop and rad A is an ideal in A. The algebra A is semisimple when rad A = {0} and radical when rad A = A. See [50, §1.5] for algebraic details.

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An element a ∈ A is nilpotent of degree n, where n ≥ 2, if an = 0, but an−1  0, and a is nilpotent if it is nilpotent of degree n for some n ≥ 2. A subset S of A is nil of degree n if each a ∈ S is nilpotent of degree at most n, and S is nilpotent of degree n if S [n] = {0}, but S [n−1]  {0}. Take a ∈ A. The spectrum, σ(a), of a is the complement in C of the resolvent set ρ(a) = {z ∈ C : zeA − a is invertible in A } ; an element a ∈ A is quasi-nilpotent if σ(a) ⊂ {0}, and the set of quasi-nilpotent elements of A is denoted by Q(A). Clearly each nilpotent element is quasi-nilpotent and rad A ⊂ Q(A). The following are standard properties of the radical: Proposition 2.1.2 Let A be an algebra. (i) For each ideal I in A, we have rad I = I ∩ rad A. In particular, I is semisimple when A is semisimple. (ii) For each ideal I in A with I ⊂ rad A, we have rad (A/I) = (rad A)/I. In particular, A/rad A is semisimple. (iii) The radical rad A contains each nil left ideal and each nil right ideal. The following result is [50, Proposition 1.5.7(ii)]. Proposition 2.1.3 Let A be a commutative algebra, and suppose that p, q ∈ I(A) with p + rad A = q + rad A in A/rad A. Then p = q. Definition 2.1.4 Let A and B be algebras. Then a homomorphism from A to B is a linear map θ : A → B such that θ(ab) = θ(a)θ(b)

(a, b ∈ A) .

An injective homomorphism is a monomorphism; a surjective homomorphism is an epimorphism; a bijective homomorphism is an isomorphism; A and B are isomorphic if there is an isomorphism from A onto B. A character on A is an epimorphism from A onto C; the family of all characters on A forms the character space of A, and this space is denoted by ΦA . Suppose that A and B are isomorphic algebras. Then we can identify ΦA and ΦB . Let A be an algebra. The theory of modules and bimodules over A is given in [50, §§1.4, 2.6]. Briefly, a left A-module is a linear space E with a bilinear map (a, x) → a · x, A × E → E, such that a · (b · x) = ab · x (a, b ∈ A, x ∈ E), and a right A-module is a linear space E with a bilinear map (a, x) → x · a, A × E → E, such that (x · a) · b = x · ab (a, b ∈ A, x ∈ E). Let (E, · ) be a left A-module. For subsets S of A and T of E, set S · T = {a · x : a ∈ S , x ∈ T } ,

S T = lin S · T ;

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75

further, such that a · (x · b) = (a · x) · b (a, b ∈ A, x ∈ E) ; thus we can write a · x · b without ambiguity. A linear subspace F of an A-bimodule that is such that a · x ∈ F and x · a ∈ F whenever a ∈ A and x ∈ F is a submodule. For example, an algebra A is itself an A-bimodule with respect to the maps given by the products in A, and a left ideal in A is a left A-module. Suppose that I is an ideal in A. Then A/I is an A-bimodule with respect to the operations given by a · (b + I) = ab + I ,

(b + I) · a = ba + I

(a, b ∈ A) .

Let A be an algebra, and suppose that E and F are both left A-modules. Then a linear map T : E → F is a left A-module homomorphism if T (a · x) = a · T x

(a ∈ A, x ∈ E) .

Similarly, we have right A-module homomorphisms, bimodule homomorphisms, left A-module isomorphisms, etc. Definition 2.1.5 Let E be a linear space. A map ∗ : E → E is a linear involution if (αx + βy)∗ = αx∗ + βy∗

(α, β ∈ C, x, y ∈ E) ,

(x∗ )∗ = x

(x ∈ E) ;

an element x in a space with a linear involution is self-adjoint if x∗ = x; the linear subspace of self-adjoint elements in E is denoted by Esa . A linear involution on an algebra A is an involution if (ab)∗ = b∗ a∗

(a, b ∈ A) .

A subset S of an algebra with an involution ∗ is ∗-closed if a∗ ∈ S (a ∈ S ); a ∗-closed ideal is a ∗-ideal. Let θ : A → B be a homomorphism between algebras A and B with involutions. Then θ is a ∗-homomorphism if θ(a)∗ = θ(a∗ ) (a ∈ A). For example, rad A is a ∗-ideal for every algebra A with an involution. Example 2.1.6 Let S be a non-empty set. For each f ∈ C S , define f (s) = f (s) (s ∈ S ) . Then the map f → f is an involution on the algebra C S . Let X be non-empty topological space. Then the restriction of this map to C b (X) is an involution on the algebra C b (X). Similarly, we obtain involutions on the algebra C 0 (K) for each non-empty, locally compact space K.

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Definition 2.1.7 Let (A, ∗ ) be an algebra with an involution. Then ⎧ n ⎫ ⎪ ⎪  ⎪ ⎪ ⎨ ⎬ a∗i ai : a1 , . . . , an ∈ A, n ∈ N⎪ A+ = ⎪ . ⎪ ⎪ ⎩ ⎭ i=1

Take a, b ∈ A+ and α, β ∈ R+ . Then αa + βb ∈ A+ , and so A+ is a cone in A. We now consider multipliers on an algebra. Let A be an algebra. For a ∈ A, we define linear maps La and Ra on A by setting La (b) = ab ,

Ra (b) = ba

(b ∈ A) .

Definition 2.1.8 Let A be an algebra. A left multiplier and right multiplier on A are linear maps L : A → A and R : A → A such that L(ab) = L(a)b ,

R(ab) = aR(b)

(a, b ∈ A) ,

respectively. A multiplier is a pair (L, R), where L and R are left and right multipliers, respectively, and aL(b) = R(a)b (a, b ∈ A) . For example, (La , Ra ) is a multiplier for each a ∈ A. The spaces of left and right multipliers on A are denoted by M  (A) and Mr (A), respectively. The space M  (A) is a unital subalgebra of the algebra L(A) and the map a → La , A → M  (A), is a homomorphism. However, we regard M r (A) as a unital subalgebra of the algebra L(A)op , so that the map a → Ra , A → M r (A), is a homomorphism. The space of all multipliers on A is called the multiplier algebra of A, and it is denoted by M(A); M(A) is a unital subalgebra of the algebra L(A) × L(A)op , so that (L1 , R1 ) · (L2 , R2 ) = (L1 L2 , R2 R1 ) for multipliers (L1 , R1 ) and (L2 , R2 ). Let A be an ideal in an algebra B. Then B is faithful over A if {b ∈ B : bA = Ab = {0}} = {0} ; the algebra A is faithful if it is faithful over itself. Let a ∈ A and (L, R) ∈ M(A). Then (L, R) (La , Ra ) = (LLa , RLa ) , (La , Ra ) (L, R) = (LRa , RRa ) , and so the map a → (La , Ra ) identifies A as an ideal in M(A) whenever A is faithful. Note that M(A) = A if and only if A has an identity. In the case where A is a commutative, faithful algebra, left and right multipliers are also right and left multipliers, respectively, and so we identify M(A) with M  (A). Definition 2.1.9 Let A be an algebra that is also a normed space for a norm · . Then (A, · ) is a normed algebra if

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77

ab ≤ a b

(a, b ∈ A) .

In the case where A has an identity eA , the normed algebra A is unital if eA = 1. A normed algebra (A, · ) is a Banach algebra if the normed space (A, · ) is a Banach space. A Banach algebra is reflexive if it is reflexive as a Banach space. A Banach ∗-algebra is a Banach algebra A that has an involution ∗ such that a∗ = a (a ∈ A). Our basic reference for the theory of Banach algebras will be the monograph [50]; other texts are [4, 25, 162, 254, 255, 272]; in particular, [254, 255] is a comprehensive treatise on Banach ∗-algebras. Let (A, · ) be a non-unital normed algebra, and set A = CeA + A, as before. We define zeA + a = |z| + a (z ∈ C, a ∈ A) . Then (A , · ) is also a normed algebra; it is a Banach algebra whenever A is a Banach algebra. Here, a closed subalgebra B of a Banach algebra A is a subalgebra of A that is a closed linear subspace, and hence a Banach algebra. However, we say that Banach algebra (B, · B ) that is a subalgebra of (A, · A ) is closed in A if it is closed as a topological subspace, and so · B ∼ · A on B. The closed subalgebra of A that is generated by a subset S of A is the closure of the set of finite sums of finite products of elements in S ; it is separable when S is countable. Hence each infinite-dimensional Banach algebra A contains a closed subalgebra that is separable and infinite dimensional; for this we take S to be a countably infinite, linearly independent subset of A. Let I be a closed ideal in a Banach algebra A. Then A/I is a Banach algebra with respect to the quotient norm. Let A be an algebra with a right identity e, and set A = eA  J, as on page 73. Then eA and J are closed linear subspaces in A, and the map a → ea is a projection on A with range eA. The following is a theorem of Loy [50, Theorem 2.2.16(i)]. Theorem 2.1.10 Let A be a separable Banach algebra. Then there exist m ∈ N and M > 0 such that, for each a ∈ A2 , there exist a1 , . . . , am , b1 , . . . , bm ∈ A with a=

m  i=1

ai bi

and

m 

ai bi ≤ M a .

i=1

Definition 2.1.11 Let A and B be Banach algebras. A Banach-algebra homomorphism is a homomorphism from A to B that is a bounded linear operator; similarly, we have a Banach-algebra monomorphism / epimorphism / isomorphism. The Banach algebras A and B are (isometrically) isomorphic if there is a (isometric) Banachalgebra isomorphism from A onto B. A (isometric) Banach-algebra embedding is a (isometric) Banach-algebra isomorphism from A onto a closed subalgebra of B.

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Let (A, · A ) and (B, · B ) be Banach algebras. Then we shall usually write A = B when A and B are isomorphic (and the isomorphism is obvious), and sometimes write (A, · A ) = (B, · B ) to indicate that A and B are isometrically isomorphic. Let A and B be Banach algebras, and suppose that B is semisimple. In the case where B is also commutative, every homomorphism from A into B is automatically continuous. In the general case, each epimorphism from A onto B is automatically continuous; this is the famous Johnson’s uniqueness-of-norm theorem [188]. This theorem, with several proofs, and other automatic continuity results are given in [50]. We now mention a class of algebras that involve Banach lattices; for a full account, see [53]. Various examples of Banach lattice algebras will be given later. We recall that the set of positive elements in a Banach lattice E is denoted by E + . Definition 2.1.12 Let A be a Banach algebra that is also a Banach lattice. Then A is a Banach lattice algebra if ab ∈ A+ (a, b ∈ A+ ). Let A and B be Banach lattice algebras. Then a map θ : A → B is a Banach lattice algebra homomorphism if θ is a Banach algebra homomorphism and a lattice homomorphism. Let A and B be Banach lattice algebras. Then a map θ : A → B that is an algebra and a lattice homomorphism is automatically continuous because it is positive. Example 2.1.13 (i) Let X be a non-empty topological space. Then the algebra (C b (X), | · |X ) is a commutative Banach ∗-algebra with respect to the involution described in Example 2.1.6, and it is a Banach lattice algebra. For a non-empty, locally compact space K, the linear subspace C 0 (K) is a closed ∗-ideal in C b (K) and it is also a Banach lattice algebra. It is standard that the character space of C 0 (K) is identified with K; see §3.1. Of course, C 0 (K) is a unital Banach algebra if and only if K is compact. Let A be a Banach algebra, and consider the Banach space C 0 (K, A). This is a Banach algebra for the product defined by ( f g)(x) = f (x)g(x)

( f, g, ∈ C 0 (K, A), x ∈ K) ;

it is commutative when A is commutative. (ii) Let E be a non-zero Banach space. Then B(E) is a unital Banach algebra with respect to the composition of operators, F (E) is an ideal in B(E), and A(E), K(E), and W(E) are each closed ideals in B(E). Let T ∈ K(E). Then the spectrum, σ(T ), of T is either a finite set or a set of the form {0}∪{ζn : n ∈ N}, where (ζn ) is a null sequence in C• . An element ζ ∈ σ(T )\{0} is such that the eigenspace {x ∈ E : T x = ζ x} is finite dimensional.

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Let A be a subalgebra of B(E) that contains F (E) and is such that A is a Banach algebra with respect to some norm, so that A is a Banach operator algebra in the sense of [50, Definition 2.5.1]. In this case, the embedding of A into B(E) is continuous [50, Proposition 2.5.5(i)]. Let A be a Banach operator algebra that is closed in B(E). Then A(E) is the minimum non-zero, closed ideal in A [50, Theorem 2.5.8(ii)]. In particular, K(E) is the minimum non-zero, closed ideal in B(E) when E has the approximation property. Let A be a closed subalgebra of B(E). Then we define Aa = {T  : T ∈ A} , so that Aa is a closed subalgebra of B(E  ) and B(E)a consists of the operators U ∈ B(E  ) such that U is weak-∗ continuous. Notice that B(E)a = B(E  ) if and only if E is reflexive. (iii) Take p with 1 ≤ p ≤ ∞. For each non-empty set S , the Banach space ( p (S ), · p ) is a commutative Banach lattice algebra for the pointwise product. Suppose that 1 < p < ∞. Then  p (S ) is a reflexive Banach algebra, and  p (S )[2] =  p (S )2 =  p/2 (S ) and



 p (S )  p (S ) =  1 (S ) .

Further,  p (S ) is an ideal in  q (S ) when 1 ≤ p ≤ q < ∞, and f g p ≤ f p g q

( f ∈  p (S ), g ∈  q (S )) .

(2.1.3)

The space  ∞ (S ) is a commutative, unital C ∗ -algebra with respect to the pointwise product, where the involution was defined in Example 2.1.6, and c 0 (S ) is a C ∗ -subalgebra of  ∞ (S ). See §2.2 for the basic theory of C ∗ -algebras. For p with 1 ≤ p ≤ ∞ and a sequence (An ) of Banach algebras, set A =  p (An ), taken with the coordinatewise product. Then A is also a Banach algebra. (iv) Let S be a semigroup, and consider the Banach space ( 1 (S ), · 1 ). There is a continuous product  , called convolution, on the space  1 (S ) that is defined to be such that δ s  δt = δ st (s, t ∈ S ). Indeed, for f, g ∈  1 (S ), define  ( f  g)(t) = { f (r)g(s) : r, s ∈ S , rs = t} (t ∈ S ) . (2.1.4) (If there are no elements r, s ∈ S with rs = t, then ( f  g)(t) = 0.) Then the structure ( 1 (S ), · 1 ,  ) is a Banach lattice algebra, called the semigroup algebra on S . Clearly the semigroup algebra is a commutative Banach algebra if and only if the semigroup is abelian. For example, set S = (Z+ , + ), and take f, g ∈  1 (S ). Then the formula for the product is n  f (k)g(n − k) (n ∈ Z+ ) . ( f  g)(n) = k=0

In particular, consider the case where G is a group. Then ( 1 (G), · 1 ,  ) is the group algebra on G; see also §4.1.

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(v) Let S be a semigroup. A weight on S is a map ω : S → R+• such that ω(st) ≤ ω(s)ω(t) (s, t ∈ S ) . In the case where S has an identity, eS , we also require that ω(eS ) = 1. A weight ω on a group G is symmetric if ω(s−1 ) = ω(s) (s ∈ G). Let ω be a weight on a semigroup S . Then the Banach space  1 (S , ω) was defined in Example 1.2.8. It is clear that ( 1 (S , ω), · ω ,  ), where the product is also given by Equation (2.1.4) when f, g ∈  1 (S , ω), is also a Banach lattice algebra, called the weighted semigroup algebra on S ; in the case where S is a group, the corresponding algebra is an example of a Beurling algebra. The algebra  1 (S , ω) is commutative if and only if S is abelian, and it is unital with identity δeS when S is a monoid. Two weights ω1 and ω2 on a semigroup S are equivalent if the two Banach algebras  1 (S , ω1 ) and  1 (S , ω2 ) are isomorphic; properties involving Arens regularity are unaffected by a change to an equivalent weight. Note that we do not necessarily suppose that a weight ω on a semigroup S is such that ω(s) ≥ 1 (s ∈ S ). However, let G be an amenable (see Definition 4.1.20) group,  on G which is equivalent to ω and let ω be a weight on G. Then there is a weight ω (s) ≥ 1 (s ∈ G) [57, Theorem 7.44]. and such that ω (vi) Let E be a non-zero Banach space. Define a product · on E ⊗ E  by requiring that (x1 ⊗ λ1 ) · (x2 ⊗ λ2 ) = x2 , λ1  x1 ⊗ λ2 (x1 , x2 ∈ E, λ1 , λ2 ∈ E  ) as in Equation (1.3.10). Then (E ⊗ E  , · π ) becomes a Banach algebra, called the nuclear algebra of E; using (1.3.10), we see that the map T : E ⊗ E  → N(E) of Equation (1.4.4) is a contractive algebra surjection, so that N(E) is a quotient of the Banach algebra E ⊗ E  and it is a Banach operator algebra on E. Now suppose that E has the approximation property. Then, by Proposition 1.4.16, the Banach algebras E ⊗ E  and N(E) are isomorphic. (vii) Let E be any non-zero Banach space, and define xy = 0 (x, y ∈ E). Then E is a Banach algebra. We say that this algebra has zero product. (viii) Let E be a Banach lattice that is Dedekind complete. Then, by Theorem 1.1.5, (B r (E), · r ) is a Dedekind complete Banach lattice, and it is also a subalgebra of B(E). For S , T ∈ B r (E), we have |S T | ≤ |S | |T |, and so S T ≤ S T . Thus (B r (E), · r ) is a Banach algebra, and it is clear that it is a Banach lattice algebra. Definition 2.1.14 Let A be a Banach algebra, and let E be a Banach space. A Banach algebra representation of A on E is a Banach algebra homomorphism from A to B(E). The left-regular representation of A is the representation a → La ,

A → B(A) .

Let A be a Banach lattice algebra, and let E be a Dedekind complete Banach lattice. A Banach lattice algebra representation of A on E is a Banach lattice algebra

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homomorphism from A to B r (E). Suppose that A is Dedekind complete. The leftregular representation of A is the representation a → La ,

A → B r (A) .

The following is a form of the Tietze extension theorem [4, Theorem 3.39]. Theorem 2.1.15 Let K be a compact space with a non-empty, closed subset L. For each g ∈ C(L), there exists f ∈ C(K) such that f | L = g and | f |K = |g|L , and so the restriction map f → f | L, C(K) → C(L), is a contractive algebra epimorphism. Each character ϕ on a Banach algebra A is continuous with ϕ ≤ 1, and so, for a Banach algebra A, we can regard ΦA as a subset of the closed unit ball A[1] of the dual space. The character space ΦA is always deemed to have the relative weak-∗ topology, σ(A , A), from A unless stated otherwise. (This is the Gel’fand topology on ΦA .) Thus limν ϕν = ϕ in ΦA if and only if limν ϕν (a) = ϕ(a) (a ∈ A). Indeed, ΦA is then a locally compact space and ΦA ∪ {0} is compact, and ΦA is compact in the case where A is unital. Definition 2.1.16 Let A be a unital Banach algebra. Then the state space of A is the set KA = {λ ∈ A : λ = eA , λ = 1} , and the elements of ex KA are the pure states of A. The space KA is a non-empty, convex, and weak-∗-compact subset of A[1] , and hence the convex hull, co ex KA , of ex KA is weak-∗ dense in KA by the Kre˘ın– Mil’man theorem, Theorem 1.2.35. Clearly ΦA ⊂ KA . We shall see in Corollary 3.6.15 that, in the case where A = C(K) for a non-empty, compact space K, the pure states on C(K) are exactly the characters on C(K), so that ΦC(K) = ex KC(K) . Let A and B be Banach algebras, and suppose that θ : A → B is a Banachalgebra homomorphism, so that θ : B → A is an operator. Then it is clear that θ (ΦB ) ⊂ ΦA ∪ {0}. Let A be a Banach algebra. Then each maximal modular left ideal of A is closed in A, and so rad A is always a closed ideal in A; A/rad A is a semisimple Banach algebra. A Banach algebra that is nil is also nilpotent [50, Theorem 2.6.34]. Take a ∈ A. The spectrum, σ(a), of a is a non-empty, compact subset of C, and the spectral radius formula states that sup{|z| : z ∈ σ(a)} = lim an 1/n n→∞

(a ∈ A) .

We shall use the following theorem of Katznelson and Tzafriri; our form is given, with a short proof, in [5, Theorem 1.2]. An element a in a Banach algebra A is power-bounded if sup{ an : n ∈ N} < ∞; in this case, there is an equivalent algebra norm ||| · ||| on A such that |||a||| ≤ 1.

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Theorem 2.1.17

Let A be

a Banach algebra, and take a ∈ A to be power-bounded. Then limk→∞

ak − ak+1

= 0 if and only if σ(a) ∩ T ⊂ {1}. Now suppose that A is a commutative Banach algebra. Then σ(a) = {ϕ(a) : ϕ ∈ ΦA } ; the maximal modular ideals of A have codimension 1, and they are exactly the kernels of the characters on A. For ϕ ∈ ΦA , we shall write Mϕ = Mϕ (A) = ker ϕ for the corresponding maximal modular ideal; clearly Φ Mϕ = ΦA \{ϕ}. In the case where A is not radical, so that ΦA  ∅, the Gel’fand transform is the algebra homomorphism and contraction G : a → a,

A → C 0 (ΦA ) ,

where a(ϕ) = ϕ(a) (ϕ ∈ ΦA ); we then have    {Mϕ : ϕ ∈ ΦA } . rad A = a ∈ A : lim an 1/n = 0 = ker G = n→∞

Thus the algebra A is semisimple if and only if for each a ∈ A such that ϕ(a) = 0 (ϕ ∈ ΦA ) implies that a = 0. We shall use the following famous Šilov’s idempotent theorem [50, Theorem 2.4.33]. Theorem 2.1.18 Let A be a commutative Banach algebra, and suppose that K is a non-empty, compact, and open subset of ΦA . Then there is a unique p ∈ I(A) such that p = χK . Example 2.1.19 Let S be a semigroup, with semigroup algebra ( 1 (S ),  ). It is standard and easily seen that, for each semi-character θ ∈ ΦS , the map   f (s)δ s → f (s)θ(s) ,  1 (S ) → C , s∈S

s∈S

is a character on the Banach algebra  1 (S ), and that each character on  1 (S ) arises in this way; the topology of pointwise convergence on ΦS coincides with the Gel’fand topology when ΦS is viewed as the character space of  1 (S ). The character on  1 (S ) that corresponds to the augmentation character, 1S , on S is the augmentation character ϕS , where   f (s)δ s → f (s) . ϕS : s∈S

s∈S

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83

Now let S be an abelian semigroup, so that  1 (S ) is a commutative Banach algebra. Then  1 (S ) is semisimple if and only if ΦS separates the points of S , in the sense that, for each s, t ∈ S with s  t, there exists θ ∈ ΦS such that θ(s)  θ(t) [170, Theorem 3.5], and this holds if and only if S is separating, in the sense that s = t whenever s, t ∈ S with st = s2 = t2 [170, Theorem 5.8]. Certainly an abelian semigroup that is either cancellative or an idempotent semigroup is separating.  Let S be an abelian semigroup, and let f = s∈S α s δ s belong to  1 (S ). Then  α s θ(s), ΦS → C , f : θ → θ( f ) = s∈S

is the Fourier transform of f , the map F : f → f is the Fourier transform, and A(ΦS ) := { f : f ∈  1 (S )} is the algebra of Fourier transforms of elements of  1 (S ). Thus F is a homomorphism which can be identified with the Gel’fand transform of  1 (S ), and A(ΦS ) is a Banach function algebra (see §3.1) on the locally compact space ΦS ; when S is separating, the map F :  1 (S ) → A(ΦS ) is an isomorphism. The algebra A(ΦS ) is unital when S is a monoid. Let S be an abelian, idempotent semigroup (i.e., a semilattice), and take θ ∈ ΦS . Then δ s (θ) = θ(s) ∈ {0, 1} (s ∈ S ). Now take θ1 , θ2 ∈ ΦS with θ1  θ2 . Then there exists s ∈ S such that {θ1 (s), θ2 (s)} = {0, 1}. The analogous definitions for group algebras on locally compact abelian groups will be given in §4.2.

Example 2.1.20 Let ω be a weight on a semigroup S , and consider the weighted semigroup algebra  1 (S , ω) of Example 2.1.13(v). Then ω is radical or semisimple if  1 (S , ω) is a radical or semisimple Banach algebra, respectively. For s ∈ S , we set ν s = inf{ω(sn )1/n : n ∈ N} = lim ω(sn )1/n , n→∞

and so ν s is the spectral radius of δ s in the Banach algebra  1 (S , ω). Now suppose that S is abelian. Then ω is semisimple if and only if S is separating and ν s > 0 (s ∈ S ). See [52, Proposition 4.8]. For example, let ω : N → [1, ∞) be any function. Then ω is a weight on the idempotent semigroups N∧ and N∨ of Example 1.1.3(ii), and so the weighted semigroup algebras ( 1 (N∧ , ω), · ω ,  ) and ( 1 (N∨ , ω), · ω ,  ) are semisimple, commutative Banach algebras. See also Examples 2.1.25, 2.4.31 and 3.2.12.

We now consider multipliers on Banach algebras. It is clear that La , Ra ∈ B(A) for a ∈ A when A is a Banach algebra. When A is a faithful Banach algebra, each

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multiplier (L, R) is such that L and R are bounded operators on A. Indeed, suppose that (an ) is a null sequence in A such that Lan → b in A. For each c ∈ A, we have cb = lim cL(an ) = lim R(c)an = 0 , n→∞

n→∞

and so b = 0 because A is faithful. By the closed graph theorem, Theorem 1.3.6, L is continuous; similarly, R is continuous. Thus M(A) is a closed, unital subalgebra of the Banach algebra B(A) × B(A)op , where the norm on the latter algebra is given by (S , T ) op = max{ S op , T op } (S , T ∈ B(A)) . In the case where the Banach algebra A is faithful, so that A is identified with an ideal in M(A), we obtain a relative norm · op on A, so that a op = max{ La , Ra }

(a ∈ A) .

Thus (A, · op ) is a normed algebra such that a op ≤ a (a ∈ A). Suppose that R ∈ M r (A), and take a, b ∈ A and λ ∈ A . Then b, R (λ · a) = Rb, λ · a = aRb, λ = R(ab), λ = b, R (λ) · a , and so R (λ · a) = R (λ) · a. When A is commutative and faithful, the space M(A) is a commutative, unital subalgebra of (B(A), · op ), and, in this case, we have a op = La op (a ∈ A). For a general discussion of multipliers, see [50, §2.5] and the text [208] of Larsen. A considerable part of this work will be concerned with approximate identities of various types in Banach algebras, and we now give the first definitions of these approximate identities. Definition 2.1.21 Let A be a Banach algebra. A left approximate identity, respectively, a right approximate identity for A is a net (eν ) in A such that lim eν a = a , ν

respectively,

lim aeν = a (a ∈ A) . ν

A net (eν ) in A is an approximate identity if it is both a left and right approximate identity. An approximate identity is sequential if it is a sequence in A indexed by N; a left or right approximate identity (eν ) is bounded if supν eν < ∞, and then supν eν is the bound of the approximate identity; an approximate identity is contractive if it has bound 1. A subset S of A has approximate units if, for each a ∈ S and each ε > 0, there exist u, v ∈ A such that a − ua + a − av < ε. We shall sometimes refer to a BLAI, a BRAI, a BAI, and a CAI for ‘bounded left approximate identity’, ‘bounded right approximate identity’, ‘bounded approximate identity’, and ‘contractive approximate identity’, respectively. A Banach algebra with an approximate identity is faithful, and each subset S of A has approximate units when A has an approximate identity. Note that a separable

2.1 Algebras and Banach algebras

85

Banach algebra with a bounded approximate identity has a sequential bounded approximate identity. Let A be a Banach algebra, and suppose that A has an identity, eA . Then eA · λ = λ · eA = λ (λ ∈ A ). Suppose that (eν ) is an approximate identity in A. For each λ ∈ A , we have (2.1.5) wk∗ – lim eν · λ = wk∗ – lim λ · eν = λ . ν

ν

Approximate identities are considered in some detail in [50, §2.9], where the following result is effectively given in Proposition 2.9.14 of the book [50]. Proposition 2.1.22 Let A be a Banach algebra.

each a1 , . . . , an ∈ A and ε > 0, there exist u, v ∈ A with

that, for

(i) Suppose

a j − ua j

+

a j − a j v

< ε ( j ∈ Nn ). Then A has an approximate identity. (ii) Take m ≥ 1. Suppose that there is a dense subset S of A such that S has approximate units contained in A[m] . Then A has a bounded approximate identity of bound m. The following is an easy consequence of Proposition 2.1.22. Corollary 2.1.23 Let A be a Banach algebra with a bounded approximate identity, and let B be a Banach algebra. Suppose that there is a Banach-algebra homomorphism θ : A → B such that θ(A) is dense in B. Then B has a bounded approximate identity. Example 2.1.24 Let (S , + ) be a cancellative, abelian semigroup that does not have an identity. Take s ∈ S . Then, for each t ∈ S , necessarily s  s + t, and so δ s − δ s  f ≥ 1 for each f ∈  1 (S ). Thus the semigroup algebra  1 (S ) does not have approximate units, and hence no approximate identity.

Example 2.1.25 Let ω : N → [1, ∞) be any function, and write Aω =  1 (N∧ , ω) for the weighted semigroup algebra of Example 2.1.20. As in Example 1.3.3, Aω   1 has the Schur property, and so Aω is weakly sequentially complete. Suppose that lim inf n→∞ ω(n) < ∞. Then there is a strictly increasing sequence (tn ) in N and M > 0 such that ω(tn ) < M (n ∈ N). Thus (δtn ) is a bounded sequence 0 in Aω . Consider an element f := nn=1 αn δn ∈ Aω , where n0 ∈ N and α1 , . . . , αn0 ∈ C. Then f = f  δtn whenever tn > n0 . It follows that (δtn ) is a bounded approximate identity for Aω , with bound M. We now claim that Aω always has an approximate identity of the form (δnk ). Indeed, we may suppose that limn→∞ ωn = ∞. Choose n1 ∈ N with ωn1 = inf n∈N ωn .

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2 Banach algebras

Now assume that n1 , . . . , nk ∈ N have been defined such that n1 < · · · < nk . Then there exists nk+1 > nk such that ωnk+1 = inf n≥nk +1 ωn . This gives us a strictly increas ing sequence (nk ) in N. Now take f = ∞ n=1 αn δn ∈ Aω . Then ∞ ∞  



f − f  δnk

ω ≤ |αn | (ω(n) + ω(nk )) ≤ 2 |αn | ω(n) → 0 as n=nk

k → ∞,

n=nk

giving the claim.

Example 2.1.26 Consider the semigroup algebra A = ( 1 (Z∨ ),  ), and set ⎧ ⎫ ⎪ ⎪   ⎪ ⎪ ⎨ ⎬ M=⎪ f = αn δn ∈ A : αn = 0⎪ , ⎪ ⎪ ⎩ ⎭ n∈Z

n∈Z

so that M is a maximal modular ideal in A and A = Cp ⊕ M, where p := δ0 ∈ I(A). Thus A and M are weakly sequentially complete. It is clear that the algebra A is not unital, that (δ−n : n ∈ Z+ ) is a contractive approximate identity for A, and also that (δ−n − δn+1 : n ∈ Z+ ) is a bounded approximate identity for M.

The following result of Ülger is from [310, Lemma 3.4]. Proposition 2.1.27 Let A be a Banach algebra with a bounded approximate identity of bound m, and take S to be a separable subset of A. Then there is a closed, separable subalgebra B of A such that B ⊃ S and B has a sequential bounded approximate identity of bound m. Further, there is a closed ideal I in A such that I ⊃ S and I has a sequential bounded approximate identity of bound m. Proof Take {an : n ∈ N} to be a dense subset of S . We inductively define a sequence (en ) in A[m] as follows. First choose e1 ∈ A[m] such that e1 a1 − a1 + a1 e1 − a1 < 1. Now suppose that e1 , . . . , en have been specified, set Yn = {a1 , . . . , an , e1 , . . . , en }, and then choose en+1 ∈ A[m] such that en+1 y − y + yen+1 − y
0. Suppose that I[m] contains a weak approximate identity for A. Then A = I, and A[m] contains a bounded approximate identity for A. Proof Take a ∈ A. By hypothesis, a belongs to the weak closure of the convex set I[m] a, and so, by Mazur’s theorem, Theorem 1.2.24(ii), a ∈ I[m] a, whence there is a sequence (bn ) in I[m] with limn→∞ bn a = a. Thus A = I. Similarly, there is a sequence (cn ) in I[m] with limn→∞ acn = a, and it now follows from Proposition 2.1.22(ii) that A has a bounded approximate identity in A[m] . Proposition 2.1.30 Let A be a Banach algebra with a bounded approximate identity of bound m. Then a op ≤ a ≤ m a op (a ∈ A) , and so · and · op are equivalent on A and A is Banach-algebra isomorphic to a closed ideal in M(A). In particular, a op = a (a ∈ A) when A has a contractive approximate identity. The following example shows that the converse of the above proposition does not hold. Example 2.1.31 Consider the semigroup algebra ( 1 (N, + ),  ). Then δ1 1 = 1. For each f ∈  1 (N), clearly f  δ1 1 = f 1 , and so f op = f 1 . However, ( 1 (N, + ),  ) does not have an approximate identity. Again, set S = (Q+• , +), define ω(s) = exp(−s2 ) (s ∈ S ), and consider the 1 weighted semigroup algebra  1 (S , ω). Then ω is a radical

weight, and so  (S , ω) is



a commutative, radical Banach

algebra.

We see that δ1/n ω ≤ 1 (n ∈ N). For each f ∈  1 (S ), clearly limn→∞

f  δ1/n

ω = f ω , and so f ω,op = f ω . However,  1 (S , ω) does not have an approximate identity. For a further example involving a maximal ideal in a uniform algebra, see Example 5.3.17.

Definition 2.1.32 Let A be a Banach algebra, and take m > 0. An approximate identity (eν ) in A is a multiplier-bounded approximate identity (MBAI) with bound m if eν a ≤ m a and aeν ≤ m a for each ν and each a ∈ A. Let A be a Banach algebra with a sequential approximate identity (en ). Then, by the uniform boundedness theorem, Theorem 1.3.7, the sequence (en ) is a multiplierbounded approximate identity.

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We shall see that many commutative Banach algebras that do not have a bounded approximate identity do have a multiplier-bounded approximate identity; the existence of a multiplier-bounded approximate identity will lead to a result on BSE norms in Theorem 5.5.17. Definition 2.1.33 Let A and B be Banach algebras, and take T ∈ B(A, B ). Then HT := {T a · b : a ∈ A[1] , b ∈ B[1] } . Thus HT is a subset of B . Proposition 2.1.34 Let A and B be non-zero Banach algebras such that B has a bounded approximate identity. (i) Suppose that HT is relatively weakly compact in B for each T ∈ B(A, B ). Then B(A, B ) = W(A, B ). (ii) Suppose that HT is relatively compact in B for each T ∈ B(A, B ). Then B(A, B ) = K(A, B ). (iii) Suppose that B has the Schur property and B(A, B ) = W(A, B ). Then HT is relatively compact in B for each T ∈ B(A, B ). Proof Take (eα ) to be a BAI in B, and suppose that T ∈ B(A, B ). Let V be the weak-∗ closure of HT in B . For each a ∈ A[1] , we have wk∗ – limα T a · eα = T a in B by Equation (2.1.5), and so the set T (A[1] ) is contained in V. (i) Since HT is relatively weakly compact in B , it follows from Proposition 1.2.11 that T (A[1] ) is relatively weakly compact, and so T is weakly compact. Hence B(A, B ) = W(A, B ). (ii) Since, by hypothesis, the set HT is relatively compact, the weak-∗ and the norm closures of HT are the same. So V is norm compact. As the set V contains the set T (A[1] ), the operator T is compact. Hence B(A, B ) = K(A, B ). (iii) Since B has the Schur property, W(A, B ) = K(A, B ) by Proposition 1.3.56, and so B(A, B ) = K(A, B ). Thus T (A[1] ) is relatively compact, and hence totally bounded, in B . For each ε > 0, there exist n ∈ N and μ1 , . . . , μn ∈ B such that  T (A[1] ) ⊂ {Bε (μi ) : i ∈ Nn } . By Proposition 1.3.72, the maps Lμ1 , . . . , Lμn are compact, and so the set  {{μi · b : b ∈ B[1] } : i ∈ Nn } Hε :=  is compact in B . Clearly HT ⊂ Hε + B[ε] , and so HT ⊂ {Hε : ε > 0}, a compact set in B . This implies that HT is relatively compact in B . We now consider the projective tensor product of two Banach algebras. Let A and B be algebras, and set A = A ⊗ B. Then there is a unique product on A with respect to which A is an algebra and such that

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89

(a1 ⊗ b1 )(a2 ⊗ b2 ) = a1 a2 ⊗ b1 b2

(a1 , a2 ∈ A, b1 , b2 ∈ B) .

The linear map that identifies A ⊗ B and B ⊗ A is an isomorphism, and so A ⊗ B and B ⊗ A are isomorphic algebras. Of course, A is commutative whenever A and B are commutative. We see that L a ⊗ b = La ⊗ Lb

(a ∈ A, b ∈ B) .

Now suppose that A and B are Banach algebras. Then A is a normed algebra with respect to the projective norm · π , and so its completion (A ⊗ B, · π ) is a Banach   algebra. Indeed, take a = mj=1 a1, j ⊗ b1, j and b = nk=1 a2,k ⊗ b2,k in A ⊗ B. Then



m  m  n n





a a



b b

a1, j a2,k ⊗ b1, j b2,k

≤ ab π ≤

1, j 2,k 1, j 2,k



j=1 k=1 j=1 k=1 π ⎛ m ⎞⎛ n ⎞ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜







⎟⎟⎟



⎜ ⎜ ⎟ ≤ ⎜⎝⎜ a1, j b1, j ⎟⎠⎟ ⎜⎜⎝ a2,k b2,k ⎟⎟⎠ , j=1

k=1

⊗ B, · π ) is a Banach algebra. and so ab π ≤ a π b π . This implies that (A Suppose that A0 , A, B0 and B are Banach algebras, and that there are Banachalgebra homomorphisms θA : A0 → A and θB : B0 → B. Then we have seen ⊗ B0 → A ⊗ B is a continuous linear operator such that that the map θA ⊗ θB : A0 θA ⊗ θB op = θA op θB op . It is clear that θA ⊗ θB is a homomorphism. It is also clear that the Banach algebras A ⊗ B and B ⊗ A are isometrically isomorphic. of Let A and B be faithful Banach algebras. Then it is clear that the map Ψ : M(A) Equation (1.4.8) induces a contraction Ψ ⊗ M(B) → M(A ⊗ B) such that



Ψ (L ⊗ M)

= L ⊗ M op for L ∈ M(A) and M ∈ M(B). op

Let A and B be Banach ∗-algebras. Then the Banach algebra (A ⊗ B, · π ) is also a Banach ∗-algebra with respect to an involution that satisfies the condition that (a ⊗ b)∗ = a∗ ⊗ b∗ (a ∈ A, b ∈ B). Example 2.1.35 Let A be a Banach algebra. In Example 1.4.2(i), we noted that there is a canonical linear isometry T :  1 ⊗ A →  1 (A) with T (α ⊗ a) = (αn a) for 1 α = (αn ) ∈  and a ∈ A. It is clear that the map T is a homomorphism, and so ⊗ A and  1 (A) are isometrically Banach-algebra isomorphic. 1

Example 2.1.36 Let S and T be semigroups, with semigroup algebras  1 (S ) and  1 (T ), respectively, and let S ×T be the product semigroup. In Example 1.4.2(ii), we ⊗  1 (T ), · π ) = ( 1 (S ×T ), · 1 ). It is clear that this identification noted that ( 1 (S ) ⊗  1 (T ), · π ) and ( 1 (S × T ), · 1 ) are is an algebra isomorphism, and so ( 1 (S ) isometrically Banach-algebra isomorphic.

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The next proposition follows immediately from Theorem 1.4.18(iii). Proposition 2.1.37 Let A and B be Banach algebras such that either A or B has the bounded approximation property as a Banach space. Then the canonical embedding of A ⊗ B into (A ⊗ B) extends to an isometric Banach-algebra embedding of    ⊗ B) . A ⊗ B into (A We shall use the following consequence of [50, Proposition 2.9.21]. Proposition 2.1.38 Let A and B be Banach algebras, and suppose that A and B each have bounded approximate identities, of bounds m and n, respectively. Then A ⊗B has a bounded approximate identity of bound mn.

 Proof Take kj=1 a j ⊗ b j in A ⊗ B, and set r = max{

a j

,

b j

: j ∈ Nk }. For each





exist u ∈ A[m] and v ∈ B[n] such that a j − ua j < ε/(3r + 1)k and

ε > 0, there

b − vb < ε/(3r + 1)k for each j ∈ N . Now u ⊗ v ≤ mn and j

j

k



k k





a j ⊗ b j − (u ⊗ v) a j ⊗ b j



j=1

j=1

π

π

k  

a − ua



b

+

a



b − vb

+

a − ua



b − vb

 < ε . ≤ j j j j j j j j j j j=1

A similar calculation on the right-hand side, and an application of Proposition 2.1.22(ii), shows that A ⊗ B has a BAI of bound mn. The converse of Proposition 2.1.38 also holds. Proposition 2.1.39 Let A and B be non-zero Banach algebras, and suppose that A ⊗ B has a bounded approximate identity. Then A and B each have a bounded approximate identity. ⊗ B → B, so that (λ ⊗ IB )(a ⊗ b) = a, λb for Proof For λ ∈ A , define λ ⊗ IB : A each a ∈ A and b ∈ B. Then (Ra (λ) ⊗ IB )(c ⊗ d) · b = c, Ra (λ)db = ca, λdb = (λ ⊗ IB )((c ⊗ d)(a ⊗ b)) for all a, c ∈ A and b, d ∈ B. Now take (zα ) to be a BAI in A ⊗ B. Then lim(Ra (λ) ⊗ IB )(zα ) · b = lim(λ ⊗ IB )(zα (a ⊗ b)) = (λ ⊗ IB )(a ⊗ b) α

α

for all a ∈ A and b ∈ B. In particular, take a ∈ A with a = 1 and λ ∈ A with λ = a, λ = 1, and set eα = (Ra (λ) ⊗ IB )(zα ) ∈ B, so that eα b → b (b ∈ B) and eα ≤ zα . Similarly, there is a bounded net ( fα ) in B such that b fα → b (b ∈ B), and so B has a BAI. Clearly A and B also have a BAI.

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Definition 2.1.40 A be a Banach algebra, and let E be a Banach space such that (E, · ) is a left A-module. Then E is a Banach left A-module if a · x ≤ a x

(a ∈ A, x ∈ E) .

Similarly, we define a Banach right A-module. A Banach A-bimodule is a left A-module and a right A-module E that is such that (a · x) · b = a · (x · b)

(a, b ∈ A, x ∈ E) .

In the case where A is commutative and E is a Banach A-bimodule with the property that a · x = x · a (a ∈ A, x ∈ E), we say that E is a Banach A-module. In particular, let (I, · I ) be a Banach algebra that is an ideal in a Banach algebra (A, · A ) such that ax I ≤ a A x I and xa I ≤ a A x I for each a ∈ A and x ∈ I. Then I is a Banach A-bimodule. For example, let E be a Banach space, and take I to be N(E) or K(E). Then I is a Banach B(E)-bimodule. Again, let S be a non-empty set, and take p, q with 1 ≤ p ≤ q < ∞. Then it follows from Equation (2.1.3) that  p (S ) is a dense Banach  q (S )-module. Let A be a Banach algebra, and let E be a Banach left A-module. Then the map ρ : A → B(E) defined by ρ(a)(x) = a · x

(a ∈ A, x ∈ E)

is a contractive homomorphism, and so is a representation of A on E. Conversely each such representation defines a Banach left A-module action of A on E. Thus the language of representations is often used instead of the language of A-modules. Suppose that E is a Banach A-bimodule. Then the dual space E  is also a Banach A-bimodule; the module operations on E  and E  are defined by x, λ · a = a · x, λ ,

x, a · λ = x · a, λ (a ∈ A, x ∈ E, λ ∈ E  )

and  Λ · a, λ =  Λ, a · λ ,

a · Λ, λ = Λ, λ · a

(a ∈ A, λ ∈ E  , Λ ∈ E  ) .

In particular, A is a Banach A-bimodule over itself, and A and A are also Banach A-bimodules, called the dual and bidual module of A, respectively. The dual module A is a Banach A-bimodule, and A · A = {λ · a : a ∈ A, λ ∈ A }

and

A A = lin A · A .

Let a ∈ A. Then we note that La (λ) = λ · a ,

Ra (λ) = a · λ (λ ∈ A ) ,

(2.1.6)

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and that

La

op = La op and

Ra

op = Ra op . Similarly, in the bidual module A , we have (2.1.7) La (M) = a · M , Ra (M) = M · a (a ∈ A, M ∈ A ) . Let A be a commutative Banach algebra. Then a · λ = λ · a (λ ∈ A ), and a · M = M · a (M ∈ A ) for each a ∈ A, so that AA = A A and AA = A A. Let A be a Banach algebra, and suppose that E and F are Banach left A-modules. Then a map T : E → F is a Banach left A-module homomorphism if T ∈ B(E, F) and T is a left A-module homomorphism, etc. The spaces E and F are isomorphic as Banach left A-modules if there is a Banach left A-module isomorphism from E onto F. Similarly, we shall consider Banach A-bimodule isomorphisms between two Banach A-bimodules, etc. Proposition 2.1.41 Let (A, · A ) be a Banach algebra, and let (I, · I ) be a dense ideal in A that is a Banach A-bimodule and such that x A ≤ x I (x ∈ I). Suppose that I has a bounded approximate identity. Then I = A. Proof Take (eα ) to be a BAI in I with bound m. For x ∈ I, we have x A ≤ x I = lim xeα I ≤ m x A , α

and so · I ∼ · A on I. Since I is dense, it follows that I = A. Proposition 2.1.42 Let A be a Banach algebra with a multiplier-bounded approximate identity (eν ) of bound m. Then eν · λ ≤ m λ

(λ ∈ A ) .

Proof Take λ ∈ A . For each a ∈ A[1] , we have |a, eν · λ| = |aeν , λ| ≤ m λ , giving the result. Proposition 2.1.43 Let A be a Banach algebra, and take F to be a Banach Abimodule in A . Then F  is a closed ideal in A. Proof We have noted that F  is a closed linear subspace of A. Take a ∈ F  and b ∈ A. Then ab, λ = a, b · λ = 0 (λ ∈ F), and so ab ∈ F  . Similarly, ba ∈ F  . Thus F  is an ideal in A. Example 2.1.44 Let E be a non-zero Banach space, and set A = (E ⊗ E  , · π ), the nuclear algebra of E, as in Example 2.1.13(vi). Then A is a Banach B(E)-bimodule for the operations · that agree with the product given in Equations (1.3.11). There is a unique continuous linear functional tr on A such that tr(x ⊗ λ) = x, λ

(x ∈ E, λ ∈ E  ) ;

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the map tr is called the trace map on A. Clearly tr(S · T ) = tr(T · S ) ,

|tr(S · T )| ≤ S π T op

(S ∈ A, T ∈ B(E)) .

As in Equation (1.4.5), the dual module of A is B(E, E  ). Suppose that E has the approximation property, so that E ⊗ E  = N(E). Then we   obtain a trace map on N(E). In this case, N(E) = B(E, E ), and the dual module operations on B(E, E  ) are given by S · T = S  ◦ T ,

T · S =T ◦ S

(S ∈ N(E), T ∈ B(E, E  )) .

(2.1.8)

Now suppose that E is reflexive and has the approximation property. In this case, the dual module of N(E) is B(E); we identify the duality. Indeed, take T ∈ B(E). Then the element in N(E) corresponding to T is ΛT , where x ⊗ λ, ΛT  = T x, λ = tr(T x ⊗ λ) (x ∈ E, λ ∈ E  ) , and so we have the trace duality: S , λT  = tr(T S ) (S ∈ N(E), T ∈ B(E)) . Take S ∈ N(E) and T ∈ B(E). Then, for each R ∈ N(E), we have R, S · ΛT  = RS , ΛT  = tr(T RS ) ,

R, ΛS T  = tr(S T R) = tr(T RS ) ,

and so S · ΛT = ΛS T . Similarly, ΛT · S = ΛT S , and hence the dual module operations on N(E) coincide with the usual product on B(E).

Definition 2.1.45 Let A be a Banach algebra, and let E be a Banach A-bimodule. Then E is essential if E = AEA and neo-unital if E = A · E · A. A very important theorem is Cohen’s factorization theorem; several (more general) forms of this theorem are given in [50, §2.9]. Theorem 2.1.46 Let A be a Banach algebra with a bounded left approximate identity of bound m. For each Banach left A-module E, each x ∈ AE, and each ε > 0, there exist a ∈ A[m] and y ∈ E with x = a · y and y − x < ε, so that AE = A · E. In particular, suppose that I is a left ideal in A that is a Banach left A-module. Then, for each x ∈ I, there exist a ∈ A and y ∈ I with x = ay. Thus the two-sided form of Cohen’s factorization theorem shows that each essential Banach A-bimodule is neo-unital whenever A has a bounded approximate identity. Corollary 2.1.47 Let A be a Banach algebra with a bounded approximate identity. Then A factors. Also AA = A · A and A A = A · A.

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Corollary 2.1.48 Let A be a Banach algebra with a bounded approximate identity. Then, for each null sequence (an ) in A, there exist a ∈ A and a null sequence (bn ) in A such that an = abn (n ∈ N). Proof Set E = c 0 (A). Then E is a Banach left A-module for the module operation given by a · (bn ) = (abn ) (a ∈ A, (bn ) ∈ E). Clearly AE = E, and so the corollary follows from Theorem 2.1.46. Corollary 2.1.49 Let A be a Banach algebra with a bounded approximate identity. Then each left and each right multiplier on A is continuous. Proof Let L ∈ M  (A), and take (an ) in c 0 (A), say an = abn (n ∈ N) for a ∈ A and (bn ) in c 0 (A), as above. Then L(an ) = L(abn ) = L(a)bn → 0 as n → ∞, and so L is continuous by Theorem 1.3.6. Similarly, each right multiplier is continuous. Thus M  (A) and M r (A) are subalgebras of B(A) and B(A)op , respectively, that are closed in B(A) when A is a Banach algebra with a bounded approximate identity.

2.2 C∗ -algebras and von Neumann algebras We shall require a few basic facts about C ∗ -algebras and von Neumann algebras, and we shall recall briefly some of these here. An important theorem for later applications will be Kaplansky’s density theorem, to be given in Theorems 2.2.11 and 2.2.14. We shall note in Theorem 2.2.12 a key theorem of Pfitzner that every C ∗ -algebra has property (V) as a Banach space. In particular, we shall discuss the projective tensor product of two C ∗ -algebras and the notion of a ‘scattered’ C ∗ algebra. The multitude of texts on C ∗ -algebras includes [85, 190, 191, 244, 257, 288, 300]. Definition 2.2.1 A Banach algebra (A, · ) is a C ∗ -algebra if it has an involution, denoted by ∗ , and if a∗ a = a 2 (a ∈ A) . A C ∗ -subalgebra of A is a subalgebra of A which is ∗-closed and norm-closed. Let A be a C ∗ -algebra. It follows immediately that a∗ = a (a ∈ A), and so a C ∗ -algebra is a Banach ∗-algebra; each C ∗ -algebra is semisimple. For example, as already stated, for each non-empty topological space X, the Banach space (C b (X), | · |X ) is a commutative, unital C ∗ -algebra, and, for a locally compact space K, the space C 0 (K) is a C ∗ -subalgebra of C b (K). An element a of a C ∗ -algebra A is normal if aa∗ = a∗ a; when A is unital, a ∈ A is unitary if aa∗ = a∗ a = eA . The unitary elements of a unital C ∗ -algebra form a group

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with respect to the product of A; this is called the unitary group, U(A), of A. For a ∈ U(A), we have a−1 = a∗ . The following two propositions give very standard properties of C ∗ -algebras. For a C ∗ -algebra A, an element a ∈ A is positive if a = a∗ and σ(a) ⊂ R+ . Proposition 2.2.2 Let A be a C ∗ -algebra, and take a ∈ A. Then the following are equivalent: (a) a ∈ A+ ; (b) a is positive; (c) a = b2 for some b ∈ Asa with ab = ba; (d) a = b∗ b for some b ∈ A. Take a ∈ A+ . Then there is a unique b ∈ A+ with b2 = a, and we set b = a1/2 , so that a1/2 is the positive square root of a. Proposition 2.2.3 Let A be a C ∗ -algebra, and let I be a closed ideal in A. Then: (i) I is a ∗-ideal, and hence I and A/I are C ∗ -algebras; (ii) there is a contractive approximate identity for I contained in I + . Definition 2.2.4 Let A and B be C ∗ -algebras. A C ∗ -homomorphism from A to B is a ∗-homomorphism; this map is a C ∗ -embedding if, further, it is an injection, and it is a C ∗ -isomorphism if, further, it is a bijection; A and B are C ∗ -isomorphic if there is a C ∗ -isomorphism from A onto B. The following theorem is given in [50, Corollary 3.2.4 and Theorem 3.2.23], etc. Theorem 2.2.5 Let A be a Banach ∗-algebra, and let B be a C ∗ -algebra. Then each ∗-homomorphism from A to B is a contraction. In the case where A is a C ∗ -algebra, the range of the ∗-homomorphism is a C ∗ -subalgebra of B. A C ∗ -embedding is an isometry. Let A and B be C ∗ -algebras that are isomorphic as algebras. Then in fact they are C -isomorphic by a theorem of Gardner [128, Theorem B]. Let A be a C ∗ -algebra, and suppose that a ∈ A is normal. Then there is a unique ∗ C -embedding θ : f → f (a) , C 0 (σ(a)\{0}) → A , ∗

such that θ(Z) = a. The map θ is the continuous functional calculus for a. In particular, we can define |a| p = (a∗ a) p/2 ∈ A+ for p > 0. For the following famous Gel’fand–Naimark theorem for commutative C ∗ -algebras, see [50, Theorem 3.2.6], etc. Theorem 2.2.6 Let A be a non-zero, commutative C ∗ -algebra. Then ΦA  ∅ and the Gel’fand transform G : A → C 0 (ΦA ) is a C ∗ -isomorphism. In the case where A is unital, the space ΦA is compact and G : A → C(ΦA ) is a unital C ∗ -isomorphism.

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Examples (i) Let X be a completely regular topological space. Then the map f → f β ,

C b (X) → C(β X) ,

is a C ∗ -isomorphism, the character space of the commutative C ∗ -algebra C b (X) is ˇ identified with the Stone–Cech compactification β X, and the dual Banach space of b C (X) is identified with M(β X). In particular, as stated earlier, throughout we shall identify  ∞ with C(β N). (ii) The canonical non-commutative, unital C ∗ -algebra is the algebra B(H), where H is a Hilbert space that has an inner product denoted by [ · , · ] and, for each T ∈ B(H), we define T ∗ ∈ B(H) by [x, T ∗ y] = [T x, y]

(x, y ∈ H) ,

so that ∗ : T → T ∗ is an involution on B(H). Then T is positive if and only if [T x, x] ≥ 0 (x ∈ H). The unitary group of B(H) is denoted by U(H).  Let H be a Hilbert space. The weak operator topology and the strong operator topology on B(H) are defined by the seminorms p x,y : T → | [T x, y] |

(x, y ∈ H)

and p x : T → T x

(x ∈ H) ,

respectively; they are denoted by wo and so, respectively. Thus (B(H), wo) and (B(H), so) are locally convex spaces. We note that B(H)[1] is wo-compact. For each so wo convex subset L of B(H), we have L = L , and so L is wo-closed if and only if it is so-closed. 

For

y ∈ H, define τy (x) = [x, y] (x ∈ H). Then τy ∈ H , the dual space of H, with 

τy = y , and each element of H has this form. Further, the map y → τy ,

H → H ,

is a conjugate-linear isometry. The space H  is a Hilbert space with respect to the inner product defined by [τy , τz ] = [z, y] (y, z ∈ H). The space B(H) is identified with the dual space of the projective tensor product H ⊗ H  , as in Equation (1.4.5), so that, given Λ ∈ (H ⊗ H  ) , the corresponding operator T Λ ∈ B(H) is specified by x ⊗ τy , Λ = T Λ x, τy  = [T Λ x, y]

(x, y ∈ H) .

The weak-∗ topology σ(B(H), H ⊗ H  ) given by this duality is also called the ultra weak topology. Since each element u of H ⊗ H  has the form u = ∞ n=1 xn ⊗yn , where ∞ xn , yn ∈ H (n ∈ N) and n=1 xn yn < ∞, it follows that the ultra-weak topology and the weak operator topology coincide on bounded subsets of B(H). The closure uw of a subset B of B(H) in the ultra-weak topology is denoted by B .

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For more information concerning these topologies on B(H), see [191, §7.4] and [244, §4.2]. Let H be a non-zero Hilbert space, and take p with 1 ≤ p < ∞. First, take T ∈ B(H)+ . The trace of T is given by  tr(T ) = {[T x, x] : x ∈ S } , where S is an orthonormal basis of H; the trace is independent of the choice of the basis S . Take p with 1 ≤ p < ∞. An operator T ∈ B(H) is of the Schatten p-class if T p := (tr(|T | p ))1/p < ∞ , and the set of Schatten p-class operators is denoted by S p (H); see [83, Chapter 4] and [244, §2.4]. Each space (S p (H), · p ) is an ideal in B(H) that is a Banach B(H)bimodule and a Banach ∗-algebra with respect to the involution on B(H). Elements of S1 (H) are called the trace-class operators, and elements of S2 (H) are called the Hilbert–Schmidt operators. Note that T ∈ S p (H) if and only if |T | ∈ S p (H) if and only if |T |1/2 ∈ S2p (H). Further, for 1 < p < ∞, we have S p (H) = Sq (H), where q = p , so that S p (H) is a reflexive Banach algebra. In fact, S2 (H) is itself a Hilbert space with respect to the inner product given by [S , T ] = tr(T ∗ S ) (S , T ∈ S2 (H)) . By [83, Theorem 4.10], a Hilbert–Schmidt operator is just a 2-summing operator, and so (S2 (H), · 2 ) = (Π2 (H), π2 ) . Further, by [83, Corollary 3.16], we have Π2 (H) = Π p (H) for 1 ≤ p < ∞. Since H has the approximation property, we can identify the nuclear algebra H ⊗ H  with N(H), as in Example 2.1.13(vi), and, by [83, Theorem 5.30(b)], we can identify S1 (H) with N(H), and so F (H) ⊂ H ⊗ H  = N(H) = S1 (H) ⊂ S2 (H) ⊂ K(H) ⊂ B(H) and F (H) is dense in (S p (H), · p ) for 1 ≤ p < ∞. Also S1 (H) = S2 (H)[2] . For T ∈ N(H), define μT : S → tr(T S ) ,

K(H) → C .

Then μT ∈ K(H) , and the map T →  μT , N(H) → K(H) , is a linear isometry, so that ⊗ H . N(H) = K(H) = H As in Example 2.1.44, we see that B(H) = S1 (H) = K(H) . It follows that the ultra-weak topology on B(H) is defined by the seminorms pT : S → |tr(T S )| , for T ∈ S1 (H).

B(H) → R+ ,

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Definition 2.2.8 Let A be a Banach ∗-algebra. A representation of A is a pair (π, H), where H is a Hilbert space and π : A → B(H) is a ∗-homomorphism. A representation (π, H) is faithful if π is an injection. Let (π, H) be a representation of A. A closed linear subspace K of H is invariant if π(A)K ⊂ K; the representation (π, H) is irreducible if the only invariant subspaces of H are {0} and H. Let A be a unital C ∗ -algebra. We have defined the state space KA and the pure states of A. In fact, an element λ ∈ A is a state if and only if λ(a∗ a) ≥ 0 (a ∈ A) and λ(eA ) = 1. As in Theorem 2.2.5, when A is a C ∗ -algebra and (π, H) is a representation of A, the map π : A → B(H) is a contraction, and π(A) is closed in B(H); in the case where A is unital, π is universal if π is unital and isometric and if each state on A has the form a → [π(a)x, x] for some x ∈ H with x = 1. See [191, Chapter 10], for example. The very famous non-commutative Gel’fand–Naimark theorem is the following; see [50, Theorem 3.2.29], [190, Theorem 4.5.6], [300, Theorem III.2.4], and all texts on C ∗ -algebras. There is also a somewhat different proof in [4, §6.6]. Theorem 2.2.9 Let A be a unital C ∗ -algebra. Then there is a Hilbert space H and a universal representation of A on H. Thus we can regard the class of C ∗ -algebras as the class of ∗-closed and normclosed subalgebras of B(H) for some Hilbert space H. The following result is the double commutant theorem of von Neumann; see [50, Theorem 3.2.32], [190, Theorem 5.3.1], [257, Theorem 2.2.2], and [300, Theorem II.3.9], for example. Theorem 2.2.10 Let H be a Hilbert space, and let A be a C ∗ -subalgebra of B(H). so wo Then A = A = Acc . We shall use the following theorem, which is Kaplansky’s density theorem; see [50, Theorem 3.2.34], [190, Theorem 5.3.5], [257, Theorem 2.3.3] and [300, Theorem II.4.8]. Theorem 2.2.11 Let H be a Hilbert space, and let A be a C ∗ -subalgebra of B(H). so Then A[1] is so-dense in (A )[1] . As mentioned on page 37, the following is a theorem of Pfitzner [260]; for a shorter proof by Fernández-Polo and Peralta, see [107]. Theorem 2.2.12 Every C ∗ -algebra A has property (V) as a Banach space.

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Definition 2.2.13 A C ∗ -algebra A is a von Neumann algebra if there is a Hilbert space H such that A is a C ∗ -subalgebra of B(H) with A closed in the weak operator topology. A C ∗ -algebra is a W ∗ -algebra if it is linearly isometric to the dual space of a Banach space. Thus a C ∗ -subalgebra of B(H) is a von Neumann algebra if and only if it is equal to its double commutant; this is the definition of a von Neumann algebra given by Takesaki in [300, Definition II.3.2], for example. A C ∗ -subalgebra of B(H) is a von Neumann algebra if and only if it is closed in the ultra-weak topology. Each von Neumann algebra is a unital Banach algebra. We see that W ∗ -algebras are defined abstractly, but von Neumann algebras are defined concretely. However, it is a seminal theorem of Sakai [287] that every abstractly defined W ∗ -algebra can be represented as a von Neumann subalgebra of B(H) for a suitable Hilbert space H; see the accounts in [191, Exercise 10.5.87], [257, Theorem 3.9.8], [288, §1.16] and [300, Chapter III, §3], for example. In the future, using standard terminology, we shall use the term ‘von Neumann algebra’ for a W ∗ -algebra, as defined in Definition 2.2.13. It follows from Proposition 1.2.53 and Theorem 2.2.12 that every von Neumann algebra is a Grothendieck space. Let R be a von Neumann algebra that is a subalgebra of B(H) for a Hilbert space H. Then a functional λ on R is normal if it is wo-continuous on R[1] ; equivalently, λ is normal if and only if (xα , λ) converges to 0 for each bounded, monotonedecreasing net (xα ) in R+ that converges to 0, and so, in the case where R = C(K) for a compact space K and R is a commutative von Neumann algebra, the normal linear functionals on R correspond to normal measures on K, as defined on page 21. The space of normal functionals on R is denoted by R∗ ; it is a closed submodule of the dual bimodule R . Further, the map T R∗ : R → R∗ of Definition 1.3.24 is a linear isometry, and so R∗ is an isometric concrete predual of R. In fact, R∗ is uniquely defined by this condition, in the sense that each Banach space E such that E   R has the property that E  R∗ and that the concrete predual specified by E as a subspace of R is always equal to R∗ . Thus R∗ is ‘the unique isometric predual’ of R. Let R be a von Neumann algebra, with R regarded as a closed ∗-subalgebra of B(H) for a Hilbert space H. Since B(H) = (H ⊗ H  ) , it is easy to see that the weak operator topology, the relative ultra-weak topology and the weak-∗ topology, σ(R, R∗ ), are equivalent on bounded subsets of R [300, §II.2]. This leads to the following form of Kaplansky’s density theorem. Theorem 2.2.14 Let R be a von Neumann algebra, and let A be a ∗-subalgebra of R that is σ(R, R∗ )-dense in R. Then A[1] is σ(R, R∗ )-dense in R[1] . Let K be a non-empty, compact space. Then there are many known equivalent conditions on K for C(K) to be a von Neumann algebra. For example, the following is part of [51, §6.4].

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Theorem 2.2.15 Let K be a non-empty, compact space. Then the following are equivalent: (a) C(K) is a von Neumann algebra; (b) C(K)  N(K) ; (c) K is a hyper-Stonean space. In this case, N(K) is the unique isometric predual space to C(K). In particular, K is Stonean when C(K) is a von Neumann algebra, but the converse is not true; see Example 2.4.24. For example, C(β N) = ( 1 ) is a von Neumann algebra. We record the following theorem of Akemann; see [50, Theorem 3.2.42]. Clauses (i) and (ii) also follow from Theorem 1.3.51(ii) and Proposition 1.2.47, respectively, because every C ∗ -algebra has property (V) by Theorem 2.2.12. Theorem 2.2.16 Let A be a C ∗ -algebra. (i) Each bounded linear operator from A into a weakly sequentially complete Banach space is weakly compact. (ii) The dual space A of A is a weakly sequentially complete Banach space. (iii) B(A, A ) = W(A, A ). As stated in Theorem 1.2.39(i), the Banach space M(K) = C0 (K) is weakly sequentially complete for every non-empty, locally compact space K. ⊗ B) = W(A, B ). Corollary 2.2.17 Let A and B be C ∗ -algebras. Then (A The following corollary of Theorem 2.2.16(ii) is also given in [300, Corollary III.5.2]. Corollary 2.2.18 The unique isometric predual of each von Neumann algebra is weakly sequentially complete. The following result follows from Theorem 1.3.57. Proposition 2.2.19 Let A and B be C ∗ -algebras such that B has the Schur property. Then (A ⊗ B) = K(A, B ) and (A ⊗ B) is weakly sequentially complete. The following theorem of Haagerup [159] extends an earlier result of Pisier; it is a consequence of the Grothendieck inequality for C ∗ -algebras. See [264, Lemma 7.3]. Theorem 2.2.20 Let A and B be two C ∗ -algebras, and suppose that A0 is a C ∗ -subalgebra of the algebra A. Then each operator in B(A0 , B ) extends to an operator in B(A, B ).

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We have introduced the notion of a scattered locally compact space on page 10, and have given characterizations of these spaces in Propositions 1.1.9, 1.2.50 and 1.3.39, and Corollary 1.3.71. We say that a commutative C ∗ -algebra of the form C 0 (K) is scattered if the corresponding locally compact space K is scattered. There is a generalization of this notion to arbitrary unital C ∗ -algebras, and we now briefly sketch this; the results will be used in §6.2. The notion of a scattered C ∗ -algebra was introduced by Jensen in [186], where further characterizations of these algebras are given, and the property has been considerably studied. We give a definition that is based on a characterization of Kusuda [206, Lemma 2.2], extending [177]. There is a recent paper of Ghasemi and Koszmider [134] that gives further characterizations of scattered C ∗ -algebras. Definition 2.2.21 A unital C*-algebra A is scattered if σ(a) is countable for each a ∈ Asa . By Proposition 1.1.9, this definition is equivalent to the previous one in the case of commutative, unital C ∗ -algebras. Theorem 2.2.22 Let A be a unital C ∗ -algebra. Then the following are equivalent: (a) A is scattered; (b) every separable, commutative C ∗ -subalgebra of A is scattered; (c) A does not contain a copy of  1 ; (d) A has the Radon–Nikodým property; (e) A =  ∞ (B(Hi ) : i ∈ I) for a family {Hi : i ∈ I} of Hilbert spaces. Proof (a) ⇔ (b) This is [206, Lemma 2.2]. (a) ⇔ (d) This is [37, Theorem 2.2]. (c) ⇔ (d) This is [136, Corollary VII.10]. (a) ⇔ (e) This is from [37] and [186]. The following corollary is immediate from the theorem because a von Neumann algebra has a unique predual. Corollary 2.2.23 Let A be a unital C ∗ -algebra that is scattered. Then A =  1 (Hi ⊗ Hi : i ∈ I) for a family {Hi : i ∈ I} of Hilbert spaces. A basic difference between commutative and non-commutative C ∗ -algebras is the Dunford–Pettis property. By Proposition 1.3.68, every commutative C ∗ -algebra has the Dunford–Pettis property, but it is rare that a non-commutative C ∗ -algebra has this property; for a discussion, see [219]. Recall from Corollary 1.3.71 that a commutative C ∗ -algebra A is such that A has the Schur property if and only if A is scattered.

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The following theorem of Lau and Ülger is from [219, Theorem 3.6], where many other equivalences are given. Theorem 2.2.24 Let A be a unital C ∗ -algebra. Then A is scattered and has the Dunford–Pettis property if and only if A has the Schur property. Proof This follows from Proposition 1.3.70 and Theorem 2.2.22. Theorem 2.2.25 Let A be a C ∗ -algebra. Then A has the Schur property if and only if  2 is not a quotient of A. Proof We may suppose that A is unital. Suppose that A has the Schur property. Then  2 is not a quotient of A by Proposition 1.3.13. Now suppose that  2 is not a quotient of A. We first claim that A is scattered. Assume to the contrary that A is not scattered. By Theorem 2.2.22, (c) ⇒ (a), A contains a copy of  1 , and so, by Corollary 1.3.64(ii),  2 is a quotient of A, a contradiction. Thus A is scattered. ⊗ Hi : i ∈ I) for a family {Hi : i ∈ I} of Hilbert By Corollary 2.2.23, A =  1 (Hi spaces. Assume first that one of the Hilbert spaces, say H, is infinite dimensional. Since H is complemented in H ⊗ H  and  2 is complemented in H, the space  2 is  complemented in A . By Proposition 1.3.29,  2 is a quotient of A, a contradiction. Hence each Hilbert space Hi is finite dimensional, and so A has the Schur property by Theorem 1.2.42. ⊗ B) is weakly Corollary 2.2.26 Let A and B be C ∗ -algebras. Then the space (A   sequentially complete if and only if A or B has the Schur property. ⊗ B) is weakly Proof Suppose first that, say, B has the Schur property. Then (A sequentially complete by Proposition 2.2.19. Conversely, suppose that (A ⊗ B) is weakly sequentially complete, and assume towards a contradiction that neither of the spaces A nor B has the Schur property. By Theorem 2.2.25, there are quotient maps q1 : A →  2 and q2 : B →  2 , ⊗  2 is a quotient map. As in Example 1.4.4,  1 ⊗ B → 2 and then q1 ⊗ q2 : A ⊗  2 , and so there is a continuous surjection is a complemented subspace of  2 ⊗ B) contains a complemented copy of T : A ⊗ B →  1 . By Theorem 1.3.34, (A ⊗ B) is weakly sequentially complete, and so  ∞ is weakly sequentially  ∞ . But (A complete, a contradiction. Hence A or B has the Schur property. Definition 2.2.27 A C ∗ -algebra A is postliminal if π(A) ⊃ K(H) for every irreducible representation (π, H) of A. Postliminal C ∗ -algebras are elsewhere also termed ‘GCR-algebras’ and ‘Type I C ∗ -algebras’; see [85, §4.3], [244, §5.6] and [257, Chapter 6] for several characterizations. Note that B(H) is not postliminal for each infinite-dimensional, separable Hilbert space H [244, p. 170]. The next result is given by Jensen in [186, Theorem 2.3].

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Proposition 2.2.28 A scattered C ∗ -algebra is postliminal. We shall also require the following result of Chu, Iochum, and Watanabe [38]. Proposition 2.2.29 A C ∗ -algebra A has the Dunford–Pettis property if and only if every irreducible representation of A is finite dimensional.

2.3 Biduals of Banach algebras Let A be a Banach algebra. In this section, we shall begin a main topic of this work, that of studying the two Arens products,  and  , on the bidual space A of A. This will lead to the notions of Arens regularity and strong Arens irregularity (in §6.1) of a Banach algebra, key themes of this book. A Banach algebra A is Arens regular when the two products  and  agree on A , and this holds if and only if each functional in A is weakly almost periodic, a property that will be used later. Here we shall give some easy results that show that Banach algebras with certain properties are or are not Arens regular; some of these results are related to the properties of mixed identities in the bidual of a Banach algebra. We shall also pay attention to Banach algebras that are ideals in their biduals; this class of Banach algebras will include several examples to be defined later, such as the Tauberian Banach sequence algebras of §3.2. Let A be a Banach algebra. As before, we regard A as a closed linear subspace of the Banach space A . Take λ ∈ A and M ∈ A , and define λ · M and M · λ in A by a, λ · M = M, a · λ ,

a, M · λ = M, λ · a

(a ∈ A) ,

so that λ · M ≤ M λ and M · λ ≤ M λ . In the case where M ∈ A, the new definitions of a, λ · M and a, M · λ coincide with the existing definitions. Next, for M, N ∈ A , define M  N, λ = M, N · λ,

M  N, λ = N, λ · M

(λ ∈ A ) ,

so that M  N, M  N ∈ A , and clearly both of the maps (M, N) → M  N and (M, N) → M  N from A × A to A are bilinear with M  N ≤ M N and M  N ≤ M N for M, N ∈ A . Again the definitions of M  N and M  N coincide with the existing definitions when either M or N belongs to A. Notice also that (M  N) · λ = M · (N · λ) (λ ∈ A ) . Take M, N ∈ A[1] , say M = wk∗ – limα aα and N = wk∗ – limβ bβ , where (aα ) and (bβ ) are nets in A[1] . Then M  N = wk∗ – lim wk∗ – lim aα bβ , α

β

M  N = wk∗ – lim wk∗ – lim aα bβ . β

α

(2.3.1)

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It follows easily from this formula that the products  and  on A are associative, and so (A ,  ) and (A , ) are both Banach algebras containing A as a closed subalgebra. This is the famous theorem of Arens [7]; see [50, Theorem 2.6.15]. Definition 2.3.1 The products  and  are the first and second Arens products, respectively, on A , and (A ,  ) is the bidual algebra of A. The bidual algebra (A ,  ) will usually be denoted just by A . Suppose that A has an identity eA . Then eA is also the identity of (A ,  ) and of  (A ,  ). Take M, N ∈ A[1] and λ ∈ A . It follows from Proposition 1.1.2 that there are sequences (am ) and (bn ) in A[1] such that M  N, λ = lim lim am bn , λ, m→∞ n→∞

M  N, λ = lim lim am bn , λ. n→∞ m→∞

(2.3.2)

Let B be a closed subalgebra of a Banach algebra A. Then it is immediate from Equation (2.3.1) that the products  and  on B are the restrictions to B of the corresponding products on A , and that (B ,  ) and (B ,  ) are weak-∗-closed subalgebras of (A ,  ) and of (A ,  ), respectively. Also, in the case where I is a closed ideal in A, the algebras (I  ,  ) and (I  ,  ) are weak-∗-closed ideals in (A ,  ) and (A ,  ), respectively, and ((A/I) ,  ) is the quotient algebra A /I  . In the case where A is a commutative Banach algebra, we have MN = NM

(M, N ∈ A ) ,

and so (A ,  ) = (A ,  ) op . Theorem 2.3.2 Let A be a Banach algebra. For each N ∈ A , the map R N : M → M  N ,

A → A ,

is always weak-∗ continuous; for each a ∈ A, the map La : M → a · M, A → A , is always weak-∗ continuous. The map LN : M → N  M is not necessarily weak-∗ continuous on A ; see Theorem 2.3.29. Similarly, LN is always weak-∗ continuous on (A , ) for each N ∈ A . The following result is [50, Proposition 2.6.25]. Proposition 2.3.3 Let A be a Banach algebra. Then (rad A ) ∩ A ⊂ rad A, and so (rad A ) ∩ A = {0} when A is semisimple. Proposition 2.3.4 Let A be a Banach algebra with a closed subalgebra B, and suppose that A2 ⊂ B. Then A  A ⊂ B .

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Proof Take M, N ∈ A , say M = limα aα and N = limβ bβ , as above. Then aα bβ ∈ B for each α, β, and so aα · N ∈ B for each α. Since R N is weak-∗-continuous and B is a weak-∗-closed subalgebra of A , it follows that M  N ∈ B , as required. Proposition 2.3.5 Let A be a Banach algebra, and suppose that M ∈ Z(A ,  ). Then M  N = M  N (N ∈ A ). Proof For each a ∈ A and λ ∈ A , we have a, λ · M = M, a · λ = M · a, λ = a · M, λ = a, M · λ , and so λ · M = M · λ. Thus, for each λ ∈ A and N ∈ A , we have M  N, λ = N  M, λ = N, M · λ = N, λ · M = M  N, λ , and hence M  N = M  N, as required. Now suppose that A is a Banach algebra and that L is a left multiplier in B(A). Then, using the notation of Equation (2.3.1), we have L (M  N) = L (lim lim aα bβ ) = lim lim L(aα bβ ) = lim lim L(aα )bβ α

α

β



β

= lim L(aα ) · N = L (M)  N α

α

β



(M, N ∈ A ) ,

so that L is a left multiplier on A . Similarly, R (M  N) = M  R (N) (M, N ∈ A )

(2.3.3)

for each right multiplier R on A, so that R is also a right multiplier on (A ,  ). Further, suppose that (L, R) ∈ M(A). Then M  L (N) = lim lim aα L(bβ ) = lim lim R(aα )bβ = R (M)  N α

β

α

β

(M, N ∈ A ) ,

and so (L , R ) ∈ M(A ). Definition 2.3.6 Let A be a Banach algebra. Then A is an ideal in A or A is an ideal in its bidual whenever A is a closed ideal in (A ,  ). In fact, A is an ideal in A if and only if A is a submodule of the Banach Abimodule A , and so A is an ideal in (A ,  ) if and only if A is an ideal in (A ,  ). For example, a unital Banach algebra A is an ideal in (A ,  ) if and only if A is reflexive. In the case where A is an ideal in its bidual, the maps LM : a → M · a and RM : a → a · M are bounded linear operators on A for each M ∈ A , and their duals are bounded linear operators on A . The following is a theorem of Watanabe.

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Theorem 2.3.7 Let A be a Banach algebra. Then A is an ideal in its bidual if and only if the maps La and Ra are weakly compact operators in B(A) for each a ∈ A. Proof Take a ∈ A. Then the operators La and Ra are both weakly compact if and only if La (A ) ⊂ A and Ra (A ) ⊂ A; by Equation (2.1.7), this holds if and only if a · M ∈ A and M · a ∈ A for each M ∈ A , and this is the case for each a ∈ A if and only if A is an ideal in (A ,  ). Corollary 2.3.8 Let A be a Banach algebra that is an ideal in its bidual. Suppose that B is a closed subalgebra of A. Then B is an ideal in its bidual. Suppose that I is a closed ideal in A. Then A/I is an ideal in its bidual. Corollary 2.3.9 Let A be a Banach algebra such that A is a Grothendieck space and that AA and A A are separable, closed linear subspaces of A . Then A is an ideal in its bidual. Proof Take a ∈ A, and consider the map La : A → A. Then La ∈ B(A , A A). Since A A is separable, it follows from Theorem 1.3.58, (a) ⇒ (c), that La is weakly compact. Hence La is weakly compact, and, similarly, Ra is weakly compact. By Theorem 2.3.7, A is an ideal in A . Definition 2.3.10 Let A be a Banach algebra. Then A is compact if the maps La and Ra are compact operators for each a ∈ A. Note that the above definition is stronger than the one given in [25, §33]. Of course, a compact Banach algebra is an ideal in its bidual, and a closed ideal in a compact algebra is also compact. Suppose that A and B are compact Banach algebras. Since La⊗b = La ⊗ Lb (a ∈ A, b ∈ B), it follows from Proposition 1.4.12 ⊗ B for each a ∈ A and b ∈ B, and hence that that La⊗b is a compact operator on A A ⊗ B is also a compact Banach algebra. Examples of compact algebras will be given in §3.2, Theorem 4.1.36, and Example 4.2.4. Corollary 2.3.11 Let A be a Banach algebra that is an ideal in its bidual. Suppose that A factors weakly and has the Dunford–Pettis property. Then A is a compact algebra. Proof Take a ∈ A. Since A factors weakly, there exist n ∈ N and elements b1 , . . . , bn , c1 , . . . , cn ∈ A with a = b1 c1 + · · · + bn cn . By Theorem 2.3.7, Lb j and Lc j are weakly compact for j ∈ Nn . Since A has the Dunford–Pettis property, it follows from Proposition 1.3.69 that La = Lb1 ◦ Lc1 + · · · + Lbn ◦ Lcn is compact. Similarly, Ra is compact. Hence A is a compact algebra.

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Corollary 2.3.12 Let A and B be Banach algebras. Then A ⊗ B is an ideal in its bidual whenever A is an ideal in its bidual and B is a compact algebra. Proof Again set L = A[1] ⊗ a B[1] , so that aco(L) = (A ⊗ B)[1] . For the result, it suffices to show that, for a ∈ A and b ∈ B, the sets La⊗b (L) = La (A[1] ) ⊗ a Lb (B[1] ) and Ra⊗b (L) = Ra (A[1] ) ⊗ a Rb (B[1] ) are both relatively weakly compact sets. By Theorem 2.3.7, La (A[1] ) and Ra (A[1] ) are relatively weakly compact sets in A for a ∈ A. By hypothesis, Lb (B[1] ) and Rb (B[1] ) are compact sets in B for b ∈ B. Hence the result follows from Proposition 1.4.11(ii). Proposition 2.3.13 Let A be a Banach algebra that is an ideal in its bidual, and take a ∈ A. Then the set a · A[1] = {a · λ : λ ∈ A[1] } is convex and weakly compact in A . Proof Clearly a · A[1] is convex. Consider a net (λα ) in A[1] . By passing to a subnet, we may suppose that wk∗ – limα λα = λ in A[1] . For each M ∈ A , we have M, a · λα  = M · a, λα  → M · a, λ = M, a · λ because M · a ∈ A, and so a · λα → a · λ weakly in A[1] . Hence a · A[1] is weakly compact. Example 2.3.14 Let E be a non-zero Banach space, and set A = (E ⊗ E  , · π ), the nuclear algebra of E, as in Example 2.1.13(vi). We claim that A is an ideal in A if and only if E is reflexive. First, suppose that E is reflexive; we shall show that Lz ∈ W(A) for each z ∈ A. Since E ⊗ E  = lin {x ⊗ λ : x ∈ E, λ ∈ E  }, it suffices to suppose that z = x0 ⊗ λ0 , where x0 ∈ E and λ0 ∈ E  . The map R : x → x, λ0  x0 ,

E → E,

has one-dimensional range, and so is compact. The map S = IE  is weakly compact because E  is reflexive, and so Lz = R ⊗ S ∈ W(A) by Proposition 1.4.12(ii), as required. Similarly, Rz ∈ W(A) for each z ∈ A, and so A is an ideal in A . The Banach operator algebra N(E) is a quotient of A, and so, by Corollary 2.3.8, N(E) is an ideal in its bidual whenever E is reflexive. Now suppose that E is a non-zero Banach space such that N(E) is an ideal in its bidual. Let (xn ) be a bounded sequence in E, and take λ0 ∈ E  with λ0  0, and then take y0 ∈ E with y0 , λ0  = 1. The sequence (xn ⊗ λ0 ) is bounded in N(E). Since Ry0 ⊗λ0 is weakly compact, and since xn ⊗ λ0 = (xn ⊗ λ0 ) (y0 ⊗ λ0 ) (n ∈ N), it follows that (xn ⊗ λ0 ) has a weakly convergent subsequence, say (xn j ⊗ λ0 ), in N(E). For each λ ∈ E  , the map y0 ⊗ λ of §1.4 belongs to N(E) , and also xk , λ = xk ⊗ λ0 , y0 ⊗ λ (k ∈ N). This shows that (xn j ) is weakly convergent in E. It follows that E is reflexive.

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Proposition 2.3.15 Let A be a semisimple Banach algebra that is an ideal in its bidual. Then rad A = {M ∈ A : A  M = {0}} . Proof Set R = rad A . By Proposition 2.1.2(i), rad A = A ∩ R, and so A ∩ R = {0} because A is semisimple. Set I = {M ∈ A : A  M = {0}}, a closed ideal in A . We have A  R ⊂ A ∩ R because A and R are ideals in A , and so R ⊂ I because R M is weak-∗ continuous on A for each M ∈ A and A is weak-∗ dense in A . Clearly I [2] = {0}, and so I is a nil ideal in A , and hence I ⊂ R by Proposition 2.1.2(iii). Thus R = I, as required. Proposition 2.3.16 Let A be a Banach algebra that is an ideal in its bidual and has a bounded approximate identity. Then A · A = A · A . Proof The BAI in A is denoted by (eα ). Take a ∈ A and λ ∈ A . Then M, a · λ · eα  = eα · M · a, λ → M · a, λ = M, a · λ (M ∈ A ) , and so a · λ · eα → a · λ weakly. Now a · λ · eα ∈ A A, and so, by Mazur’s theorem, Theorem 1.2.24(ii), a · λ ∈ A A = A · A by Theorem 2.1.46. Thus A · A ⊂ A · A. Similarly, A · A ⊂ A · A , giving the result. Let A be a Banach algebra. We note that ϕ · a = a · ϕ = ϕ(a)ϕ (a ∈ A, ϕ ∈ ΦA ), and so (2.3.4) M · ϕ = ϕ · M = M, ϕ ϕ (ϕ ∈ ΦA , M ∈ A ) . Take M, N ∈ A and ϕ ∈ ΦA . Then it follows that M  N, ϕ = M  N, ϕ = M, N · ϕ = M, ϕ N, ϕ .

(2.3.5)

We set  ϕ(M) = M, ϕ (M ∈ A ), so that  ϕ ∈ ΦA . Thus we identify ΦA with a subset of ΦA by using the map ϕ →  ϕ,

ΦA → ΦA .

This map is not necessarily continuous; this occurs if and only if the weak and weak-∗ topologies, σ(A , A ) and σ(A , A) on A , coincide on ΦA . As in Equation (1.3.8), we can write A = A ⊕ A⊥ . Let P : A → A be the Dixmier projection. Then the restriction of P to the space ΦA ∪ {0} clearly gives a continuous surjection π : (ΦA ∪ {0}, σ(A , A )) → (ΦA ∪ {0}, σ(A , A))

(2.3.6)

that is the identity on ΦA . For each ϕ ∈ ΦA ∪ {0}, the corresponding fibre in the set ΦA ∪ {0} is (2.3.7) Φ{ϕ} = {ψ ∈ ΦA ∪ {0} : π(ψ) = ϕ} .

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109

Proposition 2.3.17 Let A be a Banach algebra that is an ideal in its bidual. Then the weak and weak-∗ topologies on ΦA coincide, and so the map ϕ →  ϕ,

ΦA → ΦA ,

is an embedding. Further, ΦA = ΦA ∪ (ΦA ∩ A⊥ ). Proof Let wk∗ – limν ϕν = ϕ in ΦA , and take a ∈ A with ϕ(a) = 1. For each M ∈ A , we have M · a ∈ A, and so M, ϕν  a, ϕν  = M · a, ϕν  → M · a, ϕ = M, ϕ a, ϕ . Since a, ϕν  → a, ϕ = 1, we have M, ϕν  → M, ϕ, and so ϕν → ϕ weakly in ΦA . This shows that the weak and weak-∗ topologies on ΦA coincide and that the map ϕ →  ϕ, ΦA → ΦA , is an embedding. Now set H = ΦA ∩ A⊥ ; we shall show that ΦA = ΦA ∪ H. In the case where A is reflexive, H = ∅ and ΦA = ΦA , and hence we may suppose that A is not reflexive, and so A⊥  {0}. Take M ∈ A and Λ ∈ A⊥ . Then a, M · Λ = a · M, Λ = 0 (a ∈ A) because a · M ∈ A, and so M · Λ ∈ A⊥ . Now take ψ ∈ ΦA , say ψ = ϕ+Λ, where ϕ ∈ ΦA ∪{0} and Λ ∈ A⊥ . If Λ = 0, then ψ ∈ ΦA ; if ϕ = 0, then ψ ∈ H. Now suppose that ϕ  0; in this case, we claim that Λ = 0, which will imply that ΦA = ΦA ∪ H, as required. Indeed, for each M ∈ A , we have M · ψ = M · ϕ + M · Λ, and also M · ψ = M, ψ ψ = M, ψ ϕ + M, ψ Λ , and so M · ϕ = M, ψ ϕ. But M · ϕ = M, ϕ ϕ, and hence M, ϕ = M, ψ because ϕ  0. Thus M, Λ = 0. This holds for each M ∈ A , and so Λ = 0, as claimed. Proposition 2.3.18 Let I be an ideal in a Banach algebra (A, · A ) such that (I, · I ) is a Banach A-bimodule. Suppose that I 2 is dense in both (I, · I ) and (A, · A ). Then A is an ideal in its bidual if and only if I is an ideal in its bidual. Proof First, suppose that I is an ideal in I  . Take x ∈ I and λ ∈ I  , and define (x · λ)(a) = ax, λ (a ∈ A). Then x · λ ∈ A with x · λ A ≤ x I λ I  . Now take M ∈ A . Then | M, x · λ | ≤ M A x I λ I  , and so the map N : λ → M, x · λ, I  → C, belongs to I  . Take y ∈ I. Then N · y ∈ I  is defined by N · y, λ = M, xy · λ = M · xy, λ. Since I is an ideal in I  , it follows that M · xy ∈ I ⊂ A. Since I 2 is dense in (A, · A ), we have

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M · a ∈ A (a ∈ A), and so A is a left ideal in A . Similarly, A is a right ideal in A , and so A is an ideal in A . Second, suppose that A is an ideal in A . Take M ∈ I  , and regard M as an element of A . Then M · a ∈ A (a ∈ A), and so M · xy ∈ AI ⊂ I (x, y ∈ I). Since I 2 is dense in (I, · I ), it follows that M · z ∈ I (z ∈ I), and so I is a left ideal in I  . Similarly, I is a right ideal in I  , and so I is an ideal in I  . Throughout, we shall be interested to determine, as a major theme, whether the Banach algebras that we are considering satisfy the next definition. Definition 2.3.19 A Banach algebra A is Arens regular if the two products  and  coincide on A . In the case where A is a commutative Banach algebra, A is Arens regular if and only if (A ,  ) is commutative. Suppose that the algebra A has zero product. Then both (A ,  ) and (A ,  ) have zero products by Equation (2.3.1). Now consider the C ∗ -algebra K(H) for a Hilbert space H. Then the products  and  both coincide with the usual product on K(H) = B(H), and so K(H) is Arens regular. Proposition 2.3.20 Every reflexive Banach algebra and every Banach algebra with zero product is Arens regular. In Chapter 6, we shall define and study Banach algebras that are the furthest possible from being Arens regular, namely, those that are ‘strongly Arens irregular’; see Definition 6.1.1. The notion of what are now called the Arens products of course originates with Arens [7, 8], where a different notation is used. Arens proved, for example, that ( 1 , · ) is Arens regular (see Proposition 2.3.40 and Example 3.2.7(ii) for the product on ( 1 ) ), but that the semigroup algebra ( 1 (N),  ) is not Arens regular; in fact, it will be shown in Theorem 6.3.7 that this latter algebra is strongly Arens irregular in a strong sense. A significant early paper on the Arens regularity of semigroup algebras is that of Day [75]. In a seminal paper [40] of 1961, Civin and Yood proved that the group algebra (L1 (G),  ) is not Arens regular for each infinite locally compact abelian group G, but that every C ∗ -algebra is Arens regular and that its bidual is also a C ∗ -algebra; a different proof of the latter result, using the Vidav–Palmer theorem, is given in [25, Theorem 38.19]. For an early history of the theory of Arens products, and generalizations to related notions and more general settings, see [254, §1.4]; an account was given by Duncan and Hosseiniun in [88], and a survey was given by Filali and Singh in [117]; a more recent historical survey of Filali and Galindo is contained in [110]. See also an early paper by Grosser [154]. Let (A, · ) be an Arens regular Banach algebra. Then it follows that subalgebras of A such that they are closed in the Banach space (A, · ) and quotients of A by closed ideals are also Arens regular. Suppose that A is a non-unital Banach algebra,

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111

so that A = CeA ⊕ A. Then clearly (A ) = CeA ⊕ A , and so A is Arens regular if and only if A is Arens regular. Let A and B be Banach algebras, and suppose that θ : A → B is a continuous homomorphism. Then the maps θ : (A ,  ) → (B ,  )

and

θ : (A ,  ) → (B ,  )

are also continuous homomorphisms. It follows that, in the case where A and B are isomorphic as Banach algebras, A is Arens regular if and only if B is Arens regular. In particular, the Banach algebra A ⊗ B is Arens regular if and only if B ⊗ A is Arens regular. Proposition 2.3.21 Let (A, · A ) be a Banach algebra, and let (B, · B ) be a Banach algebra that is a subalgebra of A and such that b B ≥ b A (b ∈ B). Suppose that A is Arens regular and that bn → 0 weakly for each sequence (bn ) in B[1] such that limn→∞ bn A = 0. Then B is Arens regular. Proof The inclusion map is ι : B → A. Take M, N ∈ B . Then ι (M  N) = ι (M)  ι (N) = ι (M)  ι (N) = ι (M  N) . By Proposition 1.3.10, ι : B → A is injective, and so M  N = M  N. This shows that B is Arens regular. In general, the subalgebra B of A is not Arens regular when A is Arens regular; there are many examples that show this. For example, take A = c 0 (Z) and take B = A(Z) in later notation. Then B is a dense subalgebra of A. It will be noted that A is Arens regular, but that B is strongly Arens irregular. Similarly, bv 0 is dense in A, but bv 0 is not Arens regular, as will be shown in Example 3.2.12; also, BVC(I) is dense in C(I), but BVC(I) is not Arens regular, as will be shown in Example 3.1.35. Corollary 2.3.22 Let (A, · A ) be a Banach algebra, and let (I, · I ) be an ideal in A such that x I ≥ x A (x ∈ I) and I is a Banach A-bimodule. Suppose that A is Arens regular and that II  = I  . Then I is Arens regular. Proof Take a ∈ A, x ∈ I, and λ ∈ I  . Then | a, x · λ | = | ax, λ | ≤ a A x I λ . Now take a sequence (xn ) in I[1] ⊂ A1] such that limn→∞ xn A = 0. Then it follows that limn→∞ | xn , x · λ | = 0. Since II  = I  , this implies that xn → 0 weakly, and so, by Proposition 2.3.21, I is Arens regular. Proposition 2.3.23 Let A be a Banach algebra with a closed subalgebra B of finite codimension. Then A is Arens regular if and only if B is Arens regular.

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Proof Set A = E ⊕ B, where E is a finite-dimensional space. Then A = E ⊕ B . Take a, b ∈ E and M, N ∈ B . Then (a + M)  (b + N) = ab + a · N + M · b + M  N , with a similar formula for the product involving , and so the result follows. Proposition 2.3.24 Let A be a Banach algebra, with a concrete predual F. Suppose that R  S = R  S = 0 (R, S ∈ F ⊥ ). Then A is Arens regular. Proof By Equation (1.3.7), A = A ⊕ F ⊥ . Clearly (a + R)  (b + S) = (a + R)  (b + S) = ab + a · S + R · b whenever a, b ∈ A and R, S ∈ F ⊥ , and so A is Arens regular. Proposition 2.3.25 Let A be a Banach algebra that is Arens regular. Then A A = AA as closed linear subspaces of A . Proof Clearly AA ⊂ A A ⊂ A . Now take M ∈ A and λ ∈ A , and assume that M · λ  AA . Then there exists N ∈ A with N, M · λ = 1 and N, a · μ = 0 (a ∈ A, μ ∈ A ). But now N  M, λ = 1, and so N  M, λ = 1 because A is Arens regular. On the other hand, we have N · a, μ = 0 (a ∈ A, μ ∈ A ), and so N · a = 0 (a ∈ A). Since L N is weak-∗ continuous on (A , ), it follows that N  M = 0, the required contradiction. Thus A A = AA . Let E be a normed space, and let ∗ : E → E be an isometric linear involution on E. For λ ∈ E  , define λ∗ ∈ E  by x, λ∗  = x∗ , λ (x ∈ E) . Then the map ∗ : λ → λ∗ , E  → E  , is an isometric linear involution; this map is clearly also continuous with respect to the topology σ(E  , E). Continuing, we obtain an isometric linear involution ∗ on E  ; the restriction of this linear involution to the linear subspace E of E  is the original linear involution. Let A be a Banach ∗-algebra. Then the extended linear involution on A is not necessarily an involution on (A ,  ). The following result was first proved by Civin and Yood in [40, Theorem 6.2]. Theorem 2.3.26 Let A be a Banach ∗-algebra. Then the extended linear involution on (A ,  ) is an involution if and only if A is Arens regular.

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Proof Take M, N ∈ A . Then there are two nets (aα ) and (bβ ) in A such that M = limα aα and N = limβ bβ in (A , σ(A , A )). We have (M  N)∗ = lim lim(aα bβ )∗ = lim lim b∗β a∗α = N∗  M∗ . α

α

β

β

Thus (M  N)∗ = N∗  M∗ for all M, N ∈ A if and only if A is Arens regular. Definition 2.3.27 Let A be a Banach algebra. A linear functional λ ∈ A is [weakly] almost periodic if the map Rλ : a → a · λ ,

A → A ,

is [weakly] compact. The spaces of almost periodic and weakly almost periodic functionals on A are denoted by AP(A) and WAP(A), respectively. Clearly ΦA ⊂ AP(A) because a · ϕ = ϕ(a)ϕ (a ∈ A, ϕ ∈ ΦA ). Also AP(A) and WAP(A) are norm-closed Banach A-bimodules in A and AP(A) ⊂ WAP(A). We see from Proposition 1.3.72 that AP(A) = WAP(A) = A whenever A has the Schur property. We shall see shortly that we obtain the same classes of linear functionals if we use the operator Lλ : a → λ · a, A → A , in the above definition. Definition 2.3.28 Let A be a Banach algebra, and take λ ∈ A . Then λ satisfies the iterated-limit condition if, for each pair ((a j ), (bk )) of bounded sequences in A, necessarily lim lim a j bk , λ = lim lim a j bk , λ j→∞ k→∞

k→∞ j→∞

whenever both the iterated limits exist. The next result follows easily from Grothendieck’s criterion in Theorem 1.2.34; it originates in the work of Hennefeld [163], and was developed by Pym in [266]. The result is given in [50, Theorem 2.6.17]. Theorem 2.3.29 Let A be a Banach algebra, and take λ ∈ A . Then λ ∈ WAP(A) if and only if λ satisfies the iterated-limit condition, and so WAP(A) = WAP(Aop ). Further, WAP(A) = {λ ∈ A : M  N, λ = M  N, λ (M, N ∈ A )} , and so the following conditions on A are equivalent: (a) A is Arens regular; (b) LN : M → N  M, A → A , is weak-∗ continuous for each N ∈ A ; (c) WAP(A) = A .

(2.3.8)

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Corollary 2.3.30 Let A be a Banach algebra, and take λ ∈ A . Then λ ∈ WAP(A) if and only if the map Lλ : a → λ · a, A → A , is weakly compact. The following criterion for the Arens regularity of a Banach algebra follows immediately from Theorem 2.3.29; see [50, Theorem 2.6.17]. Corollary 2.3.31 Let A be a Banach algebra, and suppose that L is a subset of A[1] such that aco(L) = A[1] . Then A is Arens regular if and only if, for each pair ((a j ), (bk )) of sequences in L and each λ ∈ A , we have lim lim a j bk , λ = lim lim a j bk , λ

j→∞ k→∞

k→∞ j→∞

whenever both the iterated limits exist. Corollary 2.3.32 Let A be a Banach algebra. Suppose that every closed, separable subalgebra of A is Arens regular. Then A is Arens regular. The next two results are from Ülger’s paper [316, Lemma 4.1 and Proposition 4.2]. Proposition 2.3.33 Let A and B be Banach algebras, and let θ : A → B be a continuous homomorphism. Then θ (WAP(B)) ⊂ WAP(A) ∩ (ker θ)⊥ . Suppose that θ is an epimorphism. Then θ (WAP(B)) = WAP(A) ∩ (ker θ)⊥ . Proof Let λ ∈ WAP(B), so that θ (λ) ∈ A . Take M, N ∈ A . Then M  N, θ (λ) = θ (M  N), λ = θ (M)  θ (N), λ = M  N, θ (λ) . Thus θ (WAP(B)) ⊂ WAP(A). Clearly θ (B ) ⊂ (ker θ)⊥ . Now suppose that θ is an epimorphism, and take λ ∈ WAP(A) ∩ (ker θ)⊥ . By Proposition 1.3.21, there exists μ ∈ B with θ (μ) = μ ◦ θ = λ. Since θ : A → B is a surjection, elements of B have the form θ (M) for some M ∈ A , and then θ (M)  θ (N), μ = θ (M)  θ (N), μ for each M, N ∈ A , as before, whence μ ∈ WAP(B). Thus θ (WAP(B)) ⊃ WAP(A)∩(ker θ)⊥ . It follows that θ (WAP(B)) = WAP(A) ∩ (ker θ)⊥ . Corollary 2.3.34 Let A be a Banach algebras, and let I be a closed ideal in A. Then A/I is Arens regular if and only if I ⊥ ⊂ WAP(A). Proof Let q : A → A/I be the quotient map, a continuous surjection, and set B = A/I. Then q (B ) = I ⊥ and q (WAP(B)) = WAP(A)∩I ⊥ by Proposition 2.3.33. Since q is an injection, it follows that B = WAP(B) if and only if I ⊥ = WAP(A) ∩ I ⊥ , i.e., I ⊥ ⊂ WAP(A). By Theorem 2.3.29, A/I is Arens regular if and only if I ⊥ ⊂ WAP(A).

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The following result of Duncan and Hosseiniun is [88, Theorem 2]. We consider the map (2.3.9) λ : (M, N) → M  N , λ , A[1] × A[1] → C , where λ ∈ A . Proposition 2.3.35 Let A be a Banach algebra, and take λ ∈ A . (i) The functional λ ∈ AP(A) if and only if the map λ is jointly weak-∗ continuous. (ii) The functional λ ∈ WAP(A) if and only if the map λ is separately weak-∗ continuous. Proof (i) Take an element λ ∈ AP(A), so that Rλ : A → A is compact. Suppose that wk∗ – limα Mα = M and wk∗ – limβ Nβ = N in A[1] . By Theorem 1.3.45, the map

Rλ : A → A is such that limβ

Nβ · λ − N · λ

= 0. Now 

 Mα  Nβ , λ − M  N, λ ≤

Nβ · λ − N · λ

+ |Mα − M , N · λ| , and so lim α,β Mα  Nβ , λ = M  N , λ. Thus the map λ is jointly weak-∗ continuous. Conversely, suppose that λ ∈ A and that the map λ is jointly weak-∗ continuous. Let wk∗ – limα Mα = M in A[1] . Then we claim that lim Mα · λ − M · λ = 0 . α

Assume

that there exist a subnet (Mβ ) of (Mα ) and δ > 0 such

to the contrary

Mβ · λ − M · λ

> δ for all β. Then there is a net (aβ ) in A[1] such that that   Mβ · aβ , λ − M · aβ , λ > δ for all β. By passing to a subnet, we may suppose that the net (aβ ) converges weak-∗ in A[1] , and then, by the hypothesis,   limβ Mβ · aβ , λ − M · aβ , λ = 0, a contradiction. Thus the claim holds, and so Rλ is compact, i.e., λ ∈ AP(A). (ii) Suppose that λ ∈ WAP(A). Then, by the properties of the Arens product, the map λ is separately weak-∗ continuous. Conversely, suppose that the map λ is separately weak-∗ continuous, and take M, N ∈ A[1] , say N = wk∗ – limβ bβ , where (bβ ) is a net in A[1] . Then lim M · bβ , λ = M  N, λ . β

But also limβ M · bβ , λ = M  N, λ, and so M  N, λ = M  N, λ. It follows from Theorem 2.3.29 that λ ∈ WAP(A). Corollary 2.3.36 Let A be a Banach algebra, and take λ ∈ A . Then λ ∈ AP(A) if and only if the map Lλ : a → λ · a, A → A , is compact, and so AP(A) = AP(Aop ).

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Proof By Proposition 2.3.35(i), λ ∈ AP(A) if and only if the map in (2.3.9) is jointly weak-∗ continuous. Similarly, the map Lλ : a → λ · a, A → A , is compact if and only if the map (M, N) → M  N, λ, A[1] × A[1] → C, is jointly weak-∗ continuous. Now suppose that λ ∈ AP(A) ⊂ WAP(A). It follows from Theorem 2.3.29 that the two conditions involving joint continuity are the same because M  N, λ = M  N, λ (M, N ∈ A ), and so the result follows. The following result is taken from [89]. Proposition 2.3.37 Let A be a Banach algebra, and suppose that La and Ra are compact (respectively, weakly compact) for each a ∈ A. Then AA ∪ A A ⊂ AP(A) (respectively, AA ∪ A A ⊂ WAP(A) ). Proof Take a ∈ A and λ ∈ A . Then the map b → b · a · λ, A → A , is compact (respectively, weakly compact), and so a · λ ∈ AP(A) (respectively, a · λ ∈ WAP(A)). Since AP(A) and WAP(A) are closed in A , it follows that AA ⊂ AP(A) and that AA ⊂ WAP(A), respectively. Similarly, A A ⊂ AP(A) and A A ⊂ WAP(A), respectively. Corollary 2.3.38 Let A be a Banach algebra that is an ideal in its bidual. Suppose that A A = A . Then A is Arens regular. Proof By Theorem 2.3.7, La , Ra ∈ W(A) (a ∈ A), and so, by Proposition 2.3.37, A A ⊂ WAP(A). Thus WAP(A) = A , and so A is Arens regular by Theorem 2.3.29, (c) ⇒ (a). We shall see in Example 3.2.7(iii) that the converse of the above corollary does not hold. It is not a general truth that (A ,  ) is Arens regular whenever A is Arens regular: for counter-examples, see Example 6.3.29, [267] and [269]. It will be noted ⊗ c 0 is Arens regular, but that (c 0 ⊗ c 0 ) is not Arens in Example 6.2.19 that c 0 regular. Let A be a Banach algebra. We have noted that A is Arens regular if and only if A = WAP(A). The following concept, introduced by Granirer [150], considers a situation very different from A being Arens regular. Definition 2.3.39 Let A be a Banach algebra. Then A is extremely non-Arens regular when A /WAP(A) contains a closed linear subspace that has A as a quotient. More general definitions are given by Filali and Galindo in [112]. We shall note in Theorem 6.4.23 that the group algebra L1 (G) is extremely non-Arens regular for each infinite locally compact group G. The following result, in the case of  1 (An ), was given by Arikan in [9]; another proof is given in [311, Corollary 3.2].

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Proposition 2.3.40 Let (An ) be a sequence of Banach algebras, each of which is Arens regular, and take p with 1 ≤ p < ∞. Then the Banach algebras c 0 (An ) and  p (An ), with coordinatewise product, are also Arens regular. Proof First, set A = c 0 (An ), so that A is a Banach algebra and A =  ∞ (An ). It is easy to see that (Mn )  (Nn ) = (Mn  Nn ) and

(Mn )  (Nn ) = (Mn  Nn )

for (Mn ), (Nn ) ∈ A , and so it follows that A is also Arens regular. Similarly,  p (An ) is Arens regular when 1 < p < ∞. Now set A =  1 (An ), so that A is a Banach algebra. Set Fn = (An , σ(An , An )) for n ∈ N, and then set E =  ∞ (An ) = A . Fix λ ∈ A[1] , say λ = (λn ), where λn ∈ (An )[1] (n ∈ N). For n ∈ N, take Ln to be the weak closure of (An )[1] · λn in An , so that Ln ⊂ (An )[1] ; further, since An is Arens regular, it follows from Theorem 2.3.29 that WAP(An ) = An , and so the above space Ln is weakly compact in Fn .  Then L := n∈N Ln ⊂ E, and L is weakly compact in E, as in Example 1.2.25. We consider the map Rλ : a = (an ) → a · λ = (an · λn ) ,

A → A .

The image Rλ (A[1] ) of A[1] is contained in L, and so its weak closure in E is weakly compact. This shows that λ ∈ WAP(A). We have shown that WAP(A) = A , and so, by Theorem 2.3.29, A is Arens regular. In particular, ( 1 , · ) is Arens regular; see Example 3.2.7(ii) for the product on ( ) . We shall see in Example 6.3.29 that the Banach algebra  ∞ (An ) is not necessarily Arens regular in the above setting. 1 

⊗ A is Corollary 2.3.41 Let A be a Banach algebra that is Arens regular. Then  1 also Arens regular. ⊗ A and  1 (A) are isometrically Banach-algebra isoProof By Example 2.1.35,  1 morphic, and so this follows from Proposition 2.3.40. The following result is given by Ülger in [309]. The proof of the theorem, being long and quite involved, is omitted; in the special case where A does not include a copy of 1 , there is a short and different proof in [311, Corollary 4.4]. Theorem 2.3.42 Let A be a Banach algebra which is Arens regular, and suppose that K is a non-empty, compact space. Then the Banach algebra C(K, A) is also Arens regular. Let A be a Banach algebra, and let M = (Mn : n ∈ Z+ ) be a sequence in R+• such that M0 = 1, M1 ≥ 1, and

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  n Mn ≥ (n ∈ Z+ , m ∈ Z+n ) . m Mm Mn−m Take A M to be the collection of sequences a = (an : n ∈ Z+ ) in A such that a :=

∞  an n=0

Mn

< ∞,

so that (A M , · ) is a Banach space. For a = (an : n ∈ Z+ ) and b = (bn : n ∈ Z+ ) in AM , define ab = c, where c = (cn : n ∈ Z+ ) and n    n cn = am bn−m (n ∈ Z+ ) . m m=0 Then it is easily checked that (A M , · ) is a Banach algebra. The following proposition follows by essentially the same argument as that in Proposition 2.3.40. Proposition 2.3.43 Let A be a Banach algebra that is Arens regular. Then A M is Arens regular. Part of the following result was proved by Ülger in [310, Corollary 3.2]. Theorem 2.3.44 Let A be a Banach algebra with a bounded approximate identity. Then: (i) AP(A) ⊂ WAP(A) = A · WAP(A) · A ⊂ A · A · A; (ii) A · A · A = A whenever A is Arens regular; (iii) A · A = A · A = A · A · A = WAP(A) whenever A is an ideal in its bidual; (iv) A · A = A · A = A · A · A = AP(A) = WAP(A) whenever A is a compact algebra. Proof (i) Take (eα ) to be a BAI in A, say with bound m. Set E = WAP(A), so that E is a Banach A-bimodule; certainly AP(A) ⊂ E. Take λ ∈ E, and take S to be the weak closure of A[m] · λ in A , so that S is weakly compact. Since (eα · λ) is contained in S , we may suppose that (eα · λ) converges weakly in A , say to μ ∈ S . For each a ∈ A, we have a, μ = lim a, eα · λ = lim aeα , λ = a, λ , α

α

and so μ = λ. It follows from Mazur’s theorem, Theorem 1.2.24(ii), that λ ∈ AE, and so, by Cohen’s factorization theorem, E ⊂ A · E. This shows that E = A · E. Similarly, we have E = E · A. (ii) By Theorem 2.3.29, WAP(A) = A if and only if A is Arens regular, and in this case A · A · A = A by (i).

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(iii) By Proposition 2.3.16, A · A = A · A. By Theorem 2.3.7, La and Ra are weakly compact for each a ∈ A, and so AA ∪ A A ⊂ WAP(A) by Proposition 2.3.37. Hence A · A · A ⊂ WAP(A). The result now follows from (i). (iv) By Proposition 2.3.37, AA ∪ A A ⊂ AP(A), and so this is similar. The following lemma and theorem of Ülger from [314] are also given in [50, Theorem 2.9.39]. Lemma 2.3.45 Let A be an Arens regular Banach algebra. Suppose that A has a sequential bounded approximate identity (en ) of bound m and that A is weakly sequentially complete. Then A has an identity eA with eA ≤ m and such that en → eA weakly. Proof By Theorem 2.3.44(ii), A = A · A . Take λ ∈ A , say λ = a · μ, where a ∈ A and μ ∈ A . Then en , λ = en , a · μ = en a, μ → a, μ

as n → ∞ .

Thus (en ) is weakly Cauchy. Since A is weakly sequentially complete, (en ) is weakly convergent, say en → e weakly. Certainly e ∈ A[m] . Now take a ∈ A. Then aen , λ → a, λ and aen , λ → ae, λ as n → ∞ for each λ ∈ A , and so a = ae. Similarly, a = ea, and so e is the identity of A. Theorem 2.3.46 Let A be an Arens regular Banach algebra. Suppose that A has a bounded approximate identity and that A is weakly sequentially complete. Then A has an identity. Proof Suppose that the bound of the approximate identity in A is m, and assume towards a contradiction that A does not have an identity. By Proposition 2.1.27, there is a non-zero, closed subalgebra B0 of A with a sequential approximate identity in (B0 )[m] . Since B0 is Arens regular and weakly sequentially complete, it follows from Lemma 2.3.45 that B0 has an identity, say p1 ∈ (B0 )[m] . We now construct inductively a sequence (pn ) of non-zero idempotents in A[m] such that pi  p j and pi p j = pi∧ j for i, j ∈ N. Indeed, assume that p1 , . . . , pk have been defined, and set C = {a ∈ A : apk = pk a = a}, a closed subalgebra of A. Then pk is the identity of C, and so C  A; choose a0 ∈ A\C. Then there is a closed subalgebra Bk of A with p1 , . . . , pk , a0 ∈ Bk such that Bk has a sequential approximate identity in (Bk )[m] . As before, Bk has an identity, say pk+1 , in (Bk )[m] . For each j ∈ Nk , we have p j pk+1 = pk+1 p j = p j . Since a0 pk+1 = pk+1 a0 = a0 and a0  C, necessarily pk+1  pk . Hence pk+1  {p1 , . . . , pk }. This continues the inductive construction. Define P = lin {p j : j ∈ N}. Then P is a closed, commutative subalgebra of A with a sequential BAI (p j ) in P[m] . By Lemma 2.3.45, P has an identity, say eP , and pn → eP weakly as n → ∞. For j ∈ N, set

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Φ j = {ϕ ∈ ΦP : ϕ(p j ) = 1} , so that each Φ j is a clopen subset of the compact space ΦP . Clearly we have Φ j ⊂ Φ j+1 ( j ∈ N). For each ϕ ∈ ΦP , certainly ϕ(p j ) → 1 as j → ∞, and so ϕ(p j ) = 1  eventually. Thus ΦP = {Φ j : j ∈ N}, a compact space, and hence there exists k ∈ N such that ΦP = Φk . This implies that pk+1 = p k , and it follows from Proposition 2.1.3 that pk+1 = pk , a contradiction. Thus A has an identity. Theorem 2.3.47 Let A be a Banach algebra that is an ideal in its bidual and has a bounded approximate identity. (i) Suppose that A is separable. Then AA is a separable, closed linear subspace of A . (ii) Suppose that A contains a copy of  1 . Then A is not Arens regular. Proof First, suppose that (en ) is a sequential approximate identity for A. For each n ∈ N, define Hn = {en · λ : λ ∈ A[1] } . By Proposition 2.3.13, each set Hn is convex and weakly compact in A . Now define  {Hn : n ∈ N} , H = lin so that H is a closed linear subspace of A . We claim that AA ⊂ H. Indeed, assume to the contrary that there exist a ∈ A and λ ∈ A with a · λ  H. By Hahn–Banach, there exists M ∈ A with M, a · λ = 1 and M | H = 0. Take n ∈ N. For each μ ∈ A , we have M · en , μ = M, en · μ = 0, and so M · en = 0, and hence M · en a = 0. Since (en ) is an approximate identity, this implies that M · a = 0, and so M, a · λ = 0, the required contradiction. This establishes the claim. (i) Suppose that A is separable, so that A has a sequential BAI. With the above notation, it follows from Proposition 1.2.48(ii) that each Hn is separable, and so H is separable. By the claim, AA is a separable, closed linear subspace of A . (ii) Now suppose that A contains a copy of  1 , and assume towards a contradiction that A is Arens regular. Since  1 is separable, it follows from Proposition 2.1.27 that there is a closed, separable subalgebra B of A with  1 ⊂ B and such that B has a sequential BAI, say (en ). Further, B is Arens regular, and so WAP(B) = B .  Now define Hn = {en · λ : λ ∈ B[1] } for n ∈ N and H = lin {Hn : n ∈ N}. Again, H is a separable, closed linear subspace of B . By Theorem 2.3.44(ii), B = B · B . Take λ ∈ B , say λ = b · μ, where b ∈ B and μ ∈ B . Then en · λ − λ ≤ en b − b μ → 0 as n → ∞, and so λ ∈ H. Thus B = H is separable. By Proposition 1.2.50, B has the RNP, but, by Proposition 1.3.35, B does not have the RNP because B contains a copy of  1 , a contradiction. We conclude that A is not Arens regular.

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The next theorem shows that some Banach algebras are Arens regular whenever they have certain properties as a Banach space, independent of the product on the algebra. Theorem 2.3.48 Let A be a Banach algebra such that B(A, A ) = W(A, A ). Then A is Arens regular. In particular, A is Arens regular whenever A has the Schur property or A has property (V) as a Banach space. Proof Since B(A, A ) = W(A, A ), certainly WAP(A) = A , and so A is Arens regular by Theorem 2.3.29. By Proposition 1.3.72 and Corollary 1.3.52, respectively, B(A, A ) = K(A, A ) and B(A, A ) = W(A, A ) whenever A has the Schur property and A has property (V), respectively, and so A is Arens regular in both these cases. We shall now give some first examples of Banach algebras that are and are not Arens regular. For the following standard result, see [50, Corollary 3.2.37], for example. For a direct proof that all commutative C ∗ -algebras are Arens regular, see [51, Theorem 4.5.5]. Theorem 2.3.49 All Banach algebras that are linearly homeomorphic to a C ∗ algebra are Arens regular. In particular, all C ∗ -algebras A are Arens regular, and the bidual (A ,  ) is a von Neumann algebra. All closed subalgebras of a C ∗ algebra are Arens regular. Proof Let A be a Banach algebra that is linearly homeomorphic to a C ∗ -algebra. Then B(A, A ) = W(A, A ) by Theorem 2.2.16(iii), and so A is Arens regular by Theorem 2.3.48. Corollary 2.3.50 Let H be a Hilbert space. Then K(H) does not contain a copy of the Banach space  1 . Proof The C ∗ -algebra K(H) has a CAI is Arens regular by Theorem 2.3.49, and is an ideal in its bidual, B(H). Thus this follows from Theorem 2.3.47(ii). The bidual of a C ∗ -algebra is called the enveloping von Neumann algebra. Corollary 2.3.51 Let A be a Banach algebra, and let I be a closed ideal in A such that A/I is isomorphic as a Banach space to a C ∗ -algebra. Then I ⊥ ⊂ WAP(A). Proof This follows from Theorem 2.3.49 and Corollary 2.3.34. We now give an example that originates with Arens of a Banach algebra that is not Arens regular; this example will be considerably generalized later. In particular, a ‘semigroup’ version of the result will be given in Proposition 6.3.4.

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Example 2.3.52 Let G be an infinite group, and then consider the group algebra ( 1 (G),  ). It is easily seen that there are sequences (x j ) and (yk ) in G such that x1 = y1 = eG and xi1 y j1  xi2 y j2 when (i1 , j1 ) and (i2 , j2 ) are distinct points in N 2 . Set a j = δ x j and bk = δyk for j, k ∈ N, and define λ ∈  ∞ (G) by requiring that λ(x) = 1 when x = x j yk for j, k ∈ N with j < k and λ(x) = 0 otherwise. Then (a j ) and (bk ) are sequences in  1 (G)[1] and lim lim a j bk , λ = 1

j→∞ k→∞

and

lim lim a j bk , λ = 0 ,

k→∞ j→∞

and hence the iterated-limit condition in Corollary 2.3.31 fails. Thus  1 (G) is not Arens regular. It follows that  1 (G) cannot be either a closed subalgebra or a quotient of a ∗ C -algebra.

The next two examples are close to [50, Theorem 2.6.23] and are based on [311, Lemma 4.2] and [330, Theorem 3]. Example 2.3.53 Let E be a Banach space that is not reflexive, and suppose that A is a Banach operator algebra in B(E). Then A is not Arens regular.  Indeed, since E is not reflexive, there are sequences (xm ) in E[1] and (λn ) in E[1]  such that (xm , λn  : m, n ∈ N) has repeated limits that are distinct. Take Λ ∈ A such that Λ | F (E)  0. Then there exist x ∈ E and λ ∈ E  such that x ⊗ λ, Λ = 1. Set S m = xm ⊗ λ (m ∈ N) and T n = x ⊗ λn (n ∈ N). By (1.3.10), T n S m = xm , λn  x ⊗ λ

(m, n ∈ N) ,

and so (T n S m , Λ) has repeated limits that are distinct. Thus Λ  WAP(A), and so WAP(A)  A , whence A is not Arens regular by Theorem 2.3.29. The above argument shows that WAP(A) = {0} when A = A(E). In the case where E  has the BAP, A(E) has a bounded approximate identity [50, Theorem 2.9.37], but {0} = WAP(A)  A · A · A, showing that we do not always have the equality WAP(A) = A · A · A in Theorem 2.3.44(i). For related examples, see [89].

Example 2.3.54 Let E be a reflexive Banach space. Then E ⊗ E  , N(E), K(E) and A(E) are each Arens regular [50, Theorem 2.6.23]. In particular, the algebra S1 (H) of trace-class operators is Arens regular for each Hilbert space H. Suppose that E also has the approximation property. Then we identify K(E) with B(E), and then it is easily checked that both Arens products coincide with the natural product on B(E), and hence K(E) is an ideal in its bidual. We shall note in §6.1 that this is usually not the case when E is not reflexive.

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We now state two theorems that nearly characterize when a Banach algebra is Arens regular; the first theorem of Young is from [330, Theorem 1] and the second theorem of Daws is from [65]. Theorem 2.3.55 Let A be a Banach algebra that is Arens regular. Then there is a reflexive Banach space E and an isometric Banach-algebra embedding of A into B(E). Theorem 2.3.56 Let E be a super-reflexive Banach space. Then every closed subalgebra of B(E) is Arens regular. Let A be a Banach algebra. We shall now describe introverted subspaces of A . Definition 2.3.57 Let A be a Banach algebra, and let F be a closed A-submodule of (A , · ). Then the space F is faithful if a = 0 whenever a ∈ A is such that a, λ = 0 (λ ∈ F). The space F is left-introverted (respectively, right-introverted) in A if M · λ ∈ F (respectively, λ · M ∈ F) whenever M ∈ A and λ ∈ F, and F is introverted if it is both left- and right-introverted. It is immediate from the Hahn–Banach theorem that a closed submodule F of (A , · ) is left-introverted if and only if M · λ ∈ F whenever λ ∈ F and M ∈ F  . For example, A A is a left-introverted submodule of A , and A itself is introverted; the space A is always faithful, and A A is faithful whenever Aa  {0} for each a ∈ A• . The notion of left-introversion was introduced in a special case by Day in [75]; see also [215] and [326]. We note that our definition of ‘left-introverted’ generalizes that of [17, Definition 2.4]. The following characterization of left-introverted spaces was given by Lau and Loy in [215, Lemma 1.2]. Proposition 2.3.58 Let A be a Banach algebra, and let F be a closed A-submodule σ of A . Then F is left-introverted in A if and only if the weak-∗-closure A[1] · λ is contained in F for each λ ∈ F. σ

Proof Suppose that F is left-introverted, and take λ ∈ F. For each μ ∈ A[1] · λ , there is a net (aα ) in A[1] such that wk∗ – limα aα · λ = μ in A . Take a weak-∗ accumulation point M of (aα ) in A[1] ; by passing to a subnet, we may suppose that wk∗ – limα aα = M. Then a, μ = lim a, aα · λ = lim aα , λ · a = M, λ · a = a, M · λ α

α

and so μ = M · λ ∈ F. The converse is a similar calculation.

(a ∈ A) ,

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Corollary 2.3.59 Let A be a Banach algebra, and let F be a closed A-submodule of A with F ⊂ WAP(A). Then F is introverted in A . Proof Take λ ∈ F, and let S be the weak closure of A[1] · λ. Since A[1] · λ is relatively σ weakly compact in A , it follows from Proposition 1.2.11 that S = A[1] · λ . Since A[1] · λ is convex, it follows from Mazur’s theorem, Theorem 1.2.24(ii), that S is the norm-closure of A[1] · λ in A , and so S ⊂ F. By Proposition 2.3.58, F is leftintroverted. By Theorem 2.3.29, WAP(A) = WAP(Aop ), and so it is also true that F is rightintroverted, and hence introverted. The following result is part of [57, Theorem 5.4]. Theorem 2.3.60 Let A be a Banach algebra, and let F be a closed A-submodule of A . Then F ⊥ is an A-submodule of A and a weak-∗-closed left ideal in (A ,  ). Suppose, further, that F is left-introverted. Then F ⊥ is a weak-∗-closed ideal in  (A ,  ), and A /F ⊥ is a Banach algebra. Suppose, further, that A · A ⊂ F. Then A  F ⊥ = {0} and F ⊥ ⊂ rad A . Proof Clearly F ⊥ is an A-submodule of A . Take M ∈ A and N ∈ F ⊥ . For each λ ∈ F, we have a, N · λ = N, λ · a = 0 (a ∈ A) , and so N · λ = 0. Thus M  N, λ = M, N · λ = 0, and so M  N ∈ F ⊥ . Hence F ⊥ is a left ideal in A , and is clearly weak-∗ closed. Now suppose that the space F is left-introverted. For each λ ∈ F, we have N  M, λ = N, M · λ = 0, and so N  M ∈ F ⊥ . Hence F ⊥ is a right ideal in A , and so an ideal in A . It follows that A /F ⊥ is a Banach algebra. Finally, suppose that A · A ⊂ F. For each a ∈ A and λ ∈ A , we see that a, N · λ = N, λ · a = 0, and hence N · λ = 0. It follows that M  N, λ = M, N · λ = 0 , and so M  N = 0. This shows that A  F ⊥ = {0}. In particular, F ⊥ is a nilpotent ideal, and hence F ⊥ ⊂ rad A . In the notation of the next definition, F  = A /F ⊥ as a Banach space, and so we can regard F  as a quotient Banach algebra of (A ,  ). Definition 2.3.61 Let A be a Banach algebra, and let F be a closed A-submodule of A that is left-introverted. Then the space F  is the quotient Banach algebra (A ,  )/F ⊥ ; the quotient map is qF : A → F  . We have qF (M), λF  ,F = M, λA ,A

(λ ∈ F, M ∈ A ) ,

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showing that qF (M) = M | F ∈ F  . The product in F  is again denoted by  , so that M  N is defined in F  for each M, N ∈ F  by the formula: M  N, λ = M, N · λ (λ ∈ F) .

(2.3.10)

We recall that F  = A /F ⊥ is a Banach A -bimodule (with the product  ), and so we have the following formula: qF (M)  N = qF (M  N) = qF (M)  qF (N) (M, N ∈ A ) .

(2.3.11)

Consider the special case that F = A A, and take M, N ∈ A . Then a, N · λ = N, λ · a = qF (N), λ · a = a, qF (N) · λ

(a ∈ A, λ ∈ A ) ,

and so N · λ = qF (N) · λ (λ ∈ A ). It follows that M  N, λ = M, N · λ = M, qF (N) · λ = M  qF (N), λ (λ ∈ A ) . Thus

qF (M  N) = M  qF (N) (M, N ∈ A ) , (2.3.12)       giving a natural product A × A A → A A that identifies A A as a Banach left A -module. There is a natural Banach-algebra homomorphism of A into F  ; in the case where F is faithful, the map is an embedding, and we regard A as a subalgebra of (F  ,  ). Again, a · M = a  M and M · a = M  a for each a ∈ A and M ∈ F  . As usual, we define operators L M and R M in B(F  ) for M ∈ F  by the formulae: L M (N) = M  N,

R M (N) = N  M

(N ∈ F  ) .

It is easy to see that each operator R M is weak-∗ continuous on F  ; in §6.1, we shall consider when the operator L M is weak-∗ continuous on F  . Similarly, suppose that F is a closed A-submodule of A that is right-introverted. Then (F  ,  ) is a quotient Banach algebra of (A ,  ). In the case where F is introverted, we have two products on the space F  . In particular, suppose that F is a closed submodule of A such that F ⊂ WAP(A). Then F is introverted by Corollary 2.3.59 and the two products,  and , agree on F  by Theorem 2.3.29. In the following result, we regard E ⊥ as a closed linear subspace of F  ; the proof is the same as that contained in Theorem 2.3.60. Proposition 2.3.62 Let A be a Banach algebra, and let E and F be two leftintroverted subspaces of A with E ⊂ F. Then E ⊥ is a weak-∗-closed ideal in (F  ,  ). Let I be a closed ideal in a Banach algebra A, with the embedding ι : I → A. Then ι : A → I  is a continuous surjection which is an A-bimodule homomorphism. Let E be a · -closed, A-submodule of A . Then F := ι (E) is Banach A-submodule of

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I  . We use the above notation in the following proposition, taken from [59, Proposition 1.16]. Proposition 2.3.63 Let I be a closed ideal in a Banach algebra A, and suppose that E is left-introverted in A . Then F is left-introverted in I  , and there is a continuous A-bimodule monomorphism τ : F  = I  /F ⊥ → E  = A /E ⊥ . Further, τ : (F  ,  ) → (E  ,  ) is a continuous Banach-algebra embedding identifying F  as a closed ideal in E  . Proof To show that F is left-introverted in I  , we apply Proposition 2.3.58. Let λ ∈ ι (E), and let Kλ be the closure of {a · λ : a ∈ I[1] } in the topology σ(I  , I). Let (aα ) be a net in I[1] such that aα · λ → μ in (I  , σ(I  , I)). Then there λ) = λ and ι ( μ) = μ. By passing to a subnet, we exist  λ ∈ E and  μ ∈ A such that ι (  λ→ μ in (A , σ(A , A)). Since  λ ∈ E and E is left-introverted may suppose that aα ·  μ ∈ E, and so μ ∈ ι (E). Thus Kλ ⊂ F, in A , it follows from Proposition 2.3.58 that  and so ι (E) is left-introverted in I  , again by Proposition 2.3.58. Hence F is leftintroverted in I  . The existence of the specified map τ is clear. The map ι : I  → A is a continuous injection with closed range, and so ι : (I  ,  ) → (A ,  ) a Banach-algebra embedding. Since I  is a closed ideal in A , it follows that (F  ,  ) is a closed ideal in (E  ,  ). We conclude this section by relating bounded approximate identities in a Banach algebra to mixed identities in the bidual of the algebra. Definition 2.3.64 Let A be a Banach algebra. An element E of A is a mixed identity for A if E  0 and if ME = EM = M

(M ∈ A ) .

The set of mixed identities for A is denoted by EA . When A is Arens regular, a mixed identity for A is the unique identity of A . Let E ∈ EA . Then, in particular, E is a right identity for A = (A ,  ), and hence an idempotent; further, E  A is a closed right ideal in A for which E is the identity, and A = E  A  J , where J = {M − E  M : M ∈ A } = {M ∈ A : E  M = 0} ⊂ rad A , as in Equation (2.1.1). The map P : M → E  M is a projection in B(A ) with range E  A . We also have a · E = E · a = a (a ∈ A) and E, ϕ = 1 (ϕ ∈ ΦA ). Conversely, suppose that E ∈ A and a · E = E · a = a (a ∈ A). Then E ∈ EA .

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It follows from Equation (2.1.2) that  {Z(E  A ) : E ∈ EA } . Z(A ) ⊂

(2.3.13)

Proposition 2.3.65 Let A be a Banach algebra with a mixed identity E, and take L ∈ M  (A). Then L (λ) = λ · L (E) (λ ∈ A ). Further, for each λ ∈ WAP(A), the set {L (λ) : L ∈ M  (A)[1] } is relatively weakly compact in A . Proof Take λ ∈ A . For each a ∈ A, we have a, λ · L (E) = L (E), a · λ = L (E) · a, λ = L (E · a), λ = La, λ = a, L (λ) , and so L (λ) = λ · L (E). Now take λ ∈ WAP(A). By Corollary 2.3.30, the map Lλ : a → λ · a, A → A , is weakly compact, and so Lλ (A ) ⊂ A . We see that Lλ (M) = λ · M (M ∈ A ),  ), which and so the set {λ · L (E) : L ∈ M  (A)[1] } is contained in the set Lλ (E[1]    is relatively weakly compact in A . Since L (λ) = λ · L (E), it follows that the set {L (λ) : L ∈ M  (A)[1] } is relatively weakly compact in A , as required. Proposition 2.3.66 Let A be a Banach algebra, and take m > 0. Then: (i) (A ,  ) ) has a right identity in A[m] if and only if A has a bounded right approximate identity in A[m] ; (ii) A has a mixed identity in A[m] if and only if A has a bounded approximate identity in A[m] . Proof (i) Suppose that (A ,  ) ) has a right identity E ∈ A[m] . Then a net in A[m] that converges weak-∗ to E is a bounded weak right approximate identity in A[m] . By Proposition 2.1.29, A has a BRAI in A[m] . Conversely, suppose that A has a BRAI (eν ) in A[m] , and let E be a weak-∗ accumulation point of (eν ) in A[m] ; we may suppose that wk∗ – limν eν = E. For each a ∈ A and λ ∈ A , we have a, λ = lim aeν , λ = lim eν , λ · a = E, λ · a = a · E , λ , ν

ν

and so a = a · E. It follows that M = M  E for each M ∈ A , and so E is a right identity for (A ,  ). (ii) This follows easily from (i). Thus EA is the collection of weak-∗ limits of bounded approximate identities in the Banach algebra A.

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Corollary 2.3.67 Let A be a Banach algebra with a mixed identity E. Then E is the identity of (A ,  ) if and only if A = A · A. Proof Suppose that E is the identity of (A ,  ), say E = limν eν , where (eν ) is a BAI in A. Take λ ∈ A . For each M ∈ A , we have M, λ · eν  = eν , M · λ → E, M · λ = E  M, λ = M, λ , and so λ · eν → λ weakly. By Mazur’s theorem, Theorem 1.2.24(ii), λ ∈ A A. By Corollary 2.1.47, A A = A · A, and so A = A · A. Conversely, suppose that A = A · A, and take λ ∈ A , say λ = μ · a, where a ∈ A and μ ∈ A . For each M ∈ A , we have E  M, λ = E, M · (μ · a) = a, M · μ = M, λ , and so E  M = M. Thus E is the identity of (A ,  ). Examples of Banach algebras with bounded approximate identities such that A = A · A, but A  A · A , will be given after Theorem 4.1.24 and in Example 6.1.29. The following example is taken from [220, Example 2.4]. 

Example 2.3.68 We give an example of a non-unital, commutative Banach algebra A with a bounded approximate identity such that A = A · A = A · A , but such that A is not Arens regular. Let B be a unital, commutative Banach algebra that is not Arens regular; for example, take B = ( 1 (Z),  ), so that B is not Arens regular by Example 2.3.52. Set A = c 0 (B), with coordinatewise product, so that A is a non-unital, commutative Banach algebra that is also not Arens regular because it contains B as a closed subalgebra. For n ∈ N, set n

!"#$ en = (eB , . . . , eB , 0, 0, . . . ) . Then (en ) is a bounded approximate identity for A, and clearly A = A A, and so, by Cohen’s factorization theorem, A = A · A = A · A .

Let A be a non-unital Banach algebra with a bounded approximate identity such that A is weakly sequentially complete. It seems to be open whether the equality A = A · A implies that A = A · A . The proof of the following result is similar to one first given by Lau in [210, Theorem 1]. Proposition 2.3.69 Let A be a Banach algebra with a mixed identity, let F be a left-introverted subspace of A , and let T ∈ B(F). Then T is a left A-module homomorphism if and only if there exists M ∈ F  such that T λ = λ · M (λ ∈ F) .

(2.3.14)

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Proof Suppose that T has the form specified in Equation (2.3.14). Then T (a · λ) = a · λ · M = a · T λ

(a ∈ A, λ ∈ F) ,

and so T is a left A-module homomorphism. Conversely, suppose that T is an A-module homomorphism. Let E be a mixed identity in A. For each a ∈ A and λ ∈ F, we have a, T λ = E · a, T λ = E, T (a · λ) = T  (E), a · λ = a, λ · T  (E) , and so T has the required form, with M = T  (E). Proposition 2.3.70 Let A be a Banach algebra with a mixed identity E. Then EA = E + (AA )⊥ ∩ (A A)⊥ . Proof Take N ∈ (AA )⊥ ∩ (A A)⊥ . Then a · N = N · a = 0 (a ∈ A), and hence M  N = N  M = 0 (M ∈ A ). Thus E + N ∈ EA , and this shows that E + (AA )⊥ ∩ (A A)⊥ ⊂ EA . Conversely, suppose that F ∈ EA . Then a · (E − F) = (E − F) · a = 0 (a ∈ A), and so E − F ∈ (AA )⊥ ∩ (A A)⊥ , showing that EA ⊂ E + (AA )⊥ ∩ (A A)⊥ . Let A be a Banach algebra with a bounded approximate identity. In the following corollary, we consider algebras A such that also AA = A A (which implies that A · A = A · A); this condition on A is satisfied when A is commutative, when A is an ideal in its bidual (by Proposition 2.3.16), and when A is Arens regular (in which case AA = A A = A by Theorem 2.3.44(ii)). Corollary 2.3.71 Let A be a Banach algebra such that AA = A A, and suppose that E ∈ EA . Then the following are equivalent: (a) E is the unique element of EA ; (b) E is the identity of (A ,  ); (c) A = A · A. Proof (a) ⇒ (c) By Proposition 2.3.70, (A A)⊥ = {0}, and so A A = A . However A A = A · A because A has a BAI, giving (c). (b) ⇒ (a) Take F ∈ EA . Then F = E  F = E, giving (a). (b) ⇔ (c) This is Corollary 2.3.67. Proposition 2.3.72 Let A be a Banach algebra with a bounded approximate identity, and suppose that A is Arens regular. Then EA has a unique element. Proof Take E, F ∈ EA . Then E = E  F = E  F = F.

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Corollary 2.3.73 Let A be a Banach algebra that is an ideal in its bidual and has a bounded approximate identity. Then the following are equivalent: (a) EA has a unique element; (b) A has an identity; (c) A = A · A; (d) A = A A; (e) A is Arens regular. Proof (a) ⇔ (b) ⇔ (c) This follows from Corollary 2.3.71. (e) ⇒ (a) This follows from Corollary 2.3.72. (d) ⇒ (c) This is Cohen’s factorization theorem, Theorem 2.1.46. (c) ⇒ (d) This is trivial. (d) ⇒ (e) This follows from Corollary 2.3.38. The following result extends [303, Corollary 4.3]. Proposition 2.3.74 Let A be a semisimple Banach algebra with a bounded approximate identity, and suppose that A is an ideal in its bidual. Then the following are equivalent: (a) A is Arens regular; (b) A is semisimple; (c) M r (A) is Banach-algebra isomorphic to A . Proof (a) ⇒ (b) By Corollary 2.3.73, (e) ⇒ (b), A has an identity, say E. Take M ∈ rad A . By Proposition 2.3.15, M = E  M = 0, and so rad A = {0} and A is semisimple. (b) ⇒ (a), (c) Take F = A · A, as above. Then F ⊥ ⊂ rad A by Theorem 2.3.60, and so F ⊥ = {0}. Hence A = E  A , which is Banach-algebra isomorphic to M r (A), giving (c). Also A · A = A , and so A is Arens regular by Corollary 2.3.73, (c) ⇒ (e), giving (a). (c) ⇒ (a) Since M r (A) has an identity, the same is true for A , and so A is Arens regular by Corollary 2.3.73, (b) ⇒ (e). Sometimes, in the following proofs, we shall write M N, rather than M  N, for the product of two elements M, N in A . The next result is [221, Theorem 2.7]; in the statement and the proof, we set Iu = {a ∈ A : au = 0} for an element u ∈ I(A ).

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Theorem 2.3.75 Let A be a Banach algebra with a bounded approximate identity, and suppose that I is a closed ideal in A. Then I has a bounded right approximate identity if and only if there exists u ∈ I(A ) such that I = Iu and Au is a subalgebra which is closed in A . In this case, A/I and Au are isomorphic as Banach left A-modules. Proof Let E be a mixed identity for A in A . First, suppose that I has a BRAI, so that, by Proposition 2.3.66(i), I  has a right identity, say F. Set u = E − E F, so that u ∈ I(A ) by Proposition 2.1.1(ii). We have au = a − a F ∈ A

(a ∈ A) .

We claim that I = Iu . Clearly au = 0 (a ∈ I), and so I ⊂ Iu . Conversely, take a ∈ Iu . Then a = a F, and so a ∈ I  ∩ A = I by Equation (1.3.4). The claim follows. We next claim that Iu⊥ = u · A . Take λ ∈ A and a ∈ Iu , so that a, u · λ = 0, and hence u · A ⊂ Iu⊥ . Conversely, take a ∈ A and λ ∈ Iu⊥ . Then aF ∈ I  = Iu = Iu⊥⊥ , and so a F, λ = 0. Hence a, u · λ = au, λ = a − a F, λ = a, λ, and so λ = u · λ ∈ u · A , giving the second claim. Define θ : a + Iu → au, A/Iu → Au ⊂ A . Then θ is a well-defined linear isomorphism, and θ(a · (b + Iu )) = abu = a · θ(b + Iu ) (a, b ∈ A) , and hence θ is a left A-module isomorphism. Take a ∈ A. Then au = sup{|a + Iu , u · λ| : λ ∈ A[1] } ≤ a + Iu u . Also the space u · A is closed in A because u = u2 ∈ A , and the linear map λ → u · λ, A → u · A , is a continuous surjection, and so, by the open mapping theorem, Theorem 1.3.5(ii), there exists m > 0 such that u · A[m] ⊃ (u · A )[1] . Thus a + Iu = sup{|a, u · λ| : u · λ ∈ (u · A )[1] } ≤ m sup{|au, λ| : λ ∈ A[1] } = m au . It follows that au ≤ a + Iu u ≤ m u au , and so Au is closed in A , the map θ : A/I = A/Iu → Au is a Banach-space isomorphism. Hence A/I and Au are isomorphic as Banach left A-modules. Conversely, suppose that there exists u ∈ I(A ) such that I = Iu and Au is a subalgebra which is closed in A . We claim that u · A is weak-∗ closed in A . For this, set T : a → au, A → A , so that T is a bounded linear operator with closed range in A , and hence T  (A ) is weak-∗ closed in A . Since ker T = Iu , we have T  (A ) = Iu⊥ . Our aim is to show that T  (A ) = u · A .

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For each λ ∈ A , we have T  (λ) = u · λ, and so u · A ⊂ T  (A ). Since T (ab) = T a · b (a, b ∈ A), we have T  (au) = T  (a) · u = (T a)u = au2 = au = T a (a ∈ A) . Now take Λ ∈ A . Then a, u · T  (Λ) = au, T  (Λ) = T  (au), Λ = T a, Λ = a, T  (Λ) (a ∈ A) , and so T  (Λ) = u · T  (Λ), whence T  (A ) ⊂ u · A . Thus T  (A ) = u · A , as required, and hence u · A is weak-∗ closed in A , as claimed. By the claim, Iu⊥ = u · A , and so the map P : λ → u · λ, A → A , is a projection with range Iu⊥ = I ⊥ , and hence I  = I ⊥⊥ = P (A )⊥ = ker P ; in particular, P a = 0 (a ∈ I). For a ∈ I and λ ∈ A , we have a · P (E), λ = P (E), λ · a = E, P(λ · a) = E, u · λ · a = a, P(λ) · a = P a, λ · a = 0 , and so a · P (E) = 0. Set F = E − P (E), so that F ∈ I  and a · F = a · E = a (a ∈ I). This shows that F is a right identity for I  , and so I has BRAI. We note that, in the case where A is commutative, the map θ in the above proof is a Banach-algebra isomorphism. Corollary 2.3.76 Let A be a Banach algebra with a bounded approximate identity and such that A is weakly sequentially complete. Then A/I is also weakly sequentially complete for each closed ideal I of A that has a bounded right approximate identity. Proposition 2.3.77 Let A be a commutative Banach algebra with a closed ideal I that has a mixed identity. Then Z(I  ) ⊂ Z(A ). Suppose, further, that I is Arens regular and that A2 ⊂ I. Then A is Arens regular. Proof The space I  is a closed ideal in A , and I  contains a mixed identity, say E. Take R ∈ Z(I  ) and M ∈ A , so that E R = R E = R. Also E M ∈ I  , and so E M R = R E M = R M. Thus RM = EMR = RME = MER = MR. This shows that R ∈ Z(A ), and so Z(I  ) ⊂ Z(A ). Now suppose that I is Arens regular and that A2 ⊂ I, so that E is the identity of I  and A  A ⊂ I  by Proposition 2.3.4. Take M ∈ A , and set P = M − E M ∈ A . For each N ∈ A , we have P N = M N − E M N = 0 because M N ∈ I  . Thus a · P = P · a = 0 (a ∈ A), and so N P = 0. This shows that P ∈ Z(A ). Also E M ∈ I  = Z(I  ) ⊂ Z(A ), and so M = P + E M ∈ Z(A ). We have shown that A = Z(A ), and hence A is Arens regular.

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133

Let A be a Banach algebra with a mixed identity E in A , and set θ E (T ), λ = E, T  λ = T  (E), λ

(λ ∈ A )

for T ∈ B(A), so that θ E : T → θ E (T ) = T  (E), B(A) → A , is a linear operator with θ E ≤ E . The following results are based on [220, §4]. In the next theorem, we identify the Banach algebra A with the closed subalgebra {Ra : a ∈ A} of M r (A). Recall from page 76 that we regard the space M r (A) of right multipliers on an algebra A as a subalgebra of L(A)op , so that the product of R, S ∈ M r (A) is S ◦ R. Theorem 2.3.78 Let A be a Banach algebra with a mixed identity E in A . Then the map θ E : R → R (E) , M r (A) → (A ,  ) , is a Banach-algebra embedding that extends the canonical embedding κA : A → A . The range of θ E is contained in E  A . In the case where A is an ideal in its bidual, the range of θ E is equal to E  A , and A = θ E (M r (A))  (IA − L E )(A ) . Proof The map θ E is a linear operator with θ E (R) ≤ R op E (R ∈ M r (A)). Now take R ∈ M r (A). For each a ∈ A and λ ∈ A , we have R (a · E), λ = a, R λ = E, R λ · a = E, R (λ · a) = R (E), λ · a = a · R (E), λ , and so R (a · E) = a · R (E). Hence



Ra =

R (a · E)

=

a · R (E)

≤ θ E (R) a , and so R op ≤ θ E (R) . Thus θ E is a Banach-space embedding. Take R, S ∈ M r (A). Then, using Equation (2.3.3), we have θ E (RS ) = θ E (S ◦ R) = S  (R (E)) = S  (R (E)  E) = R (E)  S  (E) = θ E (R)  θ E (S ) , and so θ E is an algebra homomorphism. Take a ∈ A. Then θ E (Ra ) = Ra (E) = E · a = κA (a), and so θ E extends κA . Take R ∈ M r (A). Then θ E (R) = R (E  E) = E  R (E), and so the range of θ E is contained in E  A . Now suppose that A is an ideal in A . For each M ∈ A , define T M : a → a · M.  (N) = N  M (N ∈ A ). Thus Then T M ∈ M r (A), and T M  θ E (T M ) = T M (E) = E  M .

This shows that θ E : M r (A) → E  A is a surjection. It follows that, as on page 73, A = θ E (M r (A))  (IA − L E )(A ).

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Note that, in general, the above embedding θ E is not a canonical embedding of M r (A) into (A ,  ); it depends on the choice of the mixed identity E. Corollary 2.3.79 Let A be a commutative Banach algebra that has a mixed identity E in A . Then the map θ E : L → L (E) ,

M(A) → (A ,  ) ,

is a Banach-algebra embedding that extends the canonical embedding κA : A → A and has range contained in Z(E  A ). Proof Certainly θ E (M(A)) ⊂ E  A . Now take L ∈ M(A) and M ∈ A . Then L (E) E M = L (E M) = L (E M E) = E M L (E) because L is also a right multiplier on A , and so L (E) ∈ Z(E  A ). Thus θ E (M(A)) ⊂ Z(E  A ). The following theorem is related to [220, Theorem 3.4]. Theorem 2.3.80 Let A be a commutative Banach algebra that has a mixed identity E. Suppose that A is weakly sequentially complete and has a sequential bounded approximate identity. Then Z(A ) = A if and only if Z(E  A ) = θ E (M(A)). Proof Suppose that Z(E  A ) = θ E (M(A)), and take M ∈ Z(A ). By Equation (2.3.13), M = E M ∈ Z(E  A ), and so M ∈ θ E (M(A)), say M = L (E) for some L ∈ M(A). Thus a · M = a · L (E) = La ∈ A (a ∈ A) . Let (en ) be a sequential BAI in A, so that (en · M) is a sequence in A that converges to M weak-∗ in A . Since A is weakly sequentially complete, it follows that M ∈ A. Thus Z(A ) = A. For the converse, suppose that Z(A ) = A. Since A factors and A ⊂ Z(A ), it follows from Proposition 2.1.1(iii) that AZ(A ) = AZ(E  A ), and hence AZ(E  A ) = A2 = A. Take M ∈ Z(E  A ) and a ∈ A. Then a · M ∈ A, and so a · M = La = a · L (E) for some L ∈ M(A). Thus Z(E  A ) ⊂ θ E (M(A)). By Corollary 2.3.79, θ E (M(A)) ⊂ Z(E  A ), and so Z(E  A ) = θ E (M(A)). Theorem 2.3.81 Let A be a Banach algebra that has a mixed identity E, and let F be a faithful, left-introverted submodule of A . Then there is a Banach-algebra embedding θ = qF ◦ θ E : M r (A) → (F  ,  ) such that θ(R), λ · aF  ,F = Ra, λA,A

(a ∈ A, λ ∈ F, R ∈ M r (A))

(2.3.15)

and θ extends the canonical embedding of A into F  . In the case where F = FA, the map θ is independent of the choice of E.

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135

Proof By Theorem 2.3.78, the map θ E : R → R (E), M r (A) → (A ,  ), is a Banach-algebra embedding. Set θ = qF ◦ θ E , M r (A) → (F  ,  ), so that θ is also a Banach-algebra homomorphism. Take a ∈ A, λ ∈ F, and R ∈ M r (A). Then θ(R), λ · aF  ,F = R (E), λ · aA ,A = E, R (λ · a)A ,A = E, R (λ) · aA ,A = a, R (λ)A,A = Ra, λA,A , giving Equation (2.3.15). Since F is faithful, the map θ is an injection. Take a ∈ A. Then θ E (a) = κA (a) by Theorem 2.3.78, and so θa = a. In the case where F = FA, Equation (2.3.15) shows that θ is independent of the choice of E. The embedding θ described in the above theorem is termed the canonical embedding of M r (A) in (F  ,  ) when F = FA. Let A be a Banach algebra with a mixed identity, say E, and set A = E  A  J in the notation after Definition 2.3.64. The closed submodule F := A · A = A A of A is left-introverted and is such that F = FA; as usual, we identify the spaces A /F ⊥ and F  . For M + F ⊥ ∈ F  , define j E (M + F ⊥ ) = E  M. In the next result, we use the above notation and the maps θ E and θ from Theorems 2.3.78 and 2.3.81, respectively. Theorem 2.3.82 Let A be a Banach algebra with a mixed identity E, and set F = A A. Then J = F ⊥ , the map j E : (F  ,  ) → (E  A ,  ) is a Banach-algebra isomorphism, and (2.3.16) A = F   F ⊥ = E  A  F ⊥ . Further, (F  ,  ) has the identity qF (E) and j E ◦ θ = θ E : M r (A) → E  A . Proof Take M ∈ A , a ∈ A, and λ ∈ A . Suppose that M ∈ J. Then M, λ · a = M − E M, λ · a = a · (M − E M), λ = 0 , and so M ∈ F ⊥ . Conversely, suppose that M ∈ F ⊥ . Then a · M, λ = M, λ · a = 0 , and so E M, λ = 0, whence E M = 0, showing that M ∈ J. Thus J = F ⊥ . Now consider the specified map j E : (F  ,  ) → E  A . This map is well defined because j E (M+F ⊥ ) = 0 when M ∈ F ⊥ . It is then easily checked that j E : (F  ,  ) → (E  A ,  ) is a Banach-algebra isomorphism. Since ( j E ◦ qF )(E) = E, the identity of (E  A ,  ), the element qF (E) is the identity of (F  ,  ). Take R ∈ M r (A). Then ( j E ◦ θ)(R) = ( j E ◦ qF ◦ θ E )(R) = j E (R (E) + F ⊥ ) = E R (E) = R (E) = θ E (R) , and so j E ◦ θ = θ E .

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Example 2.3.83 Let A be a Banach algebra, set F = A A, and set W = WAP(A), so that F is a left-introverted A-submodule of A and, by Corollary 2.3.59, W is an introverted A-submodule of A . By Theorem 2.3.60, F ⊥ and W ⊥ are closed ideals in (A ,  ) and the quotient spaces F  = A /F ⊥ and W  = A /W ⊥ are Banach algebras (with products denoted by  ). The space (W  ,  ) is also a Banach algebra. Now suppose that A has a bounded approximate identity, and hence a mixed identity, say E. Then, by Theorem 2.3.44(i), W ⊂ F, and so F ⊥ ⊂ W ⊥ as closed ideals in A . It follows that W  is a quotient of the Banach algebra F  and that qW (E) is the identity of W  . As in Proposition 2.3.62, W ⊥ is a weak-∗-closed ideal in (F  ,  ).

There is a construction related to Theorem 2.3.78 under the weaker hypothesis that the Banach algebra have a multiplier-bounded approximate identity rather than a bounded approximate identity; the theorem is essentially [222, Theorem 4.1]. Theorem 2.3.84 Let A be a Banach algebra with a multiplier-bounded approximate identity. Then there is a bounded projection P in M  (A ) such that A = P(A ) ⊕ ker P and

A = P(A )  ker P .

Further, A  ker P = {0}, P(A ) ∼ (A A) and ker P = (A A)⊥ . Proof Take (eν ) to be a MBAI for A, and set Lv a = eν a (a ∈ A) for each ν. Then ⊗ A . By passing to (Lν ) is a bounded net in B(A ), which is the dual space of A a subnet, we may suppose that (Lν ) converges with respect to the weak-∗ topology on B(A ), say wk∗ – limν Lν = P. Take M ∈ A and λ ∈ A . Then Lν (M), λ = M ⊗ λ, Lν  → M ⊗ λ, P = P(M), λ , and so P(M) = lim Lν (M) = lim eν · M in (A , σ(A , A )) . ν

ν

(2.3.17)

Clearly Pa = limν eν a = a (a ∈ A). Now take M, N ∈ A . Then P(M  N) = lim eν · (M  N) = (lim eν · M)  N = P(M)  N, ν

ν

(2.3.18)

and so P ∈ M  (A ) and ker P is a right ideal in A . As a special case of (2.3.18), we see that P(a · M) = a · M (a ∈ A, M ∈ A ) . Take M ∈ A . Then

2.4 Dual Banach algebras

137

P(P(M)) = lim eν · P(M) = lim lim eν eμ · M = lim eν · M = P(M) ν

ν

μ

ν

because limμ

eν eμ − eν

= 0 for each index ν, and so P is a projection in B(A ). Further, (2.3.19) A = P(A ) ⊕ ker P as a direct sum of Banach spaces, where P(A ) = {M ∈ A : lim eν · M = M}

(2.3.20)

ker P = {M ∈ A : lim eν · M = 0} .

(2.3.21)

ν

and

ν

It follows that P(A ) and ker P are right ideals in A . We shall now show that M  N = 0 (M ∈ A , N ∈ ker P), and hence that  A  ker P = {0}, and so ker P is a closed ideal in A . First take a ∈ A and N ∈ A . For each λ ∈ A , we have a · N, λ = limν aeν · N, λ because limν aeν = a in (A, · ), and so a · N, λ = lim eν · N, λ · a = 0 . ν

Hence a · N = 0. Now take M ∈ A , say M = wk∗ – limα aα for a bounded net (aα ) in A. Since RN is weak-∗ continuous, M  N = wk∗ – limα aα · N = 0, as required. We have shown that A = P(A )  ker P. Finally, take M ∈ ker P, a ∈ A, and λ ∈ A . By (2.3.21), limν eν · M, λ · a = 0. But eν · M, λ · a = aeν , M · λ → a, M · λ = M, λ · a , 

and so M, λ · a = 0. Thus ker P ⊂ (A A)⊥ . Now take M ∈ (A A)⊥ . Then eν · M, λ = 0 for all ν and all λ ∈ A , and so, by (2.3.21), M ∈ ker P. It follows that ker P = (AA )⊥ . Since A / ker P ∼ P(A ), we have P(A ) ∼ (A A) . This completes the proof. In the special case in which A has a mixed identity E, we see that the map P defined by P(M) = E  M (M ∈ A ) is a projection in B(A ) that satisfies the specified conditions.

2.4 Dual Banach algebras We shall now introduce a further special class of Banach algebras, that of dual Banach algebras. Preliminary remarks on preduals and concrete preduals of Banach spaces were given in §1.3. Let E be a dual Banach space. Recall from Proposition 1.3.26 that each predual of E is equivalent to a concrete predual and from Proposition 1.3.27 that two concrete

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preduals, G1 and G2 , are equal as subsets of E  if and only if the topologies σ(E, G1 ) and σ(E, G2 ) are equal on E. In the first part of the section, we shall give some general results about dual Banach algebras; in particular, we shall consider when they are Arens regular or ideals in their biduals. In the second part, we shall give some examples; further examples will be given later. We shall note that a C ∗ -algebra is a dual Banach algebra if and only if it is a von Neumann algebra, and that the Banach algebra B(E) is a dual Banach algebra whenever E is a reflexive Banach space; we shall state Daws’ representation theorem for dual Banach algebras. Finally, we shall discuss when a weighted semigroup algebra  1 (S , ω) is a dual Banach algebra; these algebras will be considered further in §6.3. The theory of dual Banach algebras, especially considering their amenability properties, is given by Runde in [282, Chapter 5], following the paper [280], where the term was first used. A seminal early paper is that of Daws [67]. Let A be a Banach algebra, with dual module A , and take a closed linear subspace F of A that is a concrete predual of A, as in Definition 1.3.24, so that the contraction T F : A → F  of Equation (1.3.5) is a linear homeomorphism such that the operator T F | F : F → A is the identity map; in the current notation, we have T F (a)(λ) = λ, T F aF,F  = a, λA,A

(a ∈ A, λ ∈ F) .

(2.4.1)

We see that T F is a module homomorphism when F is a submodule of A . Definition 2.4.1 Let A be a Banach algebra. A Banach-algebra predual for A is a closed linear subspace F of A that is a concrete predual of A and an A-bimodule; such a predual is an isometric predual if the map T F is an isometry. The algebra A is a dual Banach algebra if it has a Banach-algebra predual, and A is an isometric dual Banach algebra if it has an isometric Banach-algebra predual. A Banach-algebra predual for A is unique if it is the only closed A-submodule of A with respect to which A is a dual Banach algebra. Sometimes, when F is an obvious Banach-algebra predual of a Banach algebra A, we shall say just that it is a ‘predual’. Let F be a closed linear subspace of A that is a concrete predual of A, and suppose that F contains a dense linear subspace that is an A-submodule of A . Then F is also an A-submodule of A , and so F is a Banach-algebra predual of A. Example 2.4.2 (i) Clearly every reflexive Banach algebra A is an isometric dual Banach algebra with Banach-algebra predual A . By Proposition 1.3.28, A is the unique such predual of A. (ii) The commutative Banach algebra ( 1 , · ) is obviously an isometric dual Banach algebra, with Banach-algebra predual c 0 , where c 0 is again considered as a closed linear subspace of ( 1 ) =  ∞ = C(β N), and so ( 1 ) = M(β N).

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139

We note that a Banach-algebra predual of a Banach algebra is not necessarily unique. For example, let E be any non-zero Banach space, so that A = E is a Banach algebra for the zero product, as in Example 2.1.13(vii). Then any concrete predual F of E is a closed submodule of E  (because a · λ = 0 (a ∈ A, λ ∈ F)), and so F is a Banach-algebra predual for E, showing that A is a dual Banach algebra. As we remarked on page 47, the Banach space  1 has many isometric concrete preduals no two of which are even mutually isomorphic as Banach spaces. Thus, strictly, we should refer to ‘a pair (A, F)’ when discussing dual Banach algebras. However, when the predual F that we are using is clear from the context, as will almost always be the case in this work, we shall not indicate F in the notation, and just say that ‘A is a dual Banach algebra’. In Theorems 4.1.8 and 4.1.9, we shall state that M(G), the measure algebra of a locally compact group G, is a unital, semisimple, Banach ∗-algebra and a Banach lattice algebra and that M(G) is a dual Banach algebra with isometric predual C 0 (G). That the Fourier–Stieltjes algebra B(Γ) of a locally compact group Γ is a dual Banach algebra will be noted in §4.3. Let A be a Banach algebra, and suppose that F is a closed A-submodule of A such that A  F  as a Banach space. Then it is not necessarily the case that F is a Banach-algebra predual of A. For example, let J be the Banach sequence algebra that is the James algebra, to be described in Example 3.2.10, and take F = J  . Then F is a closed submodule of J  and J  F  , but, as noted on page 47, J  is not a concrete predual of J. In fact, it will follow from Example 3.2.10 that J is not a dual Banach algebra with respect to any predual. In Example 2.4.24, we shall exhibit a compact space K such that C(K) is a dual Banach space, but not a dual Banach algebra. An example of a Fourier algebra that is isometrically the dual of a Banach space, but not a dual Banach algebra, will be mentioned in Example 4.3.36; see also Example 5.5.21. Proposition 2.4.3 Let A be a Banach algebra, and suppose that I is an ideal in A that is a Banach A-bimodule and such that there is an isometric Banach A-bimodule isomorphism T : I → A . Then I is an isometric dual Banach algebra with predual A. Proof As usual, we identify A as a closed linear subspace of A , and so the map T  | A : A → I  identifies T  (A) = F isometrically as a closed linear subspace of I  , and Equation (2.4.1) holds (with T F = T  | A and λ ∈ I). Take a ∈ A and x, y ∈ I. Then y, T  a · xI,I  = xy, T  aI,I  = a, T (xy)A,A = a, x · T yA,A = ax, T yA,A = y, T  (ax)I,I  , and so T  a · x = T  (ax) ∈ T  (A). Thus T  (A) is a left I-module in I  . Similarly, T  (A) is a right I-module in I  , and so T  (A) is a Banach I-bimodule in I  whose dual is I, as required.

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Theorem 2.4.4 Let A be a Banach algebra, and let F be a Banach space that is a concrete predual of A. Then F is a closed A-submodule of A if and only if the product in A is separately σ(A, F)-continuous. Proof Certainly the product in A is separately σ(A, F)-continuous whenever F is a closed A-submodule of A . Now suppose that the product in A is separately σ(A, F)-continuous. Take a ∈ A and λ ∈ F, and suppose that (bα ) is a net in A such that limα bα = b in (A, σ(A, F)). Then limα bα a = ba in σ(A, F). In particular, limα bα a, λA,F = ba, λA,F . This shows that limα bα , a · λA,A = b, a · λA,A , and hence that a · λ is σ(A, F)continuous on A. Thus a · λ ∈ F. Similarly, λ · a ∈ F, and so the linear subspace F is an A-submodule of A . Corollary 2.4.5 Let A be a Banach algebra, and let F be a closed A-submodule of A such that F ⊂ WAP(A). Then the Banach algebra (F  ,  ) is a dual Banach algebra with predual F. Proof The space F is a concrete predual of F  . By Corollary 2.3.59, the space F is introverted in A , and we have noted that the two products,  and  , agree on F  . Take a net (Mα ) in F  such that wk∗ – limα Mα = M in F  , and take N ∈ F  . Then, for each λ ∈ F, we have lim Mα  N, λ = lim Mα , N · λ = M, N · λ = M  N, λ α

α

because N · λ ∈ F, and, similarly, limα N  Mα , λ = N  M, λ, and so the product on (F  ,  ) = (F  ,  ) is separately σ(F  , F)-continuous. By Theorem 2.4.4, F is a closed A-submodule of F  , and so F is the Banach-algebra predual of F  . In particular, the Banach algebra (WAP(A) ,  ) is a dual Banach algebra for each Banach algebra A. See also [282, Proposition 5.1.11]. Corollary 2.4.6 Let A be a Banach algebra that is an ideal in its bidual, and set F = A A. Then (F  ,  ) is a dual Banach algebra with predual F. Proof By Proposition 2.3.37, F ⊂ WAP(A), and so the result follows from Corollary 2.4.5. Proposition 2.4.7 Let A be a dual Banach algebra with a bounded approximate identity. Then A has an identity. Proof Let (A, F) be a dual Banach algebra, and take (eν ) to be a BAI for A. Then we may suppose that there exists e ∈ A such that limν eν = e in (A, σ(A, F)). For each a ∈ A, we have a = limν eν a = ea by Theorem 2.4.4, and also a = ae, so that e is the identity of A.

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141

Corollary 2.4.8 Let (A, F) be a dual Banach algebra, and let B be a σ(A, F)-closed subalgebra of A. Then there is a Banach space H such that (B, H) is a dual Banach algebra. Proof By Corollary 1.3.22, there is a Banach space H such that H  = B and such that the two topologies σ(A, F) and σ(B, H) agree on B; by Proposition 1.3.26, we may suppose that the space H is a concrete predual of B. Thus the product on B is separately σ(B, H)-continuous, and hence, by Theorem 2.4.4, (B, H) is a dual Banach algebra. Corollary 2.4.9 Let A be a Banach algebra. Then (A ,  ) is a dual Banach algebra with Banach-algebra predual A if and only if A is Arens regular. Proof Certainly A is a concrete predual of A . By Theorem 2.4.4, the algebra A has A as a Banach-algebra predual if and only if the product map (M, N) → M  N, A × A → A , is separately weak-∗ continuous. This occurs if and only if L M (and R M ) are weak-∗ continuous on A . By Theorem 2.3.29, this happens if and only if A is Arens regular. Theorem 2.4.10 Let (A, E) be a dual Banach algebra. Then E is an introverted submodule of A , E ⊥ is a closed ideal in A , and A = A  E ⊥ .

(2.4.2)

Further, E ⊂ WAP(A). Proof We write A = A ⊕ E ⊥ as a Banach space, as in Equation (1.3.7). Take N ∈ E ⊥ and λ ∈ E. For each a ∈ A, we have a, N · λ = N, λ · a = 0 because λ · a ∈ E, and so N · λ = 0. Now take M ∈ A , say M = a + N, where a ∈ A and N ∈ E ⊥ . Then M · λ = a · λ + N · λ = a · λ ∈ E. This shows that E is left-introverted, and so, by Theorem 2.3.60, E ⊥ is a closed ideal in A , giving Equation (2.4.2). Similarly, E is right-introverted, and so introverted in A . Take λ ∈ E, and consider the map Rλ : a → a · λ, A → E ⊂ A . For M = b + N ∈ A  E ⊥ , we have b + N, a · λ = ba + N · a, λ = ba, λ + a, λ · N = ba, λ , so that M, a · λ = ba, λ. Since A is a dual algebra, and so the multiplication of A is separately σ(A, E)-continuous, it follows that the map Rλ : (A, σ(A, E)) → (A , σ(A , A )) is continuous. This shows that Rλ (A[1] ) is weakly compact in A . Thus λ ∈ WAP(A), and hence E ⊂ WAP(A). The following corollary follows from Proposition 2.3.63, and uses the notation of that result.

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Corollary 2.4.11 Let (A, E) be a dual Banach algebra, and suppose that I is a closed ideal in A, with identification ι : I → A. Set F = ι (E). Then F is leftintroverted in I  and (F  ,  ) is a closed ideal in (E  ,  ). Corollary 2.4.12 Let (A, E) be a dual Banach algebra, and let F be a closed A-submodule of A that is left-introverted and such that E ⊂ F. Then (F  ,  ) = A  E ⊥ .

(2.4.3)

Proof By Equation (2.4.2), A = A  E ⊥ . By Theorem 2.3.60, F ⊥ is a closed ideal in (A ,  ), and F ⊥ ⊂ E ⊥ , and so A /F ⊥ = A  (E ⊥ /F ⊥ ). But the quotient algebra A /F ⊥ is identified with (F  ,  ), and so Equation (2.4.3) follows, where now E ⊥ denotes the annihilator of E in F  . Corollary 2.4.13 Let (A, E) be a dual Banach algebra, and suppose that I is a closed ideal in A that is complemented in A. Then I  = I  (E ⊥ ∩ I  ). Proof Let P ∈ B(A) be the projection onto I. Then P ∈ B(A ) is a projection with range I  . By Theorem 2.4.10, A = A  E ⊥ . Take a + M ∈ A , where a ∈ A and M ∈ E ⊥ . Then P (a + M) = Pa + P (M) ∈ I ⊕ (E ⊥ ∩ I  ). The result follows. Definition 2.4.14 Let (A, F) be a dual Banach algebra. An A-invariant subspace of F is a closed linear subspace V of F such that a · λ, λ · a ∈ V (a ∈ A, λ ∈ V). In this case, V ⊥ = {a ∈ A : a · λ = λ · a = 0 (λ ∈ V)}. It follows that V ⊥ is a σ(A, F)-closed ideal in A. Indeed, a closed linear subspace V of F is A-invariant if and only if V ⊥ is an ideal in A. Proposition 2.4.15 Let (A, F) be a dual Banach algebra. Suppose that V is an A-invariant subspace of F and that V ⊥ has an identity. Then V is complemented in F. Proof The identity of V ⊥ is e. Define P : λ → λ − e · λ, F → F. Then P is a projection in B(F). Also e · λ = 0 (λ ∈ V), and so P(V) = V. Now take λ ∈ F. Then a, P(λ) = 0 (a ∈ V ⊥ ), and so P(λ) ∈ V ⊥ . By Corollary 1.2.21(i), V ⊥ = V, and so P(F) ⊂ V. It follows that P(F) = V, and so V is complemented in F. The following theorem is stated as [72, Theorem 5.1] and is based on [67, Theorem 4.4]. Theorem 2.4.16 Let A be a Banach algebra that is Arens regular, has a bounded approximate identity, and is an ideal in its bidual. Then A is the unique Banachalgebra predual of A .

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143

Proof By Corollary 2.4.9, A is a Banach-algebra predual of A . The algebra A satisfies the equivalent conditions specified in Corollary 2.3.73, and so A has an identity, say E, where E = limν eν in (A , σ(A , A )) for a bounded net (eν ) in A. Now suppose that X ⊂ A is a Banach-algebra predual of A , and take Λ ∈ X. By Equation (1.3.8), there exist λ ∈ A and ζ ∈ A⊥ such that Λ = λ + ζ (where we regard A as a linear subspace of A ). Take M ∈ A and a ∈ A. Then we have M, a · ζ = M · a, ζ = 0 because M · a ∈ A, and so a · ζ = 0, whence a · Λ = a · λ ∈ A . We may suppose that there exists F ∈ A such that F = limν eν in (A , σ(A , X)). For each a ∈ A, it follows from Theorem 2.4.4 that F · a = limν eν a in (A , σ(A , X)). Take Λ ∈ X. Then limν eν , a · Λ = F, a · Λ. Since a · Λ ∈ A , we also have limν eν , a · Λ = E, a · Λ. This implies that F · a = E · a = a. Similarly, a · F = a · E = a. Since this holds for each a ∈ A, necessarily F = E, and hence we also have limν eν = E in (A , σ(A , X)). As in Theorem 2.4.10, the map RΛ : (A , σ(A , X)) → (A , σ(A , A )) is continuous for Λ = λ + ζ in X, and hence limν eν · Λ = E · Λ = Λ weakly in A . But also limν eν · Λ = limν eν · λ = E · λ = λ weakly in A , and hence weakly in A . We conclude that Λ = λ ∈ A . Hence X ⊂ A , and so X = A by Proposition 1.3.28, showing that A is the unique Banach-algebra predual of A . The following result extends [56, Proposition 1.6]. Theorem 2.4.17 Let (A, F) be a dual Banach algebra. Then A is an ideal in its bidual if and only if AA ⊂ F and A A ⊂ F. In this case, (a + R)  (b + S) = ab

(a, b ∈ A, R, S ∈ F ⊥ ) ,

(2.4.4)

and, further, A is Arens regular, A A ⊂ F and ΦA ⊂ F. Proof Suppose that A is an ideal in A , and take a ∈ A and λ ∈ A . Since A ∼ F  , there is a bounded net (λα ) in F that converges to λ in the weak-∗ topology, σ(A , A). Since the map Ra is weakly compact, we may suppose that the net (a · λα ) is weakly convergent in F; clearly the limit of this net is a · λ, and so a · λ ∈ F. This shows that AA ⊂ F. Similarly, A A ⊂ F. Conversely, suppose that AA ⊂ F and A A ⊂ F, and take a ∈ A. Let (bα ) be a bounded net in A. By passing to a subnet, we may suppose that (bα ) converges in the topology σ(A, F), say to b ∈ A. Take λ ∈ A . Then a · λ ∈ F, and hence bα , a · λ → b, a · λ. It follows that abα → ab weakly in A, and this shows that La is weakly compact. Similarly, Ra is weakly compact, and so A is an ideal in A . Now suppose that A is an ideal in A , and take a ∈ A and S ∈ F ⊥ . Then S · a = a · S ∈ A ∩ F ⊥ = {0}. Next take M ∈ A , say M = limα aα in σ(A , A ), where (aα ) is a net in A. Then M  S = lim aα · S = 0 . α

144

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Thus (2.4.4) follows. Similarly, (a + R)  (b + S) = ab (a, b ∈ A, R, S ∈ F ⊥ ), and so A is Arens regular. It follows from Proposition 2.3.25 that A A ⊂ AA , and so A A ⊂ F. Take ϕ ∈ ΦA . Then there exists a ∈ A with ϕ(a) = 1. But now ϕ = a · ϕ ∈ A · A ⊂ F , and so ΦA ⊂ F. We shall see in Theorem 4.1.37 and Proposition 4.2.5, respectively, that the Banach sequence algebra A(Z) is an ideal in its bidual, but that it is not Arens regular, and so, by the above theorem, A(Z) is not a dual Banach algebra. On the other hand, the measure algebra M(G) of a locally compact group G is a dual Banach algebra that is not Arens regular whenever G is infinite; see Theorem 4.1.15. Corollary 2.4.18 Let A be a dual Banach algebra. Then A is an ideal in its bidual if and only if A  A ⊂ A. Proof Suppose that A is an ideal in A . Then A  A ⊂ A by the theorem. The converse is immediate. Corollary 2.4.19 Let A be a dual Banach algebra that is an ideal in its bidual. Then A is Arens regular and (A ,  ) is a dual Banach algebra that is Arens regular and also an ideal in its bidual. Suppose, further, that A is a compact algebra. Then A is a compact algebra. Proof By Theorem 2.4.17, A is Arens regular, and so, by Corollary 2.4.9, (A ,  ) is a dual Banach algebra. Take M ∈ A , say M = a + R, as above. Then L M = La and R M = Ra , and, by Theorem 2.3.7, La and Ra are weakly compact. Thus both L M and R M are weakly compact operators on A , and so A is an ideal in its bidual. By Theorem 2.4.17, (A ,  ) is Arens regular. The operators L M and R M are compact when La and Ra are compact. Proposition 2.4.20 Let (A, F) and (B, G) be dual Banach algebras. Suppose that A has the approximation property and the Radon–Nikodým property as a Banach q G. space. Then A ⊗ B is a dual Banach algebra, with Banach-algebra predual F ⊗ q G is a concrete predual of A Proof By Theorem 1.4.18(ii), F ⊗ ⊗ B. Thus it is suffiq G is a submodule of (A cient to show that F ⊗ ⊗ B) . ⊗ B) of Equation Take λ ∈ A and μ ∈ B , and consider the element λ ⊗ μ ∈ (A (1.4.1). Now take λ ∈ F and μ ∈ G, and also c ∈ A and d ∈ B. Then we claim that (c ⊗ d) · (λ ⊗ μ) = (c · λ) ⊗ (d · μ). Indeed, for each a ∈ A and b ∈ B, we have a ⊗ b, (c ⊗ d) · (λ ⊗ μ) = ac ⊗ bd, λ ⊗ μ = a ⊗ b, (c · λ) ⊗ (d · μ) ,

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145

q G is a left A and this is sufficient to give the claim. Thus F ⊗ ⊗ B-submodule of q G is a right A q G is a ⊗ B-submodule of (A ⊗ B) , and so F ⊗ (A ⊗ B) . Similarly, F ⊗ closed A ⊗ B-submodule of (A ⊗ B) , as required. Corollary 2.4.21 Let A and B be dual Banach algebras. Suppose that A is separable and has the approximation property. Then A ⊗ B is a dual Banach algebra. Proof By Proposition 1.2.50, A has the RNP, and so this follows from Proposition 2.4.20. We now give some examples of dual Banach algebras. Theorem 2.4.22 A C ∗ -algebra is a dual Banach algebra if and only if it is a von Neumann algebra. In this case, the Banach-algebra predual is unique. Proof Let R be a von Neumann algebra. Then we have noted that R∗ , the space of normal functionals on R, as on page 99, is a concrete isometric predual of R. Clearly R∗ is a submodule of R , and so R∗ is a Banach-algebra predual of R, whence R is an isometric dual Banach algebra. Conversely suppose that A is a C ∗ -algebra that is a dual Banach algebra, say with Banach-algebra predual F, so that A = A  F ⊥ , as in Equation (2.4.2). By Theorem 2.3.49, A is a C ∗ -algebra, and F ⊥ is a closed ideal in A , and so, by Proposition 2.2.3(i), F ⊥ is a ∗-ideal in A and the natural projection P : A → A is a ∗-homomorphism. By Theorem 2.2.5, P = 1, and hence, by Proposition 1.3.31, A has an isometric concrete predual. In particular, A is a von Neumann algebra, and so F is the unique Banach-algebra predual of A. The following result is essentially [300, Theorem III.2.7]. Corollary 2.4.23 Let A be a von Neumann algebra, with predual F, and let V be an A-invariant subspace of F. Then V is complemented in F. Proof The σ(A, F)-closed ideal V ⊥ has an identity, and so this follows from Proposition 2.4.15. Example 2.4.24 There are several examples in [51, §6.9] of compact spaces K such that K is not hyper-Stonean, and hence C(K) is not a dual Banach algebra, but such that C(K) is isomorphic to the dual of another Banach space. For example, let K be the Gleason cover of I as in [51, Example 6.9.10]. Then K is an infinite, separable Stonean space without any isolated points. Here N(K) = {0}, and so C(K) is not hyper-Stonean, and hence C(K) is not a von Neumann algebra. However C(K) ∼  ∞ = C(β N), so that C(K) is even isomorphically the bidual of a Banach space.

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Example 2.4.25 Let E be a reflexive Banach space, and set F = (E ⊗ E  , · π ), the nuclear algebra of E, as in Example 2.1.13(vi). As in Equation (1.4.5), the dual module of F is B(E), and so clearly F is an isometric Banach-algebra predual for B(E). In fact, it is shown by Gardella and Thiel [126, Corollary 6.5], generalizing [67, Theorem 4.4], that F is the unique Banach-algebra predual of B(E). Now suppose that E also has the approximation property, so that F = N(E). Then B(E) is an isometric dual Banach algebra, with Banach-algebra predual N(E). Since E has the Radon–Nikodým property, it follows from Theorem 1.4.18(ii) q E, identified with K(E), and so that N(E) is the dual of the Banach space E  ⊗ K(E) = B(E). By Theorem 2.4.16, N(E) is the unique Banach-algebra predual of B(E). Take x ∈ E, λ ∈ E  and T ∈ K(E). Then we see that (λ ⊗ x) · T = λ ⊗ T x

and

T · (λ ⊗ x) = T  λ ⊗ x .

It follows that, in this case, N(E) is a dual Banach algebra, with Banach-algebra q E. predual E  ⊗

Example 2.4.26 The algebra S1 (H) = K(H) of trace-class operators on a Hilbert space H is an isometric dual Banach algebra.

Let E be a reflexive Banach space with the approximation property. Then it follows from Proposition 2.4.8 that each weak-∗-closed subalgebra of B(E) is a dual Banach algebra. The following impressive theorem of Daws [67, Corollary 3.8] shows that every dual Banach algebra has this form; the theorem is expounded in [282, §5.4]. A further, related theorem is given by Daws in [70]; see also [113, Theorem 4.6]. Theorem 2.4.27 Let (A, F) be a dual Banach algebra. Then there is a reflexive Banach space E and an isometric algebra homomorphism π : A → B(E) such that π : (A, σ(A, F)) → (B(E), σ(B(E), N(E))) is continuous. Let S be a non-empty set, and let ω : S → R+• be a function. The Banach space  1 (S , ω) was defined in Example 1.2.8, and it was noted in that example that  1 (S , ω) =  ∞ (S , 1/ω) and that  1 (S , ω) is the dual of the closed linear subspace c 0 (S , 1/ω) of  ∞ (S , 1/ω), so that c 0 (S , 1/ω) is an isometric concrete predual of the space  1 (S , ω). Now suppose that S is a semigroup and that ω is a weight on S , so that we have the weighted semigroup algebra  1 (S , ω), with convolution product, as in Example 2.1.13(v). Clearly, the dual module action of s ∈ S acting on λ ∈  ∞ (S , 1/ω) is specified by Equation (1.2.10), so that again s · λ = λ ◦ R s and λ · s = λ ◦ L s for s ∈ S . The closed linear subspace c 0 (S , 1/ω) is a submodule of  ∞ (S , 1/ω) if and only if s · λ, λ · s ∈ c 0 (S , 1/ω) whenever s ∈ S and λ ∈ c 0 (S , 1/ω).

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147

Proposition 2.4.28 Let S be a weakly cancellative semigroup, and suppose that ω is a weight on S . Then ( 1 (S , ω),  ) is a dual Banach algebra with isometric Banachalgebra predual c 0 (S , 1/ω). Proof Let λ ∈ c0 (S , 1/ω) and s ∈ S . We claim that s · λ ∈ c0 (S , 1/ω). Indeed, take ε > 0. There is a finite set F ⊂ S such that |λ(t)| /ω(t) < ε/ω(s) (t ∈ S \F). Since S is weakly cancellative, F s−1 is a finite set, and |(s · λ)(r)| 0. Since A is Tauberian, there exists g P J8 pAq with g ´ f  < ε, say L “ supp g. Since A is normal and L is compact, there exists h P A with h | L “ 1L . Then g “ gh P Ar2s , and so dp f, Ar2s q < ε. Thus Ar2s “ A. Definition 3.1.11 Let A be a function algebra on K. A non-empty, closed subspace L of K is a peak set for A if there is a function f P A with f pxq “ 1 px P Lq and  f pyq < 1 py P K \ Lq; in this case, f peaks on L. A point x P K is a peak point if {x} is a peak set and a strong boundary point if, for each U P N x , there exists f P A with f pxq “ | f |K “ 1 and | f |K\U < 1. Suppose that x is a strong boundary point for A, and take U P N x and ε > 0. Then there exists f P A with f pxq “ | f |K “ 1 and | f |K\U < ε: for this, replace the specified function f by a suitably high power. In the case where K is metrizable and A is a Banach function algebra on K, every strong boundary point for A is a peak point. Definition 3.1.12 Let A be a Banach function algebra, and suppose that I is an ideal in A. Then the hull of I is the set  {ZΦA p f q : f P I} . hpIq “ {ϕ P ΦA : ϕp f q “ 0 p f P Iq} “ Hence hpIq is a closed subspace of ΦA . Suppose that I is non-zero and a Banach A-module. Then we can identify ΦI with the set ΦA \ hpIq, so that I is a natural Banach function algebra on ΦA \hpIq. Suppose that I is a closed ideal in A. Then we can identify ΦA/I with hpIq. For a closed subspace S of ΦA , set IpS q “ { f P A : f | S “ 0} (taking Ip∅q “ A), so that IpS q is a closed ideal in A; a subspace S of ΦA is a hull if S “ hpIpS qq. Also set JpS q “ { f P J8 : supp f ∩ S “ ∅} (taking Jp∅q “ J8 ), an ideal in A. The set S is a set of synthesis if JpS q “ IpS q. Thus A is strongly regular if and only if ∅ and each singleton in ΦA is a set of synthesis.

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Let I be a closed ideal in a Banach function algebra A. In general, the commutative Banach algebra A/I is not semisimple. However, suppose that S  ΦA is a hull. Then A/IpS q is a natural Banach function algebra on S , and A/IpS q is identified with the restriction algebra { f | S : f P A}. Also, we can identify ΦIpS q with ΦA \ S , so that IpS q is a natural Banach function algebra on the space ΦA \ S . Let A be a regular Banach function algebra. Then each closed subspace S of ΦA is a hull, and, for each ideal I in A, we have JpS q Ă I Ă IpS q ,

where

S “ hpIq .

(3.1.3)

Clearly each closed ideal in A is also a regular Banach function algebra. Proposition 3.1.13 Let A be a regular Banach function algebra. Then A is Tauberian if and only if hpIq  ∅ for each proper closed ideal I in A. Proof Suppose that A is Tauberian and that I is a proper closed ideal in A. Assume that hpIq “ ∅. Then A “ J8 pAq Ă I by Equation (3.1.3), a contradiction. So hpIq  ∅. Suppose that A is not Tauberian, and set I “ J8 pAq, a proper closed ideal. Take ϕ P ΦA and V P Nϕ with V compact. Then there exists f P A with f pϕq “ 1 and f | pΦA \ Vq “ 0, and so f P I. Thus ϕ  hpIq, and so hpIq “ ∅. Example 3.1.14 Let K be a non-empty, compact space. Then certainly the natural Banach function algebra CpKq is such that JpS q “ IpS q for each closed subspace S of K, and so CpKq is strongly regular. Further, each closed ideal I in CpKq has the form I “ IpS q for some closed subspace S of K. A closed subset J of the Banach lattice CpKq is an order-ideal if and only if J “ IpS q for some closed subspace S of K, and a subset B of CpKq is a band if and only if B “ IpUq for an open subset U of K [305, Proposition 3.3.42]. However the only projection bands in CpKq are of the form IpLq “ JpLq, where L is a clopen subspace of K.

Let A be a Banach function algebra, and let B be a Banach function algebra on ΦA such that A is a closed ideal in B. Then the character space of B has the form Φ B “ ΦA ∪ H “ K ∪ H ,

(3.1.4)

where K “ ΦA and H is the hull of A with respect to B; clearly K ∩ H “ ∅, and so, since H is closed in ΦB , the subspace K is open in ΦB . Further, f | H “ 0 p f P Aq. Suppose that B Ă C 0 pΦA q. We claim that K is also closed in ΦB . Indeed, assume to the contrary that there is a net pxα q in K that converges weak-∗ to ϕ P H. Take f P B with ϕp f q  0. Since B Ă C 0 pKq, it follows that limα f pxα q “ 0. But limα f pxα q “ ϕp f q, and so ϕp f q “ 0, a contradiction, giving the claim. We shall use the following result from [50, Proposition 4.1.7].

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159

Proposition 3.1.15 Let pA  · A q be a Banach function algebra, and let pB,  · B q be a Banach function algebra which is a dense subalgebra of A. Suppose that there is a constant C > 0 with  f gB ď C  f A gB

p f, g P Bq .

Then B is natural on ΦA .

  Proof Take f P B and n P N. Then  f n+1 B ď C  f n A  f B ď C  f nA  f B . Thus, for each ϕ P ΦB , we have 1/pn+1q  n+1 1/pn+1q    n/pn+1q 1/pn+1q ϕp f q “ ϕp f n+1 q ď  f B ď C 1/pn+1q  f A .  f B   Letting n Ñ 8, it follows that ϕp f q ď  f A , and so, since B is dense in A, there is a continuous extension of ϕ to an element of ΦA . Thus ΦB “ ΦA and B is natural. Let K be a non-empty, locally compact space, and suppose that A is a normal Banach function algebra on K. Take a compact subspace L of K and open sets U1 , . . . , Un with L Ă U1 ∪ · · · ∪ Un . By [278, Theorem 2.13], there exist f1 , . . . , fn P A such that f1 pxq + · · · + fn pxq “ 1 px P Lq and such that f j pKq Ă I and supp f j Ă U j for each j P Nn . The family { f1 , . . . , fn } is a partition of unity on K subordinate to {U1 , . . . , Un }. Let K and L be two non-empty, compact spaces such that there is a continuous surjection π : K Ñ L. Of course, as before, π induces an isometric Banach-algebra embedding j : CpLq Ñ CpKq defined by jpgqpxq “ gpπpxqq

pg P CpLq, x P Kq .

(3.1.5)

Let U “ {U1 , . . . , Un } be an open cover of L with the property that no proper subset of U is an open cover of L. Then there is a partition of unity subordinate to U, say {u1 , . . . , un }. Take i P Nn . Given yi P Ui and u1 , . . . , un as specified, with ui pyi q “ 1, choose x1 , . . . , xn P K such that πpxi q “ yi . The operator UU is defined to be n  UU : f Þ→ f pxi qui , CpKq Ñ CpLq . (3.1.6) i“1

Then UU P BpCpKq, CpLqq with UU  “ 1. and ε > 0. For each y P L, there Now take k P N, functions g1 , . . . , gk P CpLq,  is an open neighbourhood Uy of y such that gpzq ´ gpyq < ε pz P Uy q for each g P {g1 , . . . , gk }. Let U “ {U1 , . . . , Un } be a finite subcover of {Uy : y P L}; the point in L corresponding to Ui is yi . We then apply the above construction. For each function g P {g1 , . . . , gk } and each i P Nn , we have     gpπpxi qq ´ gpzq “ gpzq ´ gpyi q < ε pz P Ui q , and, for each z P L, we have

160

3 Banach function algebras n  n     gpzq ´ gpy q u pyq < ε UU p jpgqqpzq ´ gpzq ď ui pyq “ ε , i i i“1

i“1

  and so UU p jpgqq ´ gK < ε. 0 We have established the following theorem, which uses the above notation. For applications of the theorem, see Theorems 3.3.17 and 5.3.40. Theorem 3.1.16 Let K and L be two infinite, compact spaces such that there is a continuous surjection π : K Ñ L. Suppose that {g1 , . . . , gk } is a finite set in CpLq and that ε > 0. Then there exist an open cover U “ {U1, . . . , Un } of L and an  operator UU P BpCpKq, CpLqqr1s of the form (3.1.6) such that pUU ◦ jqpgi q ´ gi K < ε pi P 0 Nk q. Let A be a Banach function algebra with an ideal I, and take ϕ P ΦA ∪ {8}. Then a function f on ΦA belongs locally to I at ϕ if there are U P Nϕ and g P I such that f | pU ∩ ΦA q “ g | pU ∩ ΦA q. The following localization lemma is [50, Proposition 4.1.30]. Proposition 3.1.17 Let A be a regular Banach function algebra, and let I be an ideal in A. Suppose that f P C 0 pΦA q and that f belongs locally to I at ϕ for each ϕ P hpIq ∪ {8}. Then f P I. Proof There are an open cover {U1 , . . . , Un } of hpIq ∪ {8} and g1 , . . . , gn P I such that f | U j “ g j | U j p j P Nn q. Let K be a compact neighbourhood of hpIq ∪ {8} in ΦA ∪ {8} with K Ă U1 ∪ · · · ∪ Un . Since A is normal on ΦA ∪ {8}, there is a partition of unity { f1 , . . . , fn } Ă A on K that is subordinate to {U1 , . . . , Un }. Then   f f j “ g j f j p j P Nn q and p f ´ nj“1 f f j q | K “ 0, and hence f ´ nj“1 g j f j P JphpIqq Ă I. It follows that f P I. Corollary 3.1.18 Let A be a regular Banach function algebra, and take two disjoint, closed subspaces S and T of ΦA that are sets of synthesis for A and such that T is compact. Then S ∪ T is a set of synthesis for A. Proof Set L “ JpS ∪ T q, and take f P IpS ∪ T q. By Proposition 3.1.17, it suffices to show that f belongs locally to L at each ϕ P S ∪ T ∪ {8}. Since T is compact, S ∪ {8} and T are disjoint subsets of ΦA ∪ {8}, and so there are open neighbourhoods U and V of S ∪ {8} and T , respectively, in ΦA ∪ {8} with U ∩ V “ ∅; note that V is compact in ΦA . Since A is normal, there exists g P A with g | U “ 0 and g | V “ 1. There exists p fn q in JpS q with limnÑ8 fn “ f . Then fn ´ fn g P JpS ∪T q for n P N and limnÑ8 p fn ´ fn gq “ f ´ f g, and so f ´ f g P L. But p f ´ f gq | U “ f | U, and so f belongs locally to L at ϕ for each ϕ P S ∪ {8}. Also, there exists pgn q in JpT q with limnÑ8 gn “ f . Then gn g P JpS ∪ T q for n P N and limnÑ8 gn g “ f g, and so f g P L. But f g | V “ f | V, and so f belongs locally to L at each ϕ P T . The result now follows from Proposition 3.1.17.

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161

Example 4.2.18 will exhibit a strongly regular Banach function algebra A on Z8 and closed subsets S and T of Z8 such that S and T are sets of synthesis for A and S ∩ T “ {8}, but such that S ∪ T is not a set of synthesis for A. Corollary 3.1.19 Let A be a strongly regular Banach function algebra on K, and suppose that S is a closed subspace of K that is a set of synthesis for A. Then IpS q is strongly regular on K \ S . Proof By Proposition 3.1.10(i), A is natural and regular on K, and so IpS q is natural and regular on K \ S . Take x P K \ S . Since A is strongly regular, {x} is a set of synthesis for A, and so, by Corollary 3.1.18, S ∪ {x} is a set of synthesis for A. Since J x pIpS qq “ JpS ∪ {x}q, it follows that J x pIpS qq “ M x pIpS qq, and so IpS q is strongly regular at x. Theorem 3.1.20 Let A be a Banach function algebra that is Arens regular and weakly sequentially complete. Suppose that f P A is power-bounded and that f pΦA q ∩ T “ {1}. Then there exists e P IpAq such that {ϕ P ΦA : epϕq “ 1} “ {ϕ P ΦA : f pϕq “ 1} . Proof By Theorem 2.1.17, we see that   lim  f k ´ f k+1  “ 0 . kÑ8

(3.1.7)

Since f is power-bounded, the sequence p f k q is bounded in pA,  · q, and so has weak-∗ accumulation points in A , say E1 and E2 are such accumulation points. It follows from (3.1.7) that f · E1 “ E1 , and hence E1 l E1 “ E1 , so that E1 is an idempotent in A . Further, E2 l E1 “ E1 . Similarly, E1 l E2 “ E2 . Since A is Arens regular, we have E2 l E1 “ E1 l E2 , and so E1 “ E2 . This shows that the sequence p f k q has a unique weak-∗ limit in A , and hence that p f k q is a weakly Cauchy sequence in A. Since A is weakly sequentially complete, the sequence p f k q is weakly convergent in A, say with weak limit e P A. For each ϕ P ΦA , we have epϕq “ limkÑ8 f k pϕq, and so e has the required properties. Let A be a Banach function algebra, and let H be a compact subspace of ΦA . Then we define the closed ideal  AH “ { f P A : supp f Ă H} “ I ΦA \ H . (3.1.8) We note that AH “ {0} whenever int H “ ∅; when A is regular and int H  ∅, the ideal AH is a natural, regular Banach function algebra on int H. Corollary 3.1.21 Let A be a Banach function algebra that is Arens regular. Take ϕ P ΦA , and suppose that there is a compact neighbourhood H of ϕ such that AH

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3 Banach function algebras

is weakly sequentially complete and such that there exists f P Ar1s with f pϕq “ 1,     with f pψq < 1 pψ P ΦA \ {ϕ}q, and with supp f Ă H. Then ϕ is an isolated point of ΦA . Proof By hypothesis, AH is weakly sequentially complete, and AH is Arens regular because A is Arens regular; clearly f P AH is power-bounded and f pΦAH q∩T “ {1}. By Theorem 3.1.20, there is e P IpAH q with {ψ P ΦA : epψq “ 1} “ {ϕ}, and so ϕ is an isolated point of ΦA . Let pA,  · q be a Banach function algebra with multiplier algebra MpAq of A, so that A is an ideal in the algebra MpAq. Take T P MpAq, so that f pϕqpT gqpϕq “ pT f qpϕqgpϕq

p f, g P A, ϕ P ΦA q .

(3.1.9)

For ϕ P ΦA , define ϕpT q “ pT f qpϕq/ f pϕq for each f P A with f pϕq  0, so that ϕpT q is well defined and a linear functional on MpAq. For each S , T P MpAq and ϕ P ΦA , we have pT f qpϕqpS f qpϕq “ ppS T qp f qqpϕq f pϕq p f P Aq, and hence ϕpS T q “ ϕpS qϕpT q. This shows that ϕ P ΦMpAq . Clearly ϕ | A “ ϕ, and so the map ϕ Þ→ ϕ, ΦA Ñ ΦMpAq , is an embedding   onto an open subspace of ΦMpAq . Now set  T pϕq “ ϕpT q pϕ P ΦA q, so that T pϕq ď T op and T P C b pΦA q. Suppose that T | ΦA “ 0, and take f P A. Then pT f qpϕq “ 0 when f pϕq  0, and it follows from (3.1.9) that this also holds when f pϕq “ 0. Thus the map T Þ→ T | ΦA ,

MpAq Ñ C b pΦA q ,

is an algebra monomorphism, and so we can regard MpAq as a subalgebra of C b pΦA q by setting MpAq “ { f P C b pΦA q : f A Ă A} . Hence MpAq is a unital Banach function algebra on ΦA , and clearly | f |ΦA ď  f op ď  f 

p f P Aq .

(3.1.10)

Usually MpAq is not a natural Banach function algebra on ΦA . Proposition 3.1.22 Let A be a Banach function algebra that is an ideal in its bidual, has a bounded approximate identity, and is Arens regular. Then A “ MpAq is a Banach function algebra on ΦA . Proof This follows from Proposition 2.3.74. It is an interesting question to determine when the operator norm is equivalent to the given norm on a Banach function algebra. By Proposition 2.1.30, this is the case when the Banach function algebra has a bounded approximate identity; the following result gives another instance of this equivalence. Theorem 3.1.23 Let pA,  · q be a Banach function algebra such that A has the Schur property as a Banach space and A is faithful. Then  · op and  ·  are equivalent on A.

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163

Proof Assume towards a contradiction that  · op and  ·  are not equivalent on A. Then there is a sequence p fn q in A with  fn  “ 1 pn P Nq and  fn op Ñ 0 as n Ñ 8. Let N be a weak-∗ accumulation point of the sequence p fn q in A , and take a subnet p fν q of p fn q that converges weak-∗ to N. Take M P Ar1s and λ P Ar1s . For each n P N, choose gn P Ar1s with  1   xM, fn · λy ´ xgn , fn · λy  < . n Then     1 1  x fn · M, λy  <  x fn gn , λy  + ď  fn op + Ñ 0 n n

as

n Ñ 8.

Hence N l M “ wk∗ – limν fν · M in A , and so N l M “ 0. Since A is faithful, N “ 0, and hence fn Ñ 0 weakly in A. Since the space A has the Schur property, fn Ñ 0 in pA,  · q, a contradiction because  fn  “ 1 pn P Nq. Hence  · op and  ·  are equivalent. Let A be a Banach function algebra. We shall now introduce two algebras related to the multiplier algebra MpAq; the notation is taken from [224, §§4.3, 4.5], where there is a clear account of these algebras, with several more results. Examples of these algebras will be given in Example 4.2.7. Definition 3.1.24 Let A be a Banach function algebra. Then M 00 pAq “ { f P MpAq : f | pΦMpAq \ ΦA q “ 0} and M 0 pAq “ MpAq ∩ C 0 pΦA q. Clearly M 00 pAq and M 0 pAq are closed ideals in MpAq and A Ă M 00 pAq Ă M 0 pAq Ă MpAq . The hull of M 00 pAq in ΦMpAq is clearly ΦMpAq \ ΦA , and so M 00 pAq is natural on ΦA ; further, we see that M 0 pAq “ M 00 pAq whenever M 0 pAq is natural on ΦA . We shall see in Example 4.2.7 that it can be that A  M 00 pAq and that M 0 pAq is not necessarily natural on ΦA . Proposition 3.1.25 Let A be a Banach function algebra. Then A is regular if and only if M 00 pAq is regular if and only if MpAq is regular on ΦA . Proof Suppose that MpAq is regular on ΦA , and take a closed subspace S of ΦA and ϕ P ΦA \S . Then there exists f P MpAq with f pϕq “ 1 and f | S “ 0. Take g P A with gpϕq “ 1, and set h “ f g, so that h P A. Then hpϕq “ 1 and h | S “ 0. This shows that A is regular. The other implications are immediate.

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3 Banach function algebras

Let A be a regular Banach function algebra. Then the algebra MpAq is not necessarily regular; see Example 4.2.7. We now approach the definition of a Segal algebra S with respect to a Banach function algebra A. There are several somewhat different definitions of such an algebra in the literature; for example, see [50, Definition 4.1.8], [130], [180] and [270]. Let pA,  · A q be a Banach function algebra, and let I be an ideal in A that is a Banach A-module; as before, we may suppose that  f A ď  f I

p f P Iq and

 f gI ď  f A gI

p f P A, g P Iq ,

(3.1.11)

and so pI,  · I q is also a Banach function algebra and  f op,I ď  f A

p f P Iq .

(3.1.12)

For example, a Banach function algebra A is an ideal in MpAq that is a Banach MpAq-module. Proposition 3.1.26 Let A be a Banach function algebra, and let I be an ideal in A that is a Banach A-module. Suppose that A or I has a bounded approximate identity. Then MpAq | ΦI Ă MpIq . Proof Take F P MpAq and f P I. Suppose that A has a BAI. Since AI is dense in I, it follows from Cohen’s factorization theorem, Theorem 2.1.46, that there are g P A and h P I with f “ gh. Suppose that I has a BAI. Then there exist g, h P I with f “ gh. In both cases, F f “ pFgqh P AI Ă I, and so F P MpIq. Thus MpAq | ΦI Ă MpIq. Definition 3.1.27 Let A be a Banach function algebra. Then a weak Segal algebra (with respect to A) is a dense ideal in A that is a Banach A-module. Let I be a weak Segal algebra in A. Then we always suppose that Equations (3.1.11) are satisfied. Proposition 3.1.15 shows that ΦI “ ΦA , and so I is a natural Banach function algebra on ΦA . Take λ P A , and set μ “ λ | I. Then μ P I  and is such that μI  ď λA ; if μ “ 0, then λ “ 0, and so, as before, we may regard A as a linear subspace of I  . By Proposition 2.1.41, I “ A when I has a bounded approximate identity. By Proposition 3.1.26, MpAq Ă MpIq for a weak Segal algebra I with respect to a Banach function algebra A that has a bounded approximate identity. In Example 4.2.16, we shall note that it is not always true that MpAq “ MpIq. Proposition 3.1.28 Let A be a natural Banach function algebra on K, and let I be a weak Segal algebra with respect to A. Suppose that A is regular. Then I is normal and J x pIq “ J x pAq px P K8 q.

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165

Proof We first note that I is regular, and hence normal. Indeed, take a non-empty, closed subset F of K and x P K \ F. Since A is regular, there exists f P A with f pxq “ 1 and f | F “ 0. Take g P I with gpxq “ 1. Then f g P I, p f gqpxq “ 1 and f g | F “ 0, and so I is regular. Now take x P K8 . Certainly J x pIq Ă J x pAq. Suppose that f P J x pAq. Then x  supp f , and so there exists g P J x pIq with g | supp f “ 1. Then f “ f g P J x pIq, and so J x pIq “ J x pAq. Proposition 3.1.29 Let A be a Banach function algebra with a bounded approximate identity of bound m, and let I be a weak Segal algebra with respect to A. Then I has a bounded approximate identity that is a multiplier-bounded approximate identity for I of multiplier-bound m. Proof Clearly AI is dense in I, and so I is essential as a Banach A-module. By Cohen’s factorization theorem, Theorem 2.1.46, the ideal I is neo-unital, so that I “ A · I. Let peα q be a BAI for A with bound m; since I is dense in A, we may suppose that peα q is contained in I. Take f P I. Then there exist g P A and h P I with f “ gh. We have lim sup  f eα ´ f I ď lim sup geα ´ gA hI “ 0 , α

α

and so peα q is an approximate identity for I. By Equation (3.1.12), peα q is an MBAI for I with multiplier-bound m. Theorem 3.1.30 Let A be a Banach function algebra with a bounded approximate identity, and let I be a weak Segal algebra with respect to A. Then II  is a dense linear subspace of A · A “ AA . Suppose, further, that I is Arens regular. Then WAPpAq “ A · A · A. Proof Take f P I and λ P I  , and consider f · λ as a linear functional on A by setting p f · λqpgq “ x f g, λy pg P Aq. Then f · λ P A with  f · λA ď  f I λI  . By Proposition 3.1.29, there is a net peα q that is an MBAI in I and a BAI in A. Then eα f · λ P A · A and eα f · λ ´ f · λA ď eα f ´ f I λI  Ñ 0, and so f · λ P AA “ A · A . Thus II  Ă A · A . Assume towards a contradiction that II  is not dense in AA . Then there exist f P A, λ P A , and M P A such that xM, f · λy “ 1 and M | II  “ 0. For each μ P I  , the linear functional M · μ defined by pM · μqphq “ xM, h · μy ph P Iq is in I  , and so M · μ “ 0. It follows that M · μ “ 0 pμ P A q, and so M · h “ 0 ph P Aq, and then 1 “ xM, f · λy “ limxM, eα f · λy “ limxM · eα , f · λy “ 0 , α

α



the required contradiction. Thus II is dense in A · A .

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Now suppose that I is Arens regular. Take f P I and λ P I  . By Theorem 2.3.29, WAPpIq “ I  , and so the set H :“ {g f · λ : g P Ir1s } is relatively weakly compact in I  . Also H is bounded as a subset of A . Let p fm q and pgn q be sequences in Ar1s and Ir1s , respectively, such that the two iterated limits lim lim x fm , gn f · λyA,A

mÑ8 nÑ8

and

lim lim x fm , gn f · λyA,A

nÑ8 mÑ8

both exist. Since pgn f q is a bounded sequence in I and x fm , gn f · λyA,A “ x fm , gn f · λyI,I 

pm, n P Nq ,

the fact that H is relatively weakly compact in I  implies that the above two iterated limits are equal, and so, by Theorem 2.3.29, f · λ, considered as a functional on A, is weakly almost periodic. This shows that II  Ă WAPpAq. Thus A · A Ă WAPpAq, and so A · A · A Ă WAPpAq. By Theorem 2.3.44(i), WAP Ă A · A · A, and so WAPpAq “ A · A · A, as required. Definition 3.1.31 Let pA,  · A q be a Banach function algebra that has approximate units. A Banach function algebra pS ,  · S q on ΦA is a Segal algebra (with respect to A) if S is a weak Segal algebra with respect to A and if S also has approximate units. A weak Segal algebra S with respect to A is a Segal algebra when S has an approximate identity. Examples of Segal algebras will be noted in Examples 3.2.7, 3.2.8, 3.5.13, etc.; Segal algebras with respect to the group algebra of a locally compact abelian group will be discussed in §4.2. Example 4.2.17 will give an example of a weak Segal algebra with respect to ApΓq that is not a Segal algebra with respect to ApΓq. The proofs of [50, Proposition 4.1.9] and [270, Proposition 6.2.10] establish the following result. Proposition 3.1.32 Let A be a Banach function algebra that has approximate units, and let S be a Segal algebra with respect to A. Then the closed ideal structures of A and S coincide, in the sense that the map I Þ→ I ∩ S is a bijection from the set of closed ideals in A onto the set of closed ideals in S . Corollary 3.1.33 Let A be a Banach function algebra that has approximate units, and let S be a Segal algebra with respect to A. Suppose that A is strongly regular. Then S is strongly regular. Proof Set K “ ΦA , and take x P K8 . Let J be the closure of J x pS q in S . By Proposition 3.1.32, there is a closed ideal I in A with I ∩ S “ J. By Proposition

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167

3.1.28, J x pS q “ J x pAq, and so J x pAq Ă I Ă M x pAq. Thus I “ M x pAq, and so J “ M x pS q. This shows that S is strongly regular at x. It follows that S is strongly regular. Proposition 3.1.34 Let A be a Banach function algebra with approximate units, and let S be a Segal algebra with respect to A. Then S is an ideal in its bidual if and only if A is an ideal in its bidual. Proof This is a special case of Proposition 2.3.18. Let S be a Segal algebra with respect to a Banach function algebra A. We shall see in Example 4.2.16 that the fact that S is Arens regular does not imply that A is Arens regular, but we do not have an example such that A is Arens regular and S is not Arens regular. We shall finish this section by giving four further examples of Banach function algebras. Example 3.1.35 A function f on I “ ra, bs, where a < b, has bounded variation if ⎧ ⎫ ⎪ ⎪ n  ⎪ ⎪   ⎪ ⎨  ⎬  : a “ x < x < · · · < x “ b⎪  varI f :“ sup ⎪ q ´ f px q f px < 8. ⎪ j j´1 0 1 n ⎪ ⎪ ⎪ ⎪ ⎩ j“1 ⎭ The space of continuous functions on I of bounded variation is denoted by BVCpIq, and we set  f var “ | f |I + varI f p f P BVCpIqq . A function f on I is absolutely  continuous if, for each ε > 0, there exists  δ > 0 such that nj“1  f pt j q ´ f ps j q < ε whenever ps1 , t1 q, . . . , psn , tn q are pairwise disjoint, open subintervals of I with nj“1 pt j ´ s j q < δ. The space of absolutely continuous functions on I is denoted by ACpIq. Thus a function f P ACpIq if and only if there exists g P L1 pIq with  t gpsq ds pt P Iq , f ptq “ f paq + a 

and then gptq “ f ptq for almost all t P I. We then set  b  f  psq ds p f P ACpIqq .  f AC “ | f |I + a

It is shown in [50, Theorem 4.4.35] that pACpIq,  · AC q and pBVCpIq,  · var q are natural, unital, regular, self-adjoint Banach function algebras on I, with ACpIq a proper, closed subalgebra of BVCpIq. The space of restrictions to I of polynomials is dense in ACpIq. We claim that the Banach function algebra ACpIq, and hence BVCpIq, is not Arens regular; we take I “ r0, 1s for convenience.

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3 Banach function algebras

Indeed, for m P N with m ě 2, define fm to be such that fm ptq “ 1 pt P r0, 1/msq, such that fm ptq “ 0 pt P r2/m, 1sq, and such that fm is linear on r1/m, 2/ms. Then fm P ACpIq with  fm AC “ 2. Also, for n P N with n ě 3, define gn to be such that gn ptq “ 1 pt P r1/n, 2/nsq, such that gn ptq “ 0 pt P r0, 1/2ns ∪ r3/n, 1sq, and such that gn is linear on r1/2n, 1/ns and on r2/n, 3/ns. Then gn P ACpIq with gn AC “ 3. Clearly fm gn “ 0 when m > 4n and fm gn “ gn when n > 3m. Now fix a subsequence pn j q of N such that n j+1 > 2n j p j P Nq, and consider the linear functional λ on ACpIq defined by λp f q “

8   j“1

1/n j

f  psq ds

p f P ACpIqq .

1/2n j

Since the intervals r0, 1/n j+1 s and r1/2n j , 1s are disjoint for each j P N, it follows that λ P ACpIqr1s . Also limnÑ8 limmÑ8 x fm gn , λy “ 0, whereas lim lim x fm gn j , λy “ lim xgn j , λy “ 1 .

mÑ8 jÑ8

jÑ8

It follows from the iterated-limit condition in Corollary 2.3.31 that ACpIq is not Arens regular, as claimed. For a further result on these algebras, see Example 5.3.4.

Example 3.1.36 Take α such that 0 < α < 1, and consider the Banach function algebras A “ lipα I and Lipα I of Lipschitz functions on the interval I “ r0, 1s, as in [50, §4.4]. Thus Lipα I is the space of functions in CpIq such that pα p f q < 8, where  ⎧  ⎫ ⎪ ⎪ ⎪ ⎪ ⎨  f pxq ´ f pyq ⎬ pα p f q “ sup ⎪ : x, y P I, x  y , ⎪ α ⎪ ⎪ ⎩ |x ´ y| ⎭ and lipα I is the linear subspace of Lipα I consisting of the functions such that    f pxq ´ f pyq Ñ 0 as |x ´ y| Ñ 0 . |x ´ y|α For f P Lipα I, set  f α “ | f |I + pα p f q, so that Lipα I is a natural, unital, regular, selfadjoint Banach function algebra on I, and A “ lipα I is a closed subalgebra of Lipα I that is also natural, unital, regular and self-adjoint; the algebra of restrictions to I of the polynomials is dense in A. The algebras lipα I and Lipα I are called Lipschitz algebras. The Banach function algebra A is Arens regular and A “ Lipα I [50, Theorem 4.4.34]; by Corollary 2.4.9, Lipα I is a dual Banach algebra; by [56, Proposition 3.6], Lipα I is also Arens regular. For further results on this example, see Examples 5.1.23 and 5.3.3.

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Example 3.1.37 Take n P N. A function f on I “ ra, bs is n-times continuously differentiable if f  , . . . , f pnq exist on I (taking one-sided derivatives at a and b) and f pnq is continuous on I. The space of these functions is denoted by C pnq pIq, and we set n  1  pkq   f I p f P C pnq pIqq .  f n “ k! k“0 It is shown in [50, Theorem 4.4.1] that pC pnq pIq,  · n q is a natural, regular, selfadjoint, unital Banach function algebra on I and that the space of restrictions to I of polynomials is dense in C pnq pIq. Clearly C p1q pIq Ă ACpIq. It is noted in [50, Theorem 4.4.8] that each Banach function algebra C pnq pIq is Arens regular. For a result on these algebras, see Example 5.3.5.

Example 3.1.38 Again take I “ ra, bs. Set  {C pnq pIq : n P N} . C p8q pIq “ Now suppose that M “ pMn : n P Z+ q is a sequence in R+• such that M0 “ 1, M1 ě 1, and   n Mn ě pn P Z+ , m P Z+n q , m Mn Mn´m as on page 117. Then ⎧ ⎫ 8 ⎪ ⎪  ⎪ ⎪ 1  pnq  ⎨ ⎬ p8q  f I < 8⎪ DpI; pMn qq “ ⎪ f P C pIq :  f  “ . ⎪ ⎪ ⎩ ⎭ M n n“0 It is shown in [50, Theorem 4.4.12] that DpI; pMn qq is a self-adjoint, unital Banach function algebra on I; by a theorem of O’Farrell [50, Theorem 4.4.15], the space of restrictions to I of polynomials is dense in DpI; pMn qq; by [50, Theorem 4.4.22],  DpI; pMn qq is natural and regular whenever 8 n“0 Mn /Mn+1 < 8. For example, this is the case when Mn “ pn!qα pn P Z+ q, where α > 1. The algebras DpI; pMn qq (on more general sets) have been called the Dales– Davie algebras. Take A “ CpIq, so that A is a Banach algebra that is Arens regular, and let A M be the corresponding algebra of §2.3. By Proposition 2.3.43, A M is Arens regular. Clearly DpI; pMn qq can be regarded as a closed subalgebra of A M , and so DpI; pMn qq is also Arens regular. For a further result on these algebras, see Example 5.3.7.

Further Banach function algebras will be described in §3.2 and in §§4.2 ´ 4.4.

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3.2 Banach sequence algebras We shall now introduce a further important class of Banach function algebras. Banach sequence algebras that are Tauberian are key examples of Banach function algebras that are ideals in their biduals. We shall give some examples of Banach sequence algebras, and determine in some cases whether they are Arens regular; further results will be given in Chapter 6. Let S be a non-empty set, taken with the discrete topology. We recall that we write c 0 pS q and 8 pS q for the Banach spaces (with respect to the uniform norm on S ) of the functions that vanish at infinity and the bounded functions on S , respectively. The algebra of all functions on S of finite support is denoted by c 00 pS q, so that c 00 pS q “ lin {δ s : s P S } . Definition 3.2.1 A Banach sequence algebra on a non-empty set S is a Banach function algebra A on S such that c 00 pS q Ă A Ă 8 pS q . We write A0 for the closure of c 00 pS q in A. A Banach sequence algebra that is a dual Banach algebra is a dual Banach sequence algebra. Recall that, in the case where S “ N, we set Δn “ δ1 + · · · + δn pn P Nq. A natural Banach sequence algebra on a set S is contained in c 0 pS q. Let A be a Banach sequence algebra on S . Then J8 pAq “ c 00 pS q, and so A is Tauberian if and only if A0 “ A, i.e., if and only if it is strongly regular. Hence, by Proposition 3.1.10(i), a Tauberian Banach sequence algebra is natural. Clearly a natural Banach sequence algebra is always regular. Every Tauberian Banach sequence algebra on a countable set is separable. Proposition 3.2.2 Let A be a Banach sequence algebra on a set S , and suppose that A2 Ă A0 . Then A is natural on S . Proof Take ϕ P ΦA , and assume towards a contradiction that ϕ | A0 “ 0. Then ϕpαq2 “ ϕpα2 q “ 0 pα P Aq, and so ϕ “ 0, a contradiction. Thus ϕ | A0 P ΦA0 . Since A0 is natural on S , there exists s P S such that ϕpδ s q “ 1. Now take α P A. Then ϕpαq “ ϕpα · δ s q “ αpsq, and so ϕ “ ε s on A, showing that A is natural on S . It is not true that every Banach sequence algebra A on a non-empty set S such that A Ă c 0 pS q is necessarily natural: for example, see the Rajchman algebra B 0 pZq of Example 4.2.7. Proposition 3.2.3 Let A be a Tauberian Banach sequence algebra. Then A is a compact algebra, and ΦA “ ΦA ∪ pΦA ∩ AK q.

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171

Proof For each f P c 00 pΦA q, the operator L f has finite-dimensional range, and so is compact. Since c 00 pΦA q is dense in A, the operator L f is compact for each f P A, and so A is a compact algebra. By Theorem 2.3.7, A is an ideal in A , and, by Proposition 2.3.17, ΦA “ ΦA ∪ pΦA ∩ AK q. Corollary 3.2.4 Let A be a Tauberian Banach sequence algebra that is a dual Banach algebra. Then A is Arens regular, and A is a compact algebra. Proof This follows from Corollary 2.4.19 and the above proposition. For many Tauberian Banach sequence algebras A, a sequence of the form pΔnk q is a bounded approximate identity for A. However it is not the case that every Tauberian Banach sequence algebra has even an approximate identity: see Example 4.2.18. Note that a sufficient condition for a Tauberian Banach sequence algebra A on N to have a multiplier-bounded approximate identity is that there be a strictly increasing sequence pnk q in N such that limkÑ8 Pnk α “ α pα P Aq. Examples 3.2.8 and 3.2.11 will exhibit natural Banach sequence algebras A on N that are not Tauberian; in Example 3.2.8, A is compact, and so A is an ideal in its bidual, but the algebra of Example 3.2.11 is not an ideal in its bidual. Again, Example 4.2.17 will show that F8 pZq is a natural Banach sequence algebra on Z that is not Tauberian, but is compact. Finally, Example 3.2.9 will mention a natural (non-Tauberian) Banach sequence algebra on N that is even Arens regular, but is not an ideal in its bidual. Proposition 3.2.5 Let A be a natural Banach sequence algebra on N that is not Tauberian and is such that pΔn q is bounded in A. Then A is not an ideal in its bidual. Proof Take α P A\A0 , and assume towards a contradiction that Lα is weakly compact. Then the sequence pPn αq “ pΔn αq has a weakly convergent subsequence. The only possible limit of such a subsequence is α. By the Hahn–Banach theorem, there exists λ P A with λ | A0 “ 0 and xα, λy “ 1. Then xPn α, λy “ 0 pn P Nq, and so xα, λy “ 0, a contradiction. Thus Lα is not weakly compact. By Theorem 2.3.7, A is not an ideal in A . The following result is due to Kamowitz [195]. Theorem 3.2.6 Let A be a Banach function algebra on K. (i) Suppose that f P A such that L f is compact. Then ΦA \ Zp f q is countable, and f pΦA q is a countable subset of C whose only possible limit point is 0. (ii) Suppose that A is a compact algebra. Then ΦA is discrete, and A is a natural Banach sequence algebra on K “ ΦA . Proof (i) For each ϕ P ΦA , we have xg, Lf ϕy “ x f g, ϕy “ f pϕqxg, ϕy pg P Aq .

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3 Banach function algebras

Hence Lf ϕ “ f pϕqϕ, and f pϕq is an eigenvalue of Lf when f pϕq  0, and it follows that f pΦA q Ă σpLf q. Since Lf is compact, σpLf q is a countable set whose only possible limit point is 0, and so ΦA \ Zp f q is countable and f pΦA q has the same property. (ii) Take ϕ P ΦA and choose f P A with f pϕq  0. Since L f is compact, f pϕq is an isolated point of σpLf q. Assume that ϕ P ΦA is not an isolated point in ΦA . Then there is a sequence pϕn q of distinct points in ΦA \ {ϕ} such that    f pϕq ´ f pϕn q < 1/n pn P Nq , and so there exists n0 P N with f pϕq “ f pϕn q pn ě n0 q. Thus the set {ϕn ; n ě n0 } is a linearly independent set in the eigenspace corresponding to the eigenvalue f pϕq, a contradiction. Thus each point of ΦA is isolated, and so ΦA is discrete. By Šilov’s idempotent theorem, Theorem 2.1.18, δϕ P A for each ϕ P ΦA . It follows that c 00 pΦA q Ă A, and so A is a Banach sequence algebra on ΦA . Since K is determining for A, necessarily K “ ΦA , and so A is natural on K. The following are the most elementary Banach sequence algebras. Example 3.2.7 (i) Consider the Banach space pc 0 , | · |N q of null sequences, with pointwise product, so that c 0 is a Tauberian Banach sequence algebra on N and a uniform algebra. As before, c0 “ 1 and c0 “ 8 “ Cp β Nq. The algebra c 0 is Arens regular and an ideal in its bidual. By Theorem 1.3.32, c 0 is not complemented in 8 , and so c 0 is not isomorphically a dual Banach space. The sequence pΔn q of equation (1.1.4) is a contractive approximate identity in c 0 . The space c 0 is not weakly sequentially complete: for example, the sequence pΔn q is weakly Cauchy, but not weakly convergent. (ii) The Banach space 1 , with pointwise product, is a Tauberian Banach sequence algebra on N, and hence, by Proposition 3.2.3, an ideal in its bidual. Clearly 1 is a Segal algebra with respect to c 0 . As in Example 2.4.2(ii), 1 is an isometric dual Banach algebra with Banach-algebra predual c 0 and p 1 q “ 8 , and p 1 q “ Mp β Nq. Here cK0 “ MpN∗ q, and so, as in Equation (2.4.4), the product l in Mp β Nq is given by pα, μq l p β, νq “ pαβ, 0q

pα, β P 1 , μ, ν P MpN∗ qq ,

which shows that 1 is Arens regular. However, the bidual p 1 q is not a Banach function algebra. See also [40, Theorem 1.4.2] and [254, §1.4.8]. Here | · |N “  · op , and so the sequence pΔn q is a multiplier-bounded approximate identity for 1 . More generally, let S be a closed subset of β N, and set F “ {λ P Cp β Nq : λ | S “ 0} ,

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173

so that F is an introverted subspace of p 1 q “ Cp β Nq, and F  “ Mp β N \ S q, regarded as a Banach algebra. (iii) Take p, q with 1 < p < q < 8. Consider the Banach space p p ,  ·  p q; this is a reflexive, Tauberian Banach sequence algebra on N that is Arens regular and a dual Banach algebra. Again, pΔn q is a multiplier-bounded approximate identity for p , and it is clear from Equation (2.1.3) that p is a Segal algebra with respect to q ; it is also a Segal algebra with respect to c 0 .  Set A “ p , so that A “ p and AA “ 1  A. Thus the converse to Corollary 2.3.38 does not hold. Example 3.2.8 Let pω n q be a sequence in r1, 8q such that limnÑ8 ω n “ 8, and set   A “ {α “ pαn q : pω n αn q P c} and αω “ pωn αn qN pα “ pαn q P Aq . Then pA,  · ω q is a Banach sequence algebra on N. Clearly pA,  · ω q – pc, | · |N q as a Banach space, and the corresponding space J8 pAq is A0 “ {α : pω n αn q P c 0 } , a proper linear subspace of A. It follows that A0 is Tauberian, but that A is not Tauberian. The sequence pΔn q is a multiplier-bounded approximate identity for A0 with mutiplier-bound 1, but it is not a bounded approximate identity. Take β P A, and consider the map Lβ : α Þ→ pαn βn q, A Ñ A. For α P Ar1s and n P N, we have   pLβ ´ Pn Lβ qαω “ sup{ω k |αk βk | : k ě n} ď  βω sup{1/ω k : k ě n} Ñ 0 as n Ñ 8 and so Lβ “ limnÑ8 Pn Lβ in BpAq. Since each operator Pn Lβ has finite rank, the operator Lβ is compact, and so, by Theorem 3.2.6(ii), A is a natural Banach sequence algebra on N that is a compact algebra. By Theorem 2.3.7, A is an ideal in its bidual; by Theorem 2.3.49, A and A0 are Arens regular. The algebra A0 is a Segal algebra with respect to c 0 . For more on this example, see Example 5.3.1(iii).

Example 3.2.9 Let A be the remarkable example [21] of Blecher and Read. Then A is a natural Banach sequence algebra on N with the following properties: (i) A is self-adjoint, and so is dense in c0 ; (ii) A has a contractive approximate identity, and so Ar2s “ A; (iii) there exists g P A such that the singly generated subalgebra lin {gn : n P N} is dense in A, and so A is separable;

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3 Banach function algebras

(iv) the operator Lg on A is not weakly compact, and so A is not an ideal in its bidual; (v) A is not Tauberian, and A/A0 is an infinite-dimensional space; (vi) the closed ideal A0 also has a contractive approximate identity; (vii) each maximal modular ideal in A has a bounded approximate identity (but there is no upper bound to the bounds of these bounded approximate identities, and so A is not contractive); (viii) as proved in [21, Theorem 1.1], A is Banach-algebra isomorphic to a closed subalgebra of BpHq for a Hilbert space H, and hence A is Arens regular; (ix) since A is not unital, it follows from Proposition 2.4.7 that A is not a dual Banach algebra.

We do not have an easier example than the above of an Arens regular, natural Banach sequence algebra that is not an ideal in its bidual. Details of the following example are given in [2, §3.4], [50, Example 4.1.45] and [238, §4.5]; it is proved in [6] that it is a Banach sequence algebra. Example 3.2.10 Denote by P the set of elements p “ pp1 , . . . , pk q P Nk , where k ě 2, such that p1 < p2 < · · · < pk . For α “ pαn q P C N , define Npα, pq for p P P by k´1   2  2 α 2 2Npα, pq “ p j+1 ´ α p j  + α pk ´ α p1  , j“1

and set Npαq “ sup pPP Npα, pq (so that Npαq P r0, 8s). Then define J “ {α P c 0 : Npαq < 8}

and

J  “ {α P 8 : Npαq < 8} ,

so that J  “ J ‘ C1 N [238, Theorem 4.5.7]. Then pJ, Nq is a Banach space with c 00 Ă J; it is called the James space. Indeed, c 00 is dense in J, and so J is separable. It is shown in [2, Theorem 3.4.6] and in [238, §4.5] that J – J  as Banach spaces. Let α, β P J. Then it is easily checked that αβ P J and that Npαβq ď 2NpαqNp βq. Further, J  is an algebra for the pointwise product, and J is a maximal ideal in J  . For α P J  , define α J “ sup{Npαβq : β P J, Np βq ď 1} . Then the algebra pJ  ,  ·  J q is a self-adjoint, unital Banach sequence algebra on N; the algebra pJ,  ·  J q is a natural, Tauberian Banach sequence algebra on N, called the James algebra. The sequence pΔn q is a contractive approximate identity for J; for each α P C N , the sequence pΔn αq is increasing, and it converges to α J when α P J  . It follows from Corollary 2.4.7 that J is not a dual Banach algebra.

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175

As in [50, Example 4.1.45], we identify pJ  , l q and pJ  , q isometrically with pJ , · q and with pMpJq, · q, and so J is a maximal ideal in J  and J is Arens regular; cf. Corollary 2.3.73. For further discussion of this example, see Example 5.5.15. 

Example 3.2.11 Let A be the example of Feinstein described in [50, Example 4.1.46]. Indeed, for α “ pαk q P C N , set 1 k |αk+1 ´ αk | n k“1 n

pn pαq “

pn P Nq ,

ppαq “ sup{pn pαq : n P N} ,

and define A to be {α P c 0 : ppαq < 8}, so that A is easily checked to be a selfadjoint Banach sequence algebra on N for the norm given by α “ |α|N + ppαq Clearly

pα P Aq .

  A0 “ J8 pAq “ α P A : lim pn pαq “ 0 , nÑ8

a closed, separable ideal in A. Now take α, β P A and ε > 0. There exists n1 P N with |αk | ∨ | βk | < ε pk ě n1 q, 1 k |αk+1 βk+1 ´ αk βk |. For n ě n2 , we have and there exists n2 ě n1 with n2 ε > nk“1 pn pαβq ď εp1 + pn pαq + pn p βqq ď εp1 + ppαq + pp βqq , and so limnÑ8 pn pαβq “ 0, showing that A2 Ă A0 . By Proposition 3.2.2, A is a natural Banach sequence algebra on N. Note that Δn  “ 2 pn P Nq for this algebra, and so pΔn q is a bounded approximate identity for A0 . It follows that A2 “ A20 “ A0 Ă A .  For each S Ă N, set NpS q “ nPS {k P N : 2n ď k < 2n+1 }, and set  pS q p´1qk χNpS q pkq pk P Nq , αpS q “ αk . k   pS q pS q  Then αpS q P c 0 and supk k αk+1 ´ αk  ď 2, and so αpS q P Ar3s . Now let S and T be distinct subsets of N, and set β “ αpS q ´ αpT q . Choose m P S ΔT . For each k P {2m , . . . , 2m+1 ´ 2}, we have k | βk+1 ´ βk | ě 1, and so pS q

αk

“

 β ě p2m+1 p βq ě

1 2m+1

2m+1 ´2 k“2m



 2m ´ 1 1 1“ ě . 4 2m+1

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3 Banach function algebras

Since PpNq is uncountable, A is not separable. In particular, A2 “ A0  A, A is not Tauberian and A does not have an approximate identity. Note that A is an example of a unital, natural Banach function algebra defined on the compact, metrizable space N8 that is not separable. It follows from Proposition 3.2.5 that A is not an ideal in its bidual. We now claim that the Banach sequence algebra A0 contains a closed subalgebra that is Banach-algebra isomorphic to c 0 . Indeed, let pn j q be a sequence in N such that n j+1 ě 2n j p j P Nq, and consider the map θ : α “ pα j q Þ→

8 

α j δn j ,

c0 Ñ c0 .

j“1

  Clearly θ is an algebra monomorphism and θpαqN “ |α|N pα P c 0 q. Take α P c 0 . For n P N with nk ď n < nk+1 , where k P N, we have pn pθpαqq ď

k k´1 k     nj 1 1  2n j α j  ď 2 |α|N ď 2 |α|N ď 4 |α|N . nk j“1 n 2i j“1 k i“0

  Thus |α|N ď θpαq ď 5 |α|N , and so θ : c 0 Ñ A is a homeomorphic   embedding. Take α “  pα j q P c 0. For ε > 0, there exists k P N with α j  < ε p j ě kq. Set  K “ kj“1 j α j+1 ´ α j . Then, for each n ě nk , we have pn pθpαqq ď K/n + 5ε, and so limnÑ8 pn pθpαqq “ 0, showing that the range of θ is contained in A0 . Thus θ : c 0 Ñ A0 is the required isomorphic Banach-algebra embedding. Since c 0 is a linear subspace that is closed in A0 , it follows from Theorem 1.3.30 that c 0 is complemented in A0 . It also follows that A and A0 are not weakly sequentially complete. It is shown in [30] that A0 is Banach-algebra isomorphic to a closed subalgebra of an Arens regular Banach algebra B, and hence is itself Arens regular. The following simplified presentation is based on a communication from Choi. Given an interval L “ {m, m + 1, . . . , n} ⊆ N, we define varL α “

n´1   α ´ α  j j+1

pα P CN q.

j“m

For each k P N, set Lk “ {2k´1 , 2k´1 + 1, . . . , 2k } and set Bk “ pCLk ,  · var q, where  βvar “ | β|Lk + varLk β p β P CLk q . Thus each Bk is a finite-dimensional analogue of Example 3.1.35. Set B “ c 0 pBk q; this is Arens regular, by Proposition 2.3.40.  Consider the map θ : α Þ→ pα | Lk q, CN Ñ k CLk . We note that p2k ´1 pαq ě

k 2 ´1

j“2k´1

 1 j    ě varLk α pk P Nq , ´ α α j j+1 2 2k ´ 1

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177

  and hence α | Lk var ď |α|N + 2p2k ´1 pαq ď 2 α. Thus θ is a Banach-algebra homomorphism from A0 to B. We now show that θ : A0 Ñ B is bounded below. As a step towards this, we claim that pn pαq ď

  1 ppαq + 2 sup varLk α : k P N 2

pn P Nq .

For n “ 1 the claim holds trivially, since p1 pαq “ |α1 ´ α2 | “ varL1 α . For n ě 2, set r to be the unique integer such that 2r ď n < 2r+1 , and set m to be the unique integer such that 2m ď n < 2m + 1. Then r P N, 1 ď m ď n/2 and {m + 1, m + 2, . . . , n + 1} ⊆ Lr ∪ Lr+1 . Hence pn pαq “

n   j  m α j ´ α j+1  pm pαq + n n j“m+1

1 pm pαq + varLr α + varLr+1 α 2   1 ď ppαq + 2 sup varLk α : k P N . 2 ď

Thus the claim holds for all n P N. Now let α P A0 . It follows from the claim that     ppαq “ sup 2pn pαq ´ ppαq : n P N ď 4 sup varLk α : k P N .    Also, since {Lk : k P N} “ N, we see that |α|N “ sup |α|Lk : k P N . Hence     α ď 4 sup |α|Lk + varLk α : k P N “ 4 θpαqB , as required. This completes the proof that A0 is Arens regular. For further discussion of this example, see Examples 5.1.18 and 5.5.20.

The following algebra was introduced by Feinstein in [102]; some results are based on the memoir of Dales and Loy [60, §3]. Example 3.2.12 Let ω : N Ñ r1, 8q be a sequence, and set ⎧ ⎫ 8 ⎪ ⎪  ⎪ ⎪ ⎨ ⎬ 8 : ω ´ α < 8 Bω “ ⎪ α P , |α | ⎪ i i+1 i ⎪ ⎪ ⎩ ⎭ i“1

with αω “ |α|N +

8  i“1

ωi |αi+1 ´ αi |

pα P Bω q .

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3 Banach function algebras

It is again easily checked that pBω ,  · ω q is a self-adjoint Banach sequence algebra on N. In the case where ωn “ 1 pn P Nq, the elements of Bω are the sequences of bounded variation, and pBω ,  · ω q is denoted by pbv,  · bv q. For each α “ pαn q P Bω , the sequence pαn q is Cauchy, and hence convergent, say limnÑ8 αn “ α8 , and so we can regard Bω as a Banach function algebra on N8 . Take α P Bω with αn  0 pn P N8 q. Then 1/α P Bω , and so Bω is natural on N8 . Set Mω “ Bω ∩ c 0 “ {α P Bω : α8 “ 0}, a maximal ideal in Bω , so that Mω is a natural Banach sequence algebra on N. In particular, bv 0 :“ bv ∩ c 0 is a natural Banach sequence algebra on N. Clearly Mω Ă bv 0 . Consider the weighted semigroup algebra Aω “ 1 pN∧ , ωq of Example 2.1.25, so that Aω is a semisimple, commutative Banach algebra. Take β “ p βi q P Aω , and set 8  βi pn P Nq , α “ pαn q . αn “ i“n

The characters on Aω have the form β Þ→ αn for n P N, and so the character space of Aω is N; the Gel’fand transform of β is just pαn q. A sequence α “ pαn q that is a Gel’fand transform is the transform of the element β “ p βi q, where βi “ αi ´ αi+1

pi P Nq ,

and so the algebra of Gel’fand transforms of the weighted semigroup algebra Aω is exactly the Banach sequence algebra Mω (with an equivalent norm). In particular, we identify the semigroup algebra 1 pN∧ q with bv0 . As before, Mω has the Schur property, and so Mω is weakly sequentially complete. We see that, for n P N, the transform of an element δn P Aω is just Δn P Mω . As in Example 2.1.25, there is always a sequence of the form pδnk q that is an approximate identity for Aω , and then pΔnk q is an approximate identity for Mω , so that Mω is Tauberian. It follows from the uniform boundedness theorem, Theorem 1.3.7, that pΔnk q is always a multiplier-bounded approximate identity for Mω . Note that Δn ω “ 1 + ωn pn P Nq, and so the norms  · ω and  · op are not equivalent on Aω whenever ω is unbounded. Suppose that limnÑ8 ωn “ 8; choose n P N such that ωi ě M pi ě nq. We claim that Mω does not factor weakly [57, Proposition 3.3.2]. Since Mω is separable, Theorem 2.1.10 applies. Assume that Mω “ Mω2 . Then there exist m P N and M > 2  such that there exist α1 , . . . , αm , β1 , . . . , βm P Mω with δn /ωn “ m i“1 αi βi and m ď M. Then β α    i ω i ω i“1 ⎛ ⎞ m ⎜ 8  8         ⎟⎟⎟ ⎜⎜⎜   ⎜⎜⎝αi pnq  βi p jq +  βi pnq αi p jq⎟⎟⎟⎠ , 1ď i“1

and so

j“n

j“n

3.2 Banach sequence algebras

179

⎛ ⎞ m ⎜ 8  8          ⎟⎟⎟ ⎜⎜⎜   ⎜⎜⎝αi pnq ωn  βi p jq ω j +  βi pnq ωn αi p jq ω j ⎟⎟⎟⎠ M ď 2

i“1 m 

ď2

j“n

j“n

αi ω  βi ω ď 2M ,

i“1

a contradiction. Thus the claim follows. In particular, Mω does not have a bounded approximate identity. We conclude that the Banach sequence algebra Mω has a bounded approximate identity if and only if lim inf nÑ8 ωn < 8. We now consider the multiplier algebra MpMω q of Mω ; for this, we suppose that ω “ pωn q is an increasing sequence, so that ωn |αn | ď

8 

ωi |αi+1 ´ αi | ď αω

pn P Nq

i“n

for each α P Mω . Indeed, take β “ p βn q P bv, say  βbv ď 1. For each α P Mω with αω ď 1, we have αβ P c 0 and αβω ď |αβ|N +

8 

ωi | βi+1 | |αi+1 ´ αi | +

i“1

8 

ωi |αi | | βi+1 ´ βi |

i“1

⎛ ⎞ 8  ⎜⎜⎜ ⎟⎟ ď αω ⎜⎜⎝| β|N + | βi+1 ´ βi |⎟⎟⎟⎠ ď 1 , i“1

and so β P MpMω q, with  βω,op ď 1. Thus bv Ă MpMω q. Thus, in the case where the sequence ω is increasing and limnÑ8 ωn “ 8, it follows that Bω  MpMω q. Consider the case where lim inf nÑ8  ωpnq  < 8, say ωpnk q ď C for a strictly increasing sequence pnk q in N, so that Δnk  ď 1 + C pk P Nq, and take an element β “ p βn q P MpMω q. Then max{| βi | : i P Nnk } +

n k ´1

  | βi+i ´ βi | +  βnk  ď p1 + Cq  βω, op

pk P Nq .

i“1

Hence  βbv ď p1 + Cq  βω, op , and so MpMω q Ă bv. Thus MpMω q “ bv. In particular, Mpbv0 q “ bv. In this case, as in Example 3.1.7, Aω “ Mω is not Arens regular; this also follows from Theorem 2.3.47(ii). Thus these algebras are our first examples of natural, Tauberian Banach sequence algebra that are not Arens regular; we shall see in Example 6.1.24 that Mω is strongly Arens irregular. Always, Mω is a dual Banach space. As in Example 2.4.31, Mω is a dual Banach sequence algebra if and only if limnÑ8 ωpnq “ 8; in this case, Aω “ Mω is Arens regular by Corollary 2.4.19. For further discussion of these Banach sequence algebras, see Examples 5.5.21, 6.1.24 and 6.3.15(ii).

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3 Banach function algebras

Example 3.2.13 Consider the semigroup algebra A “ p 1 pZ∨ q, q and the maximal modular ideal M in A of Example 2.1.26. Set S “ Z∨ . Then it is clear that ΦS “ Z8 “ Z ∪ {8} , where the topology on ΦS has as a basis the singletons {m} for m P Z, together with the intervals pm, 8s “ {n P Z : n > m} ∪ {8} for m P Z. As in Example 3.1.7, A is  natural on Z8 . For f “ nPZ αn δn P A, we have fppmq “

m 

αn

pm P Zq ,

fpp8q “

n“´8



αn ,

nPZ

and hence M is identified as   M “ fp P ApΦS q : fpp8q “ 0 , so that M is a Banach sequence algebra on Z. For n P Z+ , the element δ´n ´ δn+1 of M has as Fourier transform the characteristic function of {´n, . . . , n} in c 00 pZq. Since pδ´n ´ δn+1 q is a bounded approximate identity for M, the algebra M is a Tauberian Banach sequence algebra on Z, identified with ⎧ ⎫ ⎪ ⎪   ⎪ ⎪ ⎨ ⎬    M“⎪ pZq : βpnq ´ βpn ´ 1q β P c < 8 . ⎪ 0 ⎪ ⎪ ⎩ ⎭ nPZ

Thus M is natural on Z and an ideal in its bidual. Now define ⎫ ⎧ ⎪ ⎪   ⎪ ⎪ ⎬ ⎨  8  βpnq ´ βpn ´ 1q < 8⎪ , B“⎪ β P pZq : ⎪ ⎪ ⎭ ⎩ nPZ

  and define  β “ | β|Z + nPZ  βpnq ´ βpn ´ 1q p β P Bq, so that B is a Banach sequence algebra on Z, and we can regard pM,  · q and pA,  · q as closed ideals in B. For each β P B, the two limits β8 “ limnÑ8 βn and β´8 “ limnÑ´8 βn both exist, and so the character space of B is {´8} ∪ Z ∪ {8}, and M has codimension 2 in B. As in Example 3.1.7, A is not Arens regular, and so, by Proposition 2.3.23, B and M are not Arens regular. In Example 6.1.24, it will be shown that these algebras are strongly Arens irregular. 

3.3 Projective tensor products of Banach function algebras

181

3.3 Projective tensor products of Banach function algebras We now consider tensor products of Banach function algebras; in particular, we shall consider the projective tensor products of Banach sequence algebras. We shall introduce the Varopoulos algebra, VpK, Lq, and determine some of its properties. We shall also discuss when projective tensor products of Banach sequence algebras are ideals in their biduals. The first result is [50, Proposition 2.3.7]. p B is a comProposition 3.3.1 Let A and B be Banach function algebras. Then A ⊗ mutative Banach algebra and ΦA ⊗p B “ ΦA × ΦB . p B; set Γ : pϕ, ψq Þ→ ϕ ⊗ ψ, ΦA × ΦB Ñ ΦA . Proof Set A “ A ⊗ Let ϕ P ΦA and ψ P ΦB . Then pϕ ⊗ ψqpa ⊗ bq “ ϕpaqψpbq pa P A, b P Bq and ϕ ⊗ ψ “ ϕ ψ. Clearly ϕ ⊗ ψ P ΦA , and the map Γ is continuous. Now take χ P ΦA , and choose a0 ⊗ b0 P A with χpa0 ⊗ b0 q “ 1. Define ϕpaq “ χpaa0 ⊗ b0 q pa P Aq ,

ψpbq “ χpa0 ⊗ bb0 q pb P Bq ,

so that ϕ and ψ are easily checked to be homomorphisms on A and B to C, respectively. Also pϕ ⊗ ψqpa ⊗ bq “ ϕpaqψpbq “ χpaa0 ⊗ b0 qχpa0 ⊗ bb0 q “ χpa ⊗ bqpχpa0 ⊗ b0 qq2 “ χpa ⊗ bq pa P A, b P Bq , and so ϕ ⊗ ψ “ χ. Since pϕ ⊗ ψqpa0 ⊗ b0 q “ 1, we have ϕ  0, and so ϕ P ΦA . Similarly ψ P ΦB . The map χ Þ→ pϕ, ψq, ΦA Ñ ΦA × ΦB , is continuous, and it is the inverse to Γ. Hence Γ : pϕ, ψq Þ→ ϕ ⊗ ψ is a homeomorphism. p B is semiAn example of Milne shows that it is not always the case that A ⊗ simple, even for uniform algebras A and B [241], but the following result is given in [304, Theorem 4]. Proposition 3.3.2 Let A and B be Banach function algebras, and suppose that the p B into A ⊗ q B is an injection. Then A ⊗ p B is a natural natural embedding of A ⊗ Banach function algebra on ΦA × ΦB . p B such that GpFq “ 0, so that, by Proposition 3.3.1, Proof Take an element F P A ⊗  pϕ ⊗ ψqpFq “ 0 pϕ P ΦA , ψ P ΦB q, say F “ 8 i“1 fi ⊗ gi , where fi P A and gi P B 8 for each i P N and i“1  fi  gi  < 8. Then ⎛8 ⎞ ⎜⎜⎜ ⎟⎟ ⎜ ψ ⎜⎝ ϕp fi qgi ⎟⎟⎟⎠ “ 0 pϕ P ΦA , ψ P ΦB q . i“1

Since B is semisimple,

8

i“1

ϕp fi qgi “ 0 pϕ P ΦA q, and so

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⎞ ⎛8 ⎟⎟ ⎜⎜⎜ ⎜ ϕ ⎜⎝ xgi , μy fi ⎟⎟⎟⎠ “ 0 pϕ P ΦA , μ P B q . i“1

Since A is semisimple,

8

8 

i“1 xgi ,

μy fi “ 0 pμ P B q, and so

xgi , μy x fi , λy “ 0 pλ P A , μ P B q .

i“1

q B, and hence, by hypothesis, F “ 0 in This shows that F “ 0 as an element of A ⊗ p B. Thus A ⊗ p B is semisimple. A⊗ p B is a natural Banach function algebra on ΦA × ΦB whenever In particular, A ⊗ either A or B has the approximation property. In this situation, we shall identify f ⊗g p B with the function in C 0 pΦA × ΦB q such that in A ⊗ p f ⊗ gqpϕ, ψq “ f pϕqgpψq

pϕ P ΦA , ψ P ΦB q .

The following result is clear. p B is a Proposition 3.3.3 Let A and B be Banach function algebras such that A ⊗ Banach function algebra. Suppose that ϕ P ΦA and ψ P ΦB are strong boundary p B. points for A and B, respectively. Then pϕ, ψq is a strong boundary point for A ⊗ We now consider projective tensor products of Banach sequence algebras. Proposition 3.3.4 Let A and B be natural Banach sequence algebras on S and T , respectively, and suppose that either A or B has the approximation property. Then p B is a natural Banach sequence algebra on S × T , and A ⊗ p B is Tauberian A⊗ whenever both A and B are Tauberian. Suppose, further, that S and T are countable and that A and B are Tauberian p B is a dual Banach algebra and is Arens and are dual Banach algebras. Then A ⊗ regular. p B is a natural Banach sequence algebra on S × T , Proof By Proposition 3.3.2, A ⊗ p B is Tauberian whenever both A and B are Tauberian. and clearly A ⊗ Now suppose that S and T are countable and that A and B are Tauberian and are p B is an ideal in its bidual and A and B are separable, dual Banach algebras. Since A ⊗ p B is a dual Banach algebra. By Corollary it follows from Corollary 2.4.21 that A ⊗ p 2.4.19, A ⊗ B is Arens regular. Proposition 3.3.5 Let A be a Banach function algebra that is an ideal in its bidual, and let B be a Tauberian Banach sequence algebra. Suppose that A or B has p B is a natural Banach function algebra on the approximation property. Then A ⊗ p B is an ideal in its bidual. ΦA × ΦB such that A ⊗

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p B is a natural Banach function algebra on ΦA × ΦB . Proof By Proposition 3.3.2, A ⊗ p B is an ideal in its bidual By Proposition 3.2.3, B is a compact algebra, and so A ⊗ by Proposition 2.3.12. p B is an ideal in its bidual It is not true that the Banach function algebra A ⊗ whenever this is true for both A and B. For example, let A and B be Banach function algebras on compact spaces such that A and B are reflexive Banach function algep B is not necessarily reflexive. Indeed, bras. Then the Banach function algebra A ⊗ p B is not necessarily even an ideal in its bidual; see Example 4.2.23. A⊗ p q , so Example 3.3.6 (i) Take p and q such that 1 < p, q < 8, and set A “ p ⊗  p q that A “ Bp , q. By Proposition 1.3.66, both p and q have the approximation property, and so, by Proposition 3.3.4, A is a natural, Tauberian Banach sequence algebra on N × N that is a dual Banach algebra. In fact, by Theorem 1.4.18(ii), the Banach-algebra  q q ,  · ε q “ Kp p , q q. It follows from Theorem 2.4.17 predual of A is F :“ p p ⊗ that A is Arens regular. By Theorem 1.4.18(iv), the Banach space A is reflexive if and only if F is   reflexive if and only if Kp p , q q “ Bp p , q q. By Pitt’s theorem [283, Corol lary 4.24(a)], this holds if and only if p > q , i.e., if and only if pq > p + q. For example, A is not reflexive when p “ q “ 2, as in Example 1.4.4. p 8 “ p ⊗ p Cp β Nq. Also set (ii) Take p with 1 < p < 8, and set A “ p ⊗ qq 1  F “ ⊗ , where q “ p . Then it again follows from Proposition 2.4.20 that A is a dual Banach function algebra with Banach-algebra predual F. We shall see in Example 6.2.8(ii) that A is Arens regular. p c 0 . Again by Proposition 3.3.2, A is a natural Banach sequence (iii) Let A “ 1 ⊗ algebra on the set N × N; clearly A is Tauberian, and hence A is an ideal in its bidual. By Corollary 2.3.41, A is Arens regular. The sequence pΔn ⊗ Δn q is a multiplier-bounded approximate identity for each of the above three examples.

We shall now define the Varopoulos algebra VpK, Lq associated with two locally compact spaces K and L. This Banach function algebra plays a very significant role in the theory of harmonic analysis. A seminal paper on the algebra is [318]; see also [145, Chapter 11]. Definition 3.3.7 Let K and L be non-empty, locally compact spaces, and set p C 0 pLq , VpK, Lq “ C 0 pKq ⊗ the projective tensor product of C 0 pKq and C 0 pLq; this algebra is called the Varopoulos algebra. We set VpKq “ VpK, Kq.

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By Proposition 3.3.1, the character space of VpK, Lq is K × L. By Proposition 1.3.66, the spaces C 0 pKq have the bounded approximation property, and so it follows from Proposition 3.3.2 that VpK, Lq is a natural Banach function algebra on K × L. Clearly the algebra VpK, Lq is self-adjoint, and so VpK, Lq is dense in C 0 pK × Lq. Our first aim is to show that VpK, Lq  CpK × Lq whenever K and L are both infinite compact spaces. We require some preliminary lemmas; in these lemmas, we suppose that n P N with n ě 2, that K and L are compact spaces, and that {x1 , . . . , xn } and {y1 , . . . , yn } are sets of distinct points in K and L, respectively. We set ζ “ e2πi/n , so that ζ n “ 1 and 1 + ζ + · · · + ζ n´1 “ 0. Lemma 3.3.8 Set Λp f, gq “

n n 1  f px j qgpyk qζ mpk´ jq ηm n2 m“1 j,k“1

p f P CpKq, g P CpLqq ,

where |ηm | “ 1 pm P Nn q. Then Λ is a bilinear functional on CpKq × CpLq with Λ “ 1. Proof Clearly Λ is a bilinear functional on CpKq × CpLq. Take f P CpKq and g P CpLq. Then     n n  n      1    ´m j mk  Λp f, gq ď 2 f px j qζ   gpyk qζ    n m“1  j“1    k“1 ⎛ n n ⎞1/2 ⎛ n n ⎞1/2 ⎟⎟⎟ ⎜⎜⎜   ⎟⎟⎟ 1 ⎜⎜⎜⎜   mpk´ jq mpk´ jq ⎟⎟⎟ ⎟⎟⎠⎟ ⎜⎜⎝⎜ f px j q f pxk qζ gpy j qgpyk qζ ď 2 ⎜⎝⎜ ⎠ n j,k“1 m“1 j,k“1 m“1 ⎞1/2 ⎛ n ⎞1/2 ⎛ n 2 ⎟⎟⎟ ⎜⎜⎜ 2 ⎟⎟⎟  1 ⎜⎜⎜⎜  ⎟ ⎜ “ ⎜⎜⎝  f px j q ⎟⎟⎠ ⎜⎜⎝ gpy j q ⎟⎟⎟⎠ ď | f |K |g|L , n j“1 j“1   and so Λ ď 1. Since Λp1K , 1L q “ 1, it follows that Λ “ 1. We now regard Λ as a bounded linear functional on VpK, Lq with Λ “ 1, so that n n 1  Fpx j , yk qζ mpk´ jq ηm pF P VpK, Lqq . ΛpFq “ 2 n m“1 j,k“1 Note that n 

kzk´1 “ pnzn+1 ´ pn + 1qzn + 1q/pz ´ 1q2

pz P C \ {1}q ,

k“1

and so n  k“1

kζ

´mpk´1q

n “ ´m ζ ´1

pm  nq ,

n  k“1

kζ ´npk´1q “

1 npn + 1q . 2

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185

Lemma 3.3.9 There is a constant C > 0 such that ΛpFq ě C log n for each n P N and each F P VpK, Lq such that Fpx j , yk q “ 1 for 1 ď j ď k ď n and Fpx j , yk q “ 0 for 1 ď k < j ď n. Proof We apply the above, taking ηm “ pζ ´m ´ 1q/ |1 ´ ζ m | pm P N \ {n}q, with ηn “ 1, so that |ηm | “ 1 pm P Nq, to obtain the following: ΛpFq “

n  1   mpk´ jq ζ η : 1 ď j ď k ď n m n2 m“1

“

n n n n  1   mpn´kq 1  ´m η kζ “ η ζ kζ ´mpk´1q m m n2 m“1 k“1 n2 m“1 k“1

“

n´1 1 n+1 1 + . n m“1 |ζ m ´ 1| 2n

  There is a constant C  > 0 such that 1/ eπiθ ´ 1 ě C  /θ for 0 < θ < 1, and so ΛpFq ě

n/2  C ě C log n m m“1

for some constant C > 0 that is independent of n. Take non-empty, open sets U and V in K and L, respectively, and take n P N. An upper-triangular set T (of size n) in U × V is formed as follows. Take sets {x1 , . . . , xn } and {y1 , . . . , yn } of distinct points in U and V, respectively, and then set T “ {pxi , y j q : 1 ď i ď j ď n} . An upper-triangular function on T is then formed as follows. Take pairwise-disjoint neighbourhoods Ui of xi in U for i P Nn and pairwise-disjoint neighbourhoods V j of y j in V for j P Nn , and consider functions of the form  { fi ⊗ g j : 1 ď i ď j ď n} , Gn “ where f1 , . . . , fn P CpKqr1s are such that fi pxi q “ 1 and supp fi Ă Ui for i P Nn and g1 , . . . , gn P CpLqr1s are such that g j px j q “ 1 and supp g j Ă V j for j P Nn . Let C > 0 be the constant specified in Lemma 3.3.9, and, for each ε > 0, choose Npεq P N such that log Npεq > 1/Cε. Lemma 3.3.10 Let K and L be compact spaces containing open subsets U and V of K and L, respectively, and take ε > 0. Suppose that |U| ∧ |V| ě Npεq. Then there exists F P VpK, Lq with Fπ “ 1, with |F|K×L < ε, and with supp F Ă U × V. Proof For n “ Npεq, there are sets {x1 , . . . , xn } and {y1 , . . . , yn } of distinct points in U and V, respectively. Let Gn be an upper-triangular function as above. By Lemma

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3 Banach function algebras

  3.3.8, ΛpGn q ď Gn π , and, by Lemma 3.3.9, ΛpGn q ě C log n. Set F “ Gn / Gn π . Then F has the required properties. Theorem 3.3.11 Let K and L be infinite, compact spaces. Then VpK, Lq  CpK × Lq . Proof It is immediate from Lemma 3.3.10 that the norms  · π and | · |K×L are not equivalent on the space VpK, Lq, giving the result. Theorem 3.3.12 Let K and L be non-empty, compact spaces. Then VpK, Lq is weakly sequentially complete if and only if K or L is scattered. Proof By Corollary 2.2.26, VpK, Lq is weakly sequentially complete if and only if MpKq or MpLq has the Schur property. By Corollary 1.3.71, this holds if and only if K or L is scattered. A more general result than the following is given in [318, Theorem 1.5.1]. Theorem 3.3.13 Let K and L be compact spaces with |K × L| ě 2. Then VpK, Lq is strongly regular, and each maximal ideal Mpx,yq in VpK, Lq has a bounded approximate identity of bound 3/2 contained in Jpx,yq . Proof We take x P K and y P L, and consider the maximal ideal M “ Mpx,yq in VpK, Lq; we also set J “ Jpx,yq .  The elements in M of the form G “ ni“1 fi ⊗ gi , where n P N, f1 , . . . , fn P CpKq, and g1 , . . . , gn P CpLq, form a dense linear subspace of M. Now G“

n n   p fi ´ fi pxq1K q ⊗ gi + fi ⊗ pgi ´ gi pyq1L q i“1

i“1

n

because i“1 fi pxqgi pyq “ 0, and so each such function G is the sum of functions of the form f ⊗ g for which either f P M x or g P My . Thus, to show that M has a bounded approximate identity of bound 3/2 and that J “ M, it follows from Proposition 2.1.22(ii) that it is sufficient to show that, for each ε > 0, each n P N, each finite sets { f1 , . . . , fn } and {g1 , . . . , gn } in CpKq and CpLq, respectively, there exists F P Jr3/2s such that    f ⊗ g ´ p f ⊗ gq · F π < ε/n whenever f P { f1 , . . . , fn } and g P {g1 , . . . , gn } are such that either f P M x or g P My ; in this case, it follows that G ´ GFπ < ε for the above function G. For each W P Npx,yq , there exist U P N x and V P Ny with U × V Ă W. By Urysohn’s lemma, there exist u P CpKqr1s with u “ 1 near x and u “ ´1 on K \ U and v P CpLqr1s with v “ 1/2 near y, v “ 1 on L \ V, and vpLq Ă r1/2, 1s. Note that |v ´ 1L |L “ 1/2. Consider the function FW P VpK, Lq defined by

3.3 Projective tensor products of Banach function algebras

187

FW “ 1K ⊗ v + u ⊗ pv ´ 1L q . We see that FW P J and that FW π ď 3/2. Now take f P CpKq and g P CpLq. Then      f ⊗ g ´ p f ⊗ gq · FW π “ p f + f uq ⊗ pg ´ gvqπ “ | f + f u|K |g ´ gv|L . Suppose that f P { f1 , . . . , fn }. Since u “ ´1 on K \ U, we can ensure that | f + f u|K is arbitrarily small for each such function f by an appropriate choice of W P Npx,yq . Similarly, suppose that g P {g1 , . . . , gn }. Since v “ 1 on L \ V, we can ensure that each|g ´ gv|L is arbitrarily small. Thus, given ε > 0, there exists FW P Jr3/2s such that  f ⊗ g ´ p f ⊗ gq · FW π < ε/n whenever f and g are the specified functions. It follows that the Banach function algebra VpK, Lq is strongly regular. We shall now show that the above bound of 3/2 for a bounded approximate identity in a maximal ideal of a Varopoulos algebra is best possible. Example 3.3.14 Let K and L both be two-point sets, so that we identify CpKq and CpLq with C 2 . The unit ball of C 2 (with the uniform norm) is the absolutely p C 2, convex hull of the two points p1, 1q and p1, ´1q, and so the unit ball of C 2 ⊗ 4 regarded as a subset of C , is the absolutely convex hull of the four points p1, 1, 1, 1q, p1, ´1, 1, ´1q, p1, 1, ´1, ´1q and p1, ´1, ´1, 1q. The identity of one of the four p C 2 is p :“ p0, 1, 1, 1q, and we have maximal ideals of C 2 ⊗ p0, 1, 1, 1q “

1 1 1 3 p1, 1, 1, 1q ´ p1, ´1, 1, ´1q ´ p1, 1, ´1, ´1q ´ p1, ´1, ´1, 1q , 4 4 4 4

and so pπ “ 3/2. Thus a bounded approximate identityin this maximal ideal has bound 3/2. It follows that each maximal ideal in an algebra VpK, Lq, where K and L are compact spaces with at least two points, has a bounded approximate identity of bound 3/2, and that the bound 3/2 is best possible.

We also give further results about VpK, Lq, taken from [319]; they will be used later. See also [145, Theorem 11.1.4]. Proposition 3.3.15 Let K and L be two non-empty, compact spaces. Suppose that U1 and U2 are non-empty, disjoint open sets in K and that V1 and V2 are non-empty, disjoint open sets in L, and take F, G P VpK, Lq with Fπ “ Gπ “ 1, with supp F Ă U1 × V1 and with supp G Ă U2 × V2 . Then F + Gπ “ 1. Proof We may suppose that F“

m  i“1

αi f1,i ⊗ g1,i ,

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3 Banach function algebras

    where f1,i P CpKq and g1,i P CpLq, where  f1,i K “ g1,i L “ 1, where supp f1,i Ă U1  and supp g1,i Ă V1 , and where αi ě 0 for i P Nm , and also where m i“1 αi “ 1. We n also suppose that G “ j“1 β j f2, j ⊗ g2, j , with analogous constraints. Then we have F +G “

n m  1  αi β j p f1,i + f2, j q ⊗ pg1,i + g2, j q + p f1,i ´ f2, j q ⊗ pg1,i ´ g2, j q . 2 i“1 j“1

        Since  f1,i + f2, j K “ g1,i + g2, j L “  f1,i ´ f2, j K “ g1,i ´ g2, j L “ 1, it follows that F + Gπ “

m  n 

αi β j “ 1 ,

i“1 j“1

as required. Define a sequence pn j q in N inductively as follows: set n1 “ 1 and then set n j+1 “ 2n j + 1 p j P Nq, so that 2 j´1 ď n j ď 3 j p j P Nq. Proposition 3.3.16 Let K and L be two infinite, compact spaces, and take j P N. Suppose thatG j is an upper-triangular function on an upper-triangular set of size n j . Then G j π ď j p j P Nq. Proof The result is trivial in the case where j “ 1. Assume inductively that the result holds for j P N, and take G j+1 to be an uppertriangular function of the above form on an upper-triangular set of size n j+1 . Then we can write G j+1 as H1 + H2 + H3 , where H1 and H2 are upper-triangular functions on upper-triangular sets of size n j and H3 is of the form H3 “ p fn j +1 + · · · + fn j+1 q ⊗ pgn j +1 + · · · + gn j+1 q for appropriate functions fn j +1 , . . . , fn j+1 , gn j +1 , . . . , gn j+1 . Thus H  3 π  “ 1. Also H1 + H2 π “ H1 π “ H2 π by Proposition 3.3.15, and so G j+1 π ď j + 1, continuing the induction. We conclude this section by exhibiting certain closed linear subspaces of VpK, Lq. In the following, we take two non-empty, compact spaces K and L, and write V “ VpK, Lq, with the norm  · π . We also take two infinite, compact spaces K0 and L0 , and set V0 “ VpK0 , L0 q, with the norm  · π, 0 , and we make the hypothesis that there are continuous surjections πK : K Ñ K0 and πL : L Ñ L0 ; in this case, we have the corresponding isometric Banach-algebra embeddings jK : CpK0 q Ñ CpKq and jL : CpL0 q Ñ CpLq, as in Equation (3.1.5). The map jK ⊗ jL : V0 Ñ V is a Banach-algebra homomorphism with  jK ⊗ jL op ď 1. The following result is essentially [318, Lemma 1.2.1]. Theorem 3.3.17 With the above notation, the map jK ⊗ jL : V0 Ñ V is an isometry, and so V0 is identified with a closed subalgebra of V.

3.4 The separating ball property

189

  Proof Set j “ jK ⊗ jL , and take h P V0 , so that  jphqπ ď hπ, 0 . Now fix ε > 0. Then there exist k P N and functions f1 , . . . , fk P CpK0 q and g1 , . . . , gk P CpK0 q such that   k    h ´ fi ⊗ gi  < ε .   i“1 π, 0

There exist U K P BpCpKq, CpK0 qqr1s and U L P BpCpLq, CpL0 qqr1s by Theorem 3.1.16 such that     ε ε pU K ◦ jK qp fi q ´ fi K < and pU L ◦ jL qpgi q ´ gi K < 0 0 k k for each i P Nk . Set U “ U K ⊗ U L P BpV, V0 qr1s . Then       pU ◦ jqp fi ⊗ gi q ´ fi ⊗ gi π, 0 ď pU K ◦ jK qp fi q ´ fi K + pU L ◦ jL qpgi q ´ gi K 0

< 2ε/k

0

pi P Nk q ,

and so k     pU ◦ jqp f ⊗ g q ´ f ⊗ g  + 2ε < 4ε . pU ◦ jqphq ´ hπ, 0 < i i i i π, 0 i“1

    Thus hπ, 0 ď pU ◦ jqphqπ, 0 + 4ε ď  jphqπ + 4ε.   It follows that  jphqπ “ hπ, 0 , giving the result. For further results about the Varopoulos algebra, see Theorems 5.3.40, 5.3.42 and 6.2.18.

3.4 The separating ball property In this section, we shall introduce some variants of the ‘separating ball property’; we shall see in Theorem 3.6.11 that some uniform algebras and in Chapter 4 that many Banach function algebras on locally compact groups do have some of these properties. This will lead us to a discussion of topologically invariant means, whose rôle will become more prominent later. The definition of the separating ball property was introduced by Ülger in [312, Definition 2.3]; the next definition elaborates this notion Definition 3.4.1 Let A be a Banach function algebra, and take ϕ P ΦA . Then: (i) A has the separating ball property at ϕ if, given ψ P ΦA ∪ {8} with ψ  ϕ, there is f P pMψ qr1s such that f pϕq “ 1; (ii) A has the weak separating ball property at ϕ if, given ψ P ΦA ∪ {8} with ψ  ϕ, there is net p fν q in pMψ qr1s such that limν fν pϕq “ 1;

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3 Banach function algebras

(iii) A has the strong separating ball property at ϕ if, given U P Nϕ , there is f P Ar1s with f pϕq “ 1 and supp f Ă U. The algebra A has the separating ball property (SBP), the weak separating ball property (WSBP) and the strong separating ball property (SSBP) if A has the corresponding property at each ϕ P ΦA . Clearly SSBP implies SBP implies WSBP. p B is a natural Banach Let A and B be Banach function algebras such that A ⊗ function algebra on ΦA × ΦB . Suppose that ϕ P ΦA and ψ P ΦB are such that A and B both have the SBP or both have the WSBP or both have the SSBP at ϕ and ψ, p B has the corresponding property at the point pϕ, ψq. respectively. Then A ⊗ Definition 3.4.2 Let A be a Banach function algebra. Then A has norm-one characters if ϕ “ 1 pϕ P ΦA q. Certainly each unital Banach function algebra has norm-one characters. On the other hand, some maximal ideals in the disc algebra do not have this property; see Example 3.6.10. The following remark is obvious. Proposition 3.4.3 Let A be a Banach function algebra with |ΦA | ě 2. Then A has the weak separating ball property if and only if Mϕ has norm-one characters for each ϕ P ΦA . Proposition 3.4.4 Let A be a Banach function algebra, and take ϕ P ΦA . Suppose that A has the separating ball property at ϕ. Then ϕ is a strong boundary point for A. Proof We may suppose that |ΦA | ě 2, and choose ψ0 P Φ A with ψ0  ϕ. Then there exists f0 P pMψ0 qr1s with f0 pϕq “ 1. Set L “ {ψ P ΦA :  f0 pψq ě 1/2}, a compact subset of ΦA . Take an open neighbourhood U of ϕ. For each ψ P L\U, there exists fψ P pMψ qr1s  with fψ pϕq “ 1, and there is Uψ P Nϕ such that  fψ pxq < 1/2 px P Uψ q. Since L \ U is compact, there are n P N and ψ1 , . . . , ψn P L \ U such that " {Uψi : i P Nn } ⊃ L \ U .   Set f “ f0 fψ1 · · · fψn , so that f P Ar1s , f pϕq “ 1 and  f pψq < 1/2 pψ P ΦA \ Uq. This shows that ϕ is a strong boundary point for A. Let A be a Banach function algebra. Recall that the projection π : ΦA ∪ {0} Ñ ΦA ∪ {0} and the fibre Φ{ϕ} were defined in Equations (2.3.6) and (2.3.7); we shall write Φ{x} for Φ{εx } , and throughout this section we shall write ΦA for the closure of ΦA in pΦA , σpA , A qq.

3.4 The separating ball property

191

The following result is close to a theorem of Ülger [312, Theorem 2.2]. Theorem 3.4.5 Let A be a Banach function algebra. (i) Take ϕ P ΦA , and suppose that A has the separating ball property at ϕ. Then there is an idempotent Eϕ P Ar1s such that f · Eϕ “ Eϕ when f P Ar1s with f pϕq “ 1 and such that xEϕ , ψy “ δϕ,ψ pψ P ΦA q. Further, ϕ is an isolated point of ΦA and x f · Eϕ ´ f pϕq Eϕ , ψy “ 0 p f P A, ψ P ΦA q .

(3.4.1)

(ii) Suppose that A has the separating ball property. Then ΦA is discrete with respect to the relative weak topology. Proof We may suppose that |ΦA | ě 2. (i) Define the set S “ { f P Ar1s : f pϕq “ 1} . Since A has the SBP at ϕ, the set S is not empty, and clearly S is a convex subset of the space A. Consider S as a subset of Ar1s , and take L to be its weak-∗ closure in A . Then L is a non-empty, compact, convex set in the locally convex space pA , σpA , A qq. For each f P S , the map T f : M Þ→ f · M ,

A Ñ A ,

is linear, and so T f | L is affine, and T f is weak-∗ continuous. Further, we have T f pS q Ă S , and hence T f pLq Ă L, for each f P S , and the operators T f commute. By the Markov–Kakutani fixed-point theorem, Theorem 1.3.74, there exists an element Eϕ P L Ă Ar1s such that f · Eϕ “ Eϕ p f P S q. By taking a net in S that converges to Eϕ weak-∗, we see that Eϕ P IpA q. Clearly we have xEϕ , ϕy “ 1 and xEϕ , ψy “ 0 pψ P ΦA \ {ϕ}q, and hence xEϕ , ψy “ δϕ,ψ pψ P ΦA q. Take ψ P ΦA \ {ϕ}. Then ψ is the limit in pΦA , σpA , A qq of a net in ΦA \ {ϕ}, and so x Eϕ , ψy “ 0. Hence ϕ is an isolated point of ΦA . Now take ψ P ΦA , and set τ “ πpψq P ΦA ∪ {0}. Suppose that τ  ϕ. Then there exists g P S with gpτq “ 0, and so xEϕ , ψy “ xg · Eϕ , ψy “ gpτq xEϕ , ψy “ 0. Hence (3.4.1) follows, and, in particular, xEϕ , ψy “ 0 pψ P ΦA \ {ϕ}q. Suppose that τ “ ϕ. Then (3.4.1) is immediate. (ii) By (i), each point ϕ P ΦA is isolated in pΦA , σpA , A qq, and so the space ΦA is weakly discrete. Corollary 3.4.6 Let A be a Banach function algebra with the separating ball property, and take f P A with f · A Ă A. Then each element in ΦA \ Zp f q is isolated in ΦA , and ΦA \ Zp f q is countable. Proof Take ϕ P ΦA \ Zp f q. By Theorem 3.4.5, there exists Eϕ P IpA q with xEϕ , ϕy “ 1 and xEϕ , ψy “ 0 pψ P ΦA \ {ϕ}q. Set g “ f · Eϕ P A. Then gpϕq  0, but gpψq “ 0 pψ P ΦA \ {ϕ}q, and so ϕ is isolated in ΦA .

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  For each n P N, set S n “ {ϕ P ΦA :  f pϕq ě 1/n}. Then S n is compact and discrete in ΦA , and so S n is finite. Hence ΦA \ Zp f q is countable. Lemma 3.4.7 Let A be a Banach function algebra, and take f P A such that f · A Ă A. Then f · A Ă lin pΦA \ Zp f qq. Proof Set Ω “ ΦA \ Zp f q. Take λ P A , and assume towards a contradiction that f · λ  lin Ω. By the Hahn–Banach theorem, there exists M P A such that xM, f · λy “ 1 and xM, ϕy “ 0 pϕ P Ωq. It follows that x f · M, ϕy “ f pϕqxM, ϕy “ 0

pϕ P ΦA q .

Since f · M P A, it follows that f · M “ 0, contradicting the fact that x f · M, λy “ 1. Thus f · A Ă lin Ω. The following theorem of Ülger is [312, Corollary 3.4]. Theorem 3.4.8 Let A be a Banach function algebra such that A has the separating ball property and is weakly sequentially complete. For each f P A, the operator L f : A Ñ A is weakly compact if and only if the closed linear subspace f · A of A is separable. Proof Take f P A. First, suppose that L f is weakly compact, so that f · A Ă A. Since A has the SBP, it follows from Corollary 3.4.6 that the set Ω :“ ΦA \ Zp f q is countable. By Lemma 3.4.7, f · A Ă lin Ω. But lin Ω is separable because Ω is countable, and so f · A is separable. Conversely, suppose that f · A is separable, say p f · λr q is a dense sequence in f · A . Let p fn q be a sequence in Ar1s . Then there is a subsequence p fnk q of p fn q such that px fnk , f · λr y : k P Nq converges in C for each r P N. It follows that, for each λ P A , the sequence px fnk , f · λy : k P Nq converges in C, and hence the sequence p f fnk q is weakly Cauchy in A. Since A is weakly sequentially complete, p f fnk q is weakly convergent, and so L f is weakly compact. Corollary 3.4.9 Let A be a separable Banach function algebra that has the separating ball property, is weakly sequentially complete, and is an ideal in its bidual. Then AA is a separable, closed linear subspace of A . Suppose, further, that A is Arens regular. Then A A is also a separable linear subspace of A . Proof Since A is an ideal in A , L f is weakly compact, and so, by Theorem 3.4.8, f · A is separable for each f P A. Since A is separable, it follows that AA is separable. Now suppose that A is Arens regular. Then A A “ AA by Proposition 2.3.25, and so A A is also separable.

3.4 The separating ball property

193

A small variation of the proof of Theorem 3.4.5 gives the following result. Theorem 3.4.10 Let A be a dual Banach function algebra, with Banach-algebra predual F Ă A , and take a non-empty, compact subset K of ΦA . Suppose that K Ă F and that ϕ “ 1 pϕ P Kq. Then there is e P IpAq with epϕq “ 1 pϕ P Kq. In the case where K “ {ϕ}, also e P Ar1s and f e “ e for each f P Ar1s with f pϕq “ 1. Proof Take ϕ P K, and again consider the set S ϕ “ { f P Ar1s : f pϕq “ 1} . Since ϕ “ 1, it follows from the Hahn–Banach theorem that S ϕ  ∅. Clearly S ϕ is convex and compact with respect to the topology σpA, Fq. For f P S ϕ , the maps L f | S ϕ : g Þ→ f g ,

Sϕ Ñ Sϕ ,

form a commuting family of affine maps, each continuous with respect to the σpA, Fq-topology, and so, again by Theorem 1.3.74, the family has a fixed point, say eϕ P S ϕ Ă Ar1s . Thus f eϕ “ eϕ for each f P S ϕ , and, in particular, eϕ P IpAq. For each ϕ P K, set T ϕ “ {ψ P ΦA : eϕ pψq “ 1}, so that T ϕ is a compact  and open subspace of ΦA such that ϕ P T ϕ . Since K Ă {T ϕ : ϕ P K} and K is compact, there exist ϕ1 , . . . , ϕn P K such that K Ă U :“ T ϕ1 ∪ · · · ∪ T ϕn . The set U is compact and open in ΦA , and so, by Šilov’s idempotent theorem, Theorem 2.1.18, the characteristic function of U, say e, belongs to IpAq and epϕq “ 1 pϕ P Kq. The following example shows that the condition that ϕ P F in the above theorem cannot be removed when obtaining the conclusion that f e “ e for each f P S ϕ . Example 3.4.11 Let A “ 8 “ Cp β Nq, so that A is dual Banach function algebra, with Banach-algebra predual 1 . Of course ΦA “ β N. Take x P β N. In the case where x P N, so that ε x P 1 , the corresponding idempotent of the above theorem is the characteristic function of x. However, when x P N∗ , so that ε x  1 , there is no idempotent e P Ar1s such that f e “ e for each f P Ar1s with f pxq “ 1.

The following is an adaptation of standard terminology; the reason for the terminology will become apparent in Chapter 4. Definition 3.4.12 Let A be a Banach function algebra, and take ϕ P ΦA . Then a topologically invariant mean at ϕ is an element mϕ P A such that xmϕ , ϕy “ 1 and

f · mϕ “ f pϕq mϕ

p f P Aq .

The set of all topologically invariant means at ϕ is denoted by TIMpϕq, with TIMpxq for TIMpε x q and TIMpϕqrms for TIMpϕq ∩ Arms , where m > 0.

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We shall write TIMpϕ, Aq if it be necessary to indicate the algebra A in the above. Note that (3.4.2) M l mϕ “ xM, ϕy mϕ pM P A q for mϕ P TIMpϕq; in particular, mϕ is an idempotent in A . It is clear that, in the case where TIMpϕq is non-empty, it is a convex and weak-∗ closed subspace of A , and it is a right-zero semigroup with respect to the Arens product l because m1 l m2 “ m2 for m1 , m2 P TIMpϕq. It follows that lin TIMpϕq is a left ideal in A . Proposition 3.4.13 Let A be a Banach function algebra, let I be a non-zero, closed ideal in A, and take ϕ P ΦI . Then TIMpϕ, Iq “ TIMpϕ, Aq. Proof Fix g P I with gpϕq “ 1. Let mϕ P TIMpϕ, Aq. Then clearly mϕ “ g · mϕ P I  and mϕ P TIMpϕ, Iq, so that TIMpϕ, Aq Ă TIMpϕ, Iq. Let mϕ P TIMpϕ, Iq, so that mϕ P I  Ă A and xmϕ , ϕy “ 1. Take f P A. Then f g P I and f · mϕ “ f g · mϕ “ p f gqpϕqmϕ “ f pϕq mϕ , and so mϕ P TIMpϕ, Aq. Thus TIMpϕ, Iq Ă TIMpϕ, Aq, giving the result. Proposition 3.4.14 Let A be a Banach function algebra, and take ϕ P ΦA . (i) Any two elements of TIMpϕq coincide on WAPpAq. In particular, TIMpϕq consists of at most one element when A is Arens regular. (ii) We have TIMpϕq · A Ă AA .     (iii) We have TIMpϕq ď AA . Proof (i) Take m1 , m2 P TIMpϕq. Then m1 l m2 “ m2 and m2 l m1 “ m1 . But xm1 l m2 , λy “ xm2 l m1 , λy pλ P WAPpAqq by Theorem 2.3.29, and so m1 and m2 agree on WAPpAq, and hence on A when A is Arens regular, where we are using Theorem 2.3.29, (a) ⇒ (c). (ii) Take mϕ P TIMpϕq and λ P A , and assume towards a contradiction that mϕ · λ  AA . Then there exists M P A such that xM, mϕ · λy “ 1 and such that f · M “ 0 p f P Aq. Since x f · M, ϕy “ f pϕq xM, ϕy for all f P A and there exists f P A with f pϕq “ 1, it follows that xM, ϕy “ 0. However, using Equation (3.4.2), we have xM, ϕy xmϕ , λy “ xM l mϕ , λy “ xM, mϕ · λy “ 1 , giving the required contradiction. Hence TIMpϕq · A Ă AA .      (iii) The  claimed result is trivial if TIMpϕq ď 1, and so we may suppose that TIMpϕq ě 2. Take mϕ P TIMpϕq and λ P A . Then

3.4 The separating ball property

195

x f, mϕ · λy “ x f · mϕ , λy “ f pϕq x mϕ , λy p f P Aq . Now take m1 , m2 P TIMpϕq with m1  m2 . Then it follows exists λ P A   that there  with m1 · λ  m2 · λ. This implies that TIMpϕq · A  ě TIMpϕq, and so the result follows from (ii). Proposition 3.4.15 Let A be a Banach function algebra with |ΦA | ě 2 such that A has a bounded approximate identity with bound m, and take ϕ P ΦA . (i) Suppose that TIMpϕqr1s  ∅. Then Mϕ has a bounded approximate identity with bound m + 1. (ii) Suppose that Mϕ has a bounded approximate identity. Then TIMpϕq  ∅. Proof By Proposition 2.3.66(ii), Arms contains a mixed identity, say E. (i) Now take mϕ P TIMpϕqr1s , and set Fϕ “ E ´ mϕ , so that Fϕ P pMϕ qrm+1s . For each f P Mϕ , we have f · Fϕ “ f · E ´ f pϕq mϕ “ f , and so Fϕ is a mixed identity for Mϕ . By Proposition 2.3.66(ii), Mϕ has a BAI of bound m + 1. (ii) Let Eϕ be a mixed identity for Mϕ , and define mϕ “ E ´ E l Eϕ P A . Then xmϕ , ϕy “ 1. Now take M P A . Since A “ Mϕ ‘ C E, we have M “ N+xM, ϕy E, where N P Mϕ , and so M l mϕ “ M ´ M l Eϕ “ N + xM, ϕy E ´ pN + xM, ϕy Eq l Eϕ “ xM, ϕy E ´ xM, ϕy E l Eϕ “ xM, ϕy mϕ . Thus mϕ P TIMpϕq, and so TIMpϕq  ∅. The next theorem is based on a result of Renaud [271, Corollary 3]. In the statement, the family Nϕ is a net with respect to reverse inclusion. Theorem 3.4.16 Let A be a strongly regular Banach function algebra. Take ϕ P ΦA , and let V to be a subnet of Nϕ consisting of relatively compact sets. Suppose that there exists m > 0 such that, for each V P V, there exists fV P Arms with fV pϕq “ 1 and supp fV Ă V. Then each weak-∗ accumulation point of the net { fV : V P V} in A belongs to TIMpϕqrms . In particular, TIMpϕqrms  ∅. Proof Take f P A with f pϕq “ 1. We first claim that lim  f fV ´ fV  “ 0 .

VPV

Indeed, take V0 to be a compact neighbourhood of ϕ, and take ε > 0. Since A is regular, and hence, by Proposition 3.1.10(i), normal, there exists g P A with g | V0 “ 1. Since f ´ g P Mϕ and A is strongly regular at ϕ, there is h P Jϕ with  f ´ g ´ h < ε. For each sufficiently small V P V (with  V Ă V0 ), we have  fV h “ 0 and fV g “ fV , and so  f fV ´ fV  “ p f ´ g ´ hq fV  < ε, and this implies the claim.

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3 Banach function algebras

Let mϕ be a weak-∗ accumulation point of the net { fV : V P V} in Arms . It follows from the claim that f · mϕ “ mϕ for each f P A with f pϕq “ 1. Clearly xmϕ , ϕy “ 1. For f P A with f pϕq  0, certainly f · mϕ “ f pϕq mϕ . For f P A with f pϕq “ 0, take g P A with gpϕq “ 1. Then pg ´ f q · mϕ “ mϕ “ g · mϕ , and so f · mϕ “ 0 “ f pϕqmϕ . We conclude that f · mϕ “ f pϕq mϕ for each f P A. Hence mϕ P TIMpϕqrms . Corollary 3.4.17 Let A be a strongly regular Banach function algebra such that AA Ă WAPpAq. Take  ϕPΦ  A , and suppose that A has the strong separating ball property at ϕ. Then TIMpϕq “ 1.   Proof By Theorem 3.4.16, TIMpϕq ě 1. Now suppose that m1 , m2 P TIMpϕq. Take f P A with f pϕq “ 1. For each λ P A , we have f · λ P WAPpAq, and so, by Proposition 3.4.14(i), xm1 , f · λy “ xm2 , f · λy, whence  x f · m1 , λy “ x f · m2 , λy. But f · m1 “ m1 and f · m2 “ m2 , and so m1 “ m2 . Thus TIMpϕq “ 1. Corollary 3.4.18 Let A be a Banach function algebra. Suppose that A is strongly regular, has the strong separating ball property, is weakly sequentially complete, and has a bounded approximate identity. Then A is Arens regular if and only if A is finite dimensional. Proof We may suppose that |ΦA | ě 2. By Theorem 3.4.16, TIMpϕqr1s  ∅ for each ϕ P ΦA , and so, by Proposition 3.4.15(i), each maximal ideal of A has a BAI. Now suppose that A is Arens regular. Since A is weakly sequentially complete, Theorem 2.3.46 applies to show that each maximal ideal in A has an identity and A has an identity. Hence ΦA is compact and each point in ΦA is isolated, and so ΦA is finite, which implies that A is finite dimensional. The converse is immediate. Let K be an infinite, locally compact space. Then the algebra C 0 pKq is strongly regular, has the strong separating ball property, has a contractive approximate identity, and is Arens regular, but manifestly it is not a finite-dimensional space. Thus, by the corollary, C 0 pKq is not weakly sequentially complete; this shows the importance of the fact that A is weakly sequentially complete in the above corollary. We recall that the subalgebra AH of A was defined in Equation (3.1.8). Theorem 3.4.19 Let A be a strongly regular Banach function algebra such that AH is weakly sequentially complete for each compact subspace H of ΦA . Take ϕ P ϕ is an ΦA , and suppose that A has the strong separating ball property at ϕ. Then  isolated point of ΦA if and only if {ϕ} is a Gδ -subset of ΦA and TIMpϕq “ 1.

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197

Proof Recall that the characteristic function of {ϕ} is χϕ . Suppose that ϕ is an isolated point of ΦA . Then certainly {ϕ} is a Gδ -subset of ΦA . By Šilov’s idempotent theorem, Theorem 2.1.18, χϕ belongs to A, and clearly χϕ P TIMpϕq. Now take mϕ P TIMpϕq. Then mϕ “ χϕ · mϕ “ mϕ · χϕ “ χϕ , and  so χϕ is the unique element of TIMpϕq, whence TIMpϕq  “ 1.  Now suppose that {ϕ} is a Gδ -subset of ΦA and that TIMpϕq “ 1. Then Nϕ has a countable basis, say pVn q, where we may suppose that H :“ V1 is compact and that Vn+1 Ă Vn pn P Nq. For each n P N, choose fn P Ar1s such that fn pϕq “ 1 and supp fn Ă Vn . By Theorem 3.4.16,  each weak-∗ accumulation point of the sequence p fn q belongs to TIMpϕq. But TIMpϕq “ 1, and so this sequence has a unique weak-∗ accumulation point, say mϕ . This shows that limnÑ8 fn “ mϕ in pA , σpA , A qq. The sequence p fn q is contained in the closed ideal AH and AH is weakly sequentially complete, and hence mϕ P A. Clearly mϕ “ χϕ , and so ϕ is an isolated point of ΦA . Theorem 3.4.20 Let A be a strongly regular Banach function algebra such that ΦA is metrizable, such that A has the strong separating ball property, and such that AH is weakly sequentially complete for each compact subset H of ΦA . Suppose that I is a weak Segal algebra with respect to A such that I is Arens regular. Then ΦA is discrete. Proof Take ϕ P ΦA . Since ΦA is metrizable, {ϕ} is a Gδ -subset of ΦA .  Since the  algebra I is Arens regular, it follows from Proposition 3.4.14(i) that TIMpϕ, Iq ď 1. Take f P I with f pϕq “ 1, and take sequences pVn q in Nϕ and p fn q in Ar1s as in Theorem 3.4.19. Set H “ V1 . By Proposition 3.1.28, I is normal, and so there exists g P I with g | H “ 1, say gI “ m. For n P N, set hn “ fn g, so that hn P Irms , hn pϕq “ 1, and supp hn Ă Vn . We have limnÑ8  f hn ´ hn I “ 0, and so, as in Theorem 3.4.16, TIMpϕ, Iqrms  ∅. Hence there is a unique element in TIMpϕ, Iq. It follows that the sequence phn q is weakly Cauchy in I, and hence phn q is weakly Cauchy in AH because we can regard A as a linear subspace of I  . Since AH is weakly sequentially complete, the sequence phn q converges in A, and its limit is χϕ . Thus ϕ is an isolated point in ΦA . We have shown that each point of ΦA is isolated, and so ΦA is discrete. For an application of the above result to the Figà-Talamanca–Herz algebras A p pΓq, see Theorem 6.4.2(ii).

3.5 Pointwise approximate identities In this section, we shall introduce pointwise versions of bounded approximate identities and of contractive approximate identities. We are particularly interested in

198

3 Banach function algebras

Banach function algebras that have a contractive pointwise approximate identity; we shall characterize these algebras in Theorem 3.5.7, and use this characterization later. A Banach function algebra A is pointwise contractive if A and every non-zero, maximal modular ideal in A has a contractive pointwise approximate identity; we shall characterize pointwise contractive uniform algebras in Theorem 3.6.32. In Example 3.5.13, we shall exhibit a natural, pointwise contractive Banach function algebra on I that is not a uniform algebra. These classification results will be extended in Theorem 5.4.12. The material of this section is mostly drawn from [64], where more results are given. The following definition originates with Jones and Lahr [189]; see also [63, 179, 200, 297]. However our terminology is different. Definition 3.5.1 Let A be a Banach function algebra. A net peν q in A is a pointwise approximate identity (PAI) if lim eν pϕq “ 1 pϕ P ΦA q ; ν

the PAI is bounded, with bound m > 0, if supν eν  ď m, and then peν q is a bounded pointwise approximate identity (BPAI); a bounded pointwise approximate identity of bound 1 is a contractive pointwise approximate identity (CPAI). The algebra A is pointwise contractive if A and each of its non-zero, maximal modular ideals have a contractive pointwise approximate identity. A unital Banach function algebra certainly has a contractive pointwise approximate identity. A Banach function algebra A is pointwise contractive if and only if, F of Φ for each ϕ P ΦA ∪ {8}, each non-empty, finite subset   A with ϕ  F, and each ε > 0, there exists f P Mϕ with  f  < 1 + ε and 1 ´ f pψq < ε pψ P Fq. Certainly each contractive Banach function algebra is pointwise contractive. Note that a bounded pointwise approximate identity is not necessarily an approximate identity. Further, a non-unital Banach function algebra that is reflexive cannot have a bounded pointwise approximate identity. A pointwise contractive Banach function algebra has the weak separating ball property, but the converse is not true. For example, consider the Banach sequence algebra 1 : this algebra satisfies even the strong separating ball property, but it does not have a bounded pointwise approximate identity. Clearly each Banach function algebra has a pointwise approximate identity. Our Example 3.2.11 exhibited a natural Banach sequence algebra A on N such that pΔn q is a bounded pointwise approximate identity, but A has no approximate identity. Example 3.6.37 will give a natural, pointwise contractive uniform algebra that is not contractive, and Example 3.6.42 will give a natural uniform algebra with a contractive pointwise approximate identity, but no approximate identity. The Banach function algebra M in Example 3.5.13 is pointwise contractive and has an approximate identity, but does not have a bounded approximate identity.

3.5 Pointwise approximate identities

199

Let pA,  · A q and pB,  · B q be natural Banach function algebras on a non-empty, locally compact space, and suppose that B is a dense subalgebra of A and that  f A ď  f B p f P Bq. Then A is pointwise contractive whenever B is pointwise contractive. Proposition 3.5.2 Let A be a Banach function algebra such that |ΦA | ě 2, and take ϕ P ΦA . Suppose that A and Mϕ have bounded pointwise approximate identities. Then there exists M P A with xM, ϕy “ 1 and xM, ψy “ 0 pψ P ΦA \ {ϕ}q. Proof Let peα q be a BPAI in A, and let pdβ q be a BPAI in Mϕ . Let E and D be weak-∗ accumulation points of these nets in A . Set M “ E ´ D P A . Then M has the required properties. Proposition 3.5.3 Let A be a Banach function algebra, and let I be a weak Segal algebra with respect to A. Suppose that A has the separating ball property and that I has a contractive pointwise approximate identity. Then I has the weak separating ball property. Proof We may suppose that |ΦA | ě 2. Let peα q be a CPAI in I, and take ϕ, ψ P ΦA with ϕ  ψ. Since A has the SBP, there exists f P Ar1s with f pϕq “ 0 and f pψq “ 1. It follows that p f eα q is a net in Ir1s with p f eα qpϕq “ 0 for each α and with limα p f eα qpψq “ 1. Hence I has the WSBP. Recall that the space of semi-characters on a semigroup S is denoted by ΦS ; as in Example 3.1.7, we identify the semigroup algebra p 1 pS q, q with the Banach function algebra ApΦS q when S is abelian and separating. Proposition 3.5.4 Let S be an abelian semigroup that is separating, and suppose that there is a net psα q in S such that limα ϕpsα q “ 1 pϕ P ΦS q. Then the Banach function algebra ApΦS q has a contractive pointwise approximate identity. Proof For each α, set fα “ δ sα P 1 pS q, so that fpα P ApΦS qr1s . Then clearly limα fpα pϕq “ limα ϕpsα q “ 1 pϕ P ΦS q, so that p fpα q is the required CPAI. The semigroup pQ+• , + q is abelian, separating and cancellative, and so, by Proposition 2.4.28, the semigroup algebra 1 pQ+• q is a dual Banach function algebra, with isometric Banach-algebra predual c 0 pQ+• q. It follows from Example 2.1.24 that this algebra does not have an approximate identity, but the following theorem, which was given by Jones and Lahr in [189], shows that it has a contractive pointwise approximate identity. Thus, for Banach function algebras, we cannot replace ‘bounded approximate identity’ by ‘bounded pointwise approximate identity’ in Proposition 2.4.7. It seems to be an open question whether this algebra factors. Theorem 3.5.5 The Banach function algebra ApΦQ+• q has a contractive pointwise approximate identity.

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3 Banach function algebras

Proof Set S “ pQ+• , + q, so that ApΦQ+• q is identified with the semigroup algebra 1 pS q. We first make a remark. Take r P N and z1 , . . . , zr P T, say z j “ eiθ j for j P Nr , and take ε P p0, 1q. There exists m0 P N with dpm0 θ j , 2π Zq < ε/2 p j P Nr q. For each j P Nr , define   U j “ {w “ |w| eiθ : θ ´ θ j  < ε/4m0 , 1 ´ ε/4 < |w|m0 ď 1} , itk take w  1 , . . . , wm0 P U j , say wk “ |wk | e for k P Nm0 , and set u “ w1 · · · wm0 . We have t1 + · · · + tm0 ´ m0 θ j  < ε/4 and |u| > 1 ´ ε/4, and hence  ε ε    |u ´ 1| ď u ´ zmj 0  + zmj 0 ´ 1 < + “ ε . 2 2

Take r P N and ϕ1 , . . . , ϕr P ΦS . We first claim that there are a strictly increasing sequence pnk q in N and z1 , . . . , zr P T such that ϕi p1/nkq Ñ zi  for each i P Nr . Indeed, for each i P Nr , set ζi “ ϕi p1q P D \ {0}, so that ϕi p1/nq “ |ζi |1/n Ñ 1 as n Ñ 8. Thus we may successively take strictly increasing subsequences of N, with final sequence pnk q such that each sequence pϕi p1/nk qq converges to a point of T, giving the claim. Let F be the family of finite subsets of ΦS , and consider the directed set N × F , where pn1 , F1 q ď pn2 , F2 q for n1 , n2 P N and F1 , F2 P F if n1 ď n2 and F1 Ă F2 . Take ν “ pn, Fq P N × F , say F “ {ϕ1 , . . . , ϕr }, and take a strictly increasing sequence pnk q in N and z1 , . . . , zr P T as specified in the claim. By the remark, there exist m0 P N and subsets U1 , . . . , Ur of D such that z j P U j and |u ´ 1| < 1/n pu P U j q for each j P Nr . Then there exists Kν P N such that ϕ j p1/nk q P U j p j P Nr q for k ě Kν . It follows that    1  ϕ j pm0 /nk q ´ 1 “ ϕ j p1/nk qm0 ´ 1 < n

pk ě Kν , j P Nr q .

Define qν “ m0 /nKν P S to obtain a net pqν q in S . Take ϕ P S and ε > 0, choose n0 P N such that 1/n0 < ε, and set F0 “ {ϕ}. Then ν0 :“ pn0 , F0 q P N × F . Take ν “ pn, Fq P N × F with ν ě ν0 . Then we have  ϕpqν q ´ 1 < 1/n0 < ε. We have shown that limν ϕpqν q “ 1 pϕ P ΦS q, and so, by Proposition 3.5.4, ApΦS q has a contractive pointwise approximate identity. There is a small variation of ‘pointwise contractive’ that we shall mention. Definition 3.5.6 Let A be a Banach function algebra. Then A is strongly pointwise characters in ΦA , and contractive if, for each ε > 0, each set {ϕ1 , . . ., ϕn } of distinct  each ζ1 , . . . , ζn P D, there exists f P Ar1s with ζi ´ f pϕi q < ε pi P Nn q. Clearly a strongly pointwise contractive Banach function algebra is pointwise contractive. We shall show in Theorem 3.6.31 that a pointwise contractive uniform algebra is strongly pointwise contractive; Examples 3.5.13 and 3.5.14 are

3.5 Pointwise approximate identities

201

strongly pointwise contractive Banach function algebras that are not uniform algebras; Example 3.6.33 is a pointwise contractive Banach function algebra that is not strongly pointwise contractive. The first main result of this section is a characterization of Banach function algebras that have a contractive pointwise approximate identity. The characterization will be applied in Theorems 4.4.11, 5.1.31 and 5.1.35. Theorem 3.5.7 Let A be a Banach function algebra. Then A has a contractive pointwise approximate identity if and only if λ “ 1 pλ P co ΦA q. Proof Certainly λ ď 1 pλ P co ΦA q. Suppose that A has CPAI. Then there exists an element M P A with M “ 1 and xM, ϕy “ 1 pϕ P ΦA q. Take λ P co ΦA . Then xM, λy “ 1, and so λ ě 1. Hence λ “ 1. Conversely, suppose that λ “ 1 pλ P co ΦA q. Then, for each λ P co ΦA , there exists M P A with M “ xM, λy “ 1. In particular, take n P N and a subset  S “ {ϕ1 , . . . , ϕn } of ΦA . Set λS “ p nj“1 ϕ j q/n P co ΦA . Since λS  “ 1, there   exists MS P A with MS  “ xMS , λS y “ 1. Since xMS , ϕy ď 1 pϕ P ΦA q, the latter is possible only if xMS , ϕ j y “ 1 p j P Nn q. Thus, for each non-empty, finite subset S of ΦA , there is an element MS P Ar1s with xMS , ϕy “ 1 pϕ P Fq, and this easily gives a CPAI in A. Proposition 3.5.8 Let A be a Banach function algebra with a bounded pointwise approximate identity. Then ΦA is weakly closed in A . Proof Since ΦA ∪ {0} is closed in pA , σpA , Aqq, and hence in pA , σpA , A qq, the weak closure of ΦA is either ΦA or ΦA ∪ {0}. By hypothesis, there exists M in A with xM, ϕy “ 1 pϕ P ΦA q, and so xM, λy “ 1 for each λ in the σpA , A q-closure of ΦA . Hence 0 does not belong to this weak closure, and so ΦA is weakly closed. Proposition 3.5.9 Let A be a reflexive Banach function algebra with |ΦA | ě 2. (i) Take ϕ P ΦA , and suppose that Mϕ has a bounded pointwise approximate identity. Then Mϕ has an identity and ϕ is an isolated point of ΦA . (ii) Suppose that Mϕ has a bounded pointwise approximate identity for each ϕ P ΦA . Then A is finite dimensional. (iii) Suppose that ΦA is connected. Then the algebra A is unital if and only if there exists ϕ P ΦA with ϕ “ 1. Proof (i) Since A is reflexive, the closed balls of Mϕ are weakly compact, and so the BPAI in Mϕ has a weakly convergent subnet, with weak limit f P Mϕ , say. It is clear that f pψq “ 1 pψ P ΦA \ {ϕ}q, and so f is the identity of Mϕ . Since f P C 0 pΦA q, the point ϕ is an isolated point of ΦA . (ii) By (i), ΦA is compact and each point is isolated, and so ΦA is finite. Hence A is finite dimensional.

202

3 Banach function algebras

(iii) Certainly ϕ “ 1 pϕ P ΦA q when A is unital. Now take ϕ P ΦA with ϕ “ 1. Since A is reflexive, it follows from the Hahn– Banach theorem that there exists f P A with  f  “ 1 and f pϕq “ 1. Define K “ co { f n : n P N} , so that K is convex and weakly compact in A with K Ă Ar1s , and gpϕq “ 1 pg P Kq. The map L f | K : K Ñ K is weakly continuous on K, and so, by the Schauder– Tychonoff fixed-point theorem, Theorem 1.3.73, this map has a fixed point: there exists g P K with f g “ g. Clearly hg “ g ph P Kq, and, in particular, g2 “ g, so that g P IpAq. Since gpϕq “ 1 and ΦA is connected, the element g is the identity of A. Also g “ 1, and so A is unital. Proposition 3.5.10 Let A be a pointwise contractive Banach function algebra. (i) Take F and G to be disjoint, non-empty,   finite subsets of ΦA , and take ε > 0. Then there exists f P Ar1s such that 1 ´ f pϕq < ε pϕ P Fq and f pψq “ 0 pψ P Gq. α1 , . . . , αn P D, and take ε > 0. (ii) Let ϕ1 , . . . , ϕn be distinct points  in ΦA , take  Then there exists f P Ar4s such that  f pϕi q ´ αi  < ε pi P Nn q. Proof (i) Set k “ |G|,  and choose  η P p0, ε/kq. For each ψ P G, there exists fψ P Ar1s with fψ pψq “ 0 and 1 ´ fψ pϕq < η pϕ P Fq. Now define # f “ { fψ : ψ P G} . Then clearly f P Ar1s and f pψq “ 0 pψ P Gq. For each ϕ P F, we have      1 ´ f pϕq ď {1 ´ fψ pϕq : ψ P G} < kη < ε , as required. (ii) First suppose that α1 , . . . , αn P r0, 1s, say 0 ď α1 ď · · · ď αn ď 1, and set α0 “ 0. By (i), for each j P Nn , there exists a function f j P A with  f j  ď α j ´ α j´1 , with f j pϕi q “ 0 pi “ 1, . . . , j ´ 1q, and with   ε  f j pϕi q ´ pα j ´ α j´1 q < pi “ j, . . . , nq . 4n   Define f “ f1 + · · · + fn . Then f P Ar1s and  f pϕi q ´ αi  < ε/4 pi P Nn q. Now consider the general case, where α1 , . . . , αn P D. For j P Nn , there exist α1, j , α2, j , α3, j , α4, j P r0, 1s such that α j “ α1, j ´ α 2, j + ipα3, j ´ α4, j q. For each i “ 1, 2, 3, 4, choose fi P Ar1s such that  fi pϕ j q ´ αi, j  < ε/4  p j P Nn q, and then set f “ f1 ´ f2 + ip f3 ´ f4 q, so that f P Ar4s and  f pϕ j q ´ α j  < ε p j P Nn q. Proposition 3.5.11 Let A be a Banach function algebra that is an ideal in its bidual. Suppose that A has a bounded pointwise approximate identity contained in a closed ideal I. Then A “ I, and A has a bounded approximate identity.

3.5 Pointwise approximate identities

203

Proof Let M P I  be a weak-∗ accumulation point of a BPAI in I. Take f P A. Then f · M P A because A is an ideal in A . For each ϕ P ΦA , we have xM, ϕy “ 1, and so x f · M, ϕy “ x f, ϕy. Thus f · M “ M · f “ f . It follows that M is a mixed identity for A, and so A has a BAI. Also A “ A l M Ă I  , and so A “ I. Corollary 3.5.12 Let A be a Tauberian Banach sequence algebra. Then A has a bounded pointwise approximate identity if and only if it has a bounded approximate identity. We conclude this section with two examples, taken from [63, §5], of a pointwise contractive and of a contractive Banach function algebra, respectively, neither of which is a uniform algebra. These examples will be mentioned in Examples 5.1.36 and 5.4.13 and in Example 5.4.16, respectively. Example 3.5.13 Consider the set A of functions f P CpIq such that   1  f ptq ´ f p0q Ip f q :“ dt < 8 . t 0 Clearly A is a self-adjoint, linear subspace of CpIq containing the polynomials, and so A is uniformly dense in CpIq. Note that A contains each f P CpIq with supp f Ă p0, 1s. For f P A, define  f  “ | f |I + Ip f q . Clearly pA,  · q is a normed space; we shall show that it is complete. Indeed, take p fn q to be a Cauchy sequence in pA,  · q. Then p fn q is uniformly convergent, and so there exists f P CpIq such that | fn ´ f |I Ñ 0 as n Ñ 8. Take ε > 0. Then there exists n0 P N such that | fm ´ fn |I + Ip fm ´ fn q < ε pm, n ě n0 q . By Fatou’s lemma, Ip fm ´ f q ď lim inf nÑ8 Ip fm ´ fn q ď ε for each m ě n0 . We see that Ip f q ď Ip fn0 q + ε, and so f P A; further,  fm ´ f  ď 2ε pm ě n0 q, and so p fn q converges to f in pA,  · q. This shows that pA,  · q is a Banach space. Our first claim is that the set A is a subalgebra of CpIq and, further, that we have  f g ď  f  g p f, g P Aq. Indeed, for f, g P A, we have       p f gqptq ´ p f gqp0q ď | f |I gptq ´ gp0q + |g|I  f ptq ´ f p0q pt P Iq, and so Ip f gq ď | f |I Ipgq + |g|I Ip f q, which implies the claim. Also 1I  “ 1, and so pA,  · q is a unital Banach function algebra on I. Set M “ { f P A : f p0q “ 0}, so that M is a maximal ideal in A with Φ M “ p0, 1s, and, further, (3.5.1)  f g ď | f |I g p f P C b pp0, 1sq, g P Mq , so that MpMq “ C b pp0, 1sq and  f op “ | f |I p f P Mq.

204

3 Banach function algebras

Our second claim is that A is natural on I. By Proposition 3.1.4(ii), it suffices to show that each f P A with ZI p f q “ ∅ is invertible in A. We may suppose that f p0q “ 1, say f “ 1 + g, where g P M, and that 1/ f “ 1 + h, where h P CpIq  and hp0q “ 0. Choose δ > 0 such that gptq < 1/2 pt P r0, δsq. Since hptq ď    $δ  2 gptq pt P r0, δsq, we have 0 phptq/tq dt < 8; clearly     1  hptq 1 dt ď |h|I log < 8, t δ δ and so h P M and 1/ f P A, giving the claim. Our third claim is that the norm of A is not equivalent to the uniform norm. To see this, take n P N, and define fn “ nZ on r0, 1/ns and fn “ 1 on r1/n, 1s. Then fn P A and  1  1/n 1 dt “ 2 + log n , n dt +  fn  “ 1 + 0 1/n t whereas | fn |I “ 1. This gives the claim. Our fourth claim is that the algebra of polynomials (restricted to I) is dense in A. Indeed, to see this, it suffices to show that, given f P J0 and ε > 0, there is a polynomial p such that  f ´ p < ε. For this, we first define gptq “ f ptq/t pt P Iq (with gp0q “ 0), so that g P CpIq. There is a polynomial q such that |g ´ q|I < ε/2. Set pptq “ tqptq pt P Iq, so that p is a polynomial; clearly  f ´ p < ε, as required. Our fifth claim is that A is strongly regular. Indeed, take t P I and n P N, and define gn to be the restriction to I of the function which is 0 on the interval rt ´ 1/n, t + 1/ns, which is equal to 1 outside the interval rt ´ 2/n, t + 2/ns, and which is linear on rt ´ 2/n, t ´ 1/ns and rt + 1/n, t + 2/ns, so that pgn q is a sequence in Jt pAq. Now take f P Mt . Then  f ´ f gn  Ñ 0 as n Ñ 8. This is immediate for t > 0. In  the case where t “ 0, fix ε > 0 and take δ > 0 such that  f ptq < ε p0 ď t ď δq and   δ  f ptq dt < ε . t 0 Then, for n > 1/δ, we see that  f ´ f gn  < 2ε. The claim follows. Thus the natural Banach function algebra A is strongly regular. The above sequence pgn q (for t “ 0) is a multiplier-bounded approximate identity for M and a contractive approximate identity for C 0 pp0, 1sq, and so M is a Segal algebra with respect to C 0 pp0, 1sq. For t P p0, 1s, we see that gn  “ 1 + Op1/nq as n Ñ 8, and so the sequence pgn / gn  : n P Nq is a contractive approximate identity r2s for Mt . This implies that Mt “ Mt “ Jt . The maximal ideal M does not have a bounded approximate identity. Indeed, our sixth claim is the slightly stronger fact that M 2 has infinite codimension in M. To  see this, first take f P M 2 , say f “ kj“1 g j h j , where g1 , . . . , gk , h1 , . . . , hk P M, and set k      u“ pg  + h q . j

j“1

j

3.5 Pointwise approximate identities

205

  Then u P M and  f ptq ď uptq2 pt P Iq. We apply this with fα defined by fα ptq “

1 plogp1/tqqα

pt P p0, 1sq ,

with fα p0q “ 0, for α > 0. Then fα P M whenever α > 1. Suppose that fα P M 2 .   Then there exists uα P M such that  fα ptq ď uα ptq2 pt P Iq, and hence uα ptq ě 1/plogp1/tqqα/2 pt P p0, 1sq. This implies that α > 2, and so fα P M \ M 2 for α P p1, 2s. It follows easily that the set { fα + M 2 : α P p1, 2s} is linearly independent in M/M 2 , giving the claim. Our seventh claim is that M does have a contractive pointwise approximate identity and, in fact, that A is strongly pointwise contractive. To see this, first take t P I and take fn,t to be the restriction to I of the function such that fn,t ptq “ 1, such that fn,t “ 0 outside the interval rt ´ 1/n, t + 1/ns, and such that fn,t is linear on rt ´ 1/n, ts and on rt, t + 1/ns. Now suppose that F is a finite subset of I \ {t}, say F “ {t1 , . . . , tk }. Take i P Nn . If ti  0, set gn,i “ fn,ti pn P Nq, and, if ti “ 0, set gn,i “ 1 ´ fn,t pn P Nq. For sufficiently large n P N, we have gn,i P Mt , gn,i pti q “ 1, and supp pgn,i ´ gn,i p1qq Ă p0, 1s and supp pgn,i ´ gn,i p1qq is contained in an interval of length 2/n, so that Ipgn,i q “ Op1/nq as n Ñ 8. Now take ζ1 , . . . , ζn P T, and set  gF,n “ ki“1 ζi gn,i . Then clearly fF,n pti q “ ζi pi P Nn q and  fF,n  “ 1 + Op1/nq as n Ñ 8, and so A is strongly pointwise contractive. Our final claim is that M and, hence, A are Arens regular. For n P N, define In “ r1/pn + 1q, 1/ns, and set An “ CpIn q, with the norm   1/n  f ptq dt p f P An q .  f n “ | f |In + t 1/pn+1q Then An is Banach function algebra on In and An is Banach-algebra isomorphic to pCpIn q, | · |In q, and so An is Arens regular. By Proposition 2.3.40, 1 pAn q is Arens regular. Consider the closed subalgebra of 1 pAn q that consists of the functions p fn q in 1 pAn q such that fn+1 p1/pn + 1qq “ fn p1/pn + 1qq pn P Nq. Then this subalgebra is also Arens regular. But this subalgebra is clearly identified with M, and so M is Arens regular, giving the claim.

Example 3.5.14 We now construct a Banach function algebra A on the unit circle T, but, for notational convenience, we identify CpTq with the subalgebra of Cpr´1, 1sq consisting of the functions f with expp´πi f p´1qq “ exppπi f p1qq. Addition and subtraction in r´1, 1s are taken modulo r´1, 1s. The main point that we shall prove is that A is contractive, but not equivalent to a uniform algebra. We fix a constant α with 1 < α < 2. For t P r´1, 1s, the shift S t f of f P Cpr´1, 1sq by t is defined by pS t f qpsq “ f ps ´ tq ps P r´1, 1sq ;

206

3 Banach function algebras

the oscillation of f is   ω f ptq “ sup{ f psq ´ pS t f qpsq : s P r´1, 1s} and

 Ω f ptq “  f ´ S t f 1 “

1

´1

pt P r´1, 1sq ,

   f psq ´ f ps ´ tq ds pt P r´1, 1sq .

Then we define

 Ip f q “

1 ´1

Ω f ptq dt . |t|α

We note that the function t Þ→ Ω f ptq, r´1, 1s Ñ R+ , is continuous, and so Ip f q is well defined (in r0, 8sq. Also Ω f “ Ω1´ f , and so Ip f q “ Ip1 ´ f q. We now define A to be the space of functions f P CpTq with Ip f q < 8, and set  f  “ | f |T + Ip f q p f P Aq . Certainly A is a self-adjoint, linear subspace of CpTq, and pA,  · q is a normed space. Take p fn q to be a Cauchy sequence in A. Then there exists f P CpTq with fn Ñ f uniformly on T as n Ñ 8. We have Ω fn ptq Ñ Ω f ptq as n Ñ 8 for each t P r´1, 1s, and so it again follows from Fatou’s lemma that fn Ñ f in pA,  · q as n Ñ 8. Thus pA,  · q is a Banach space. We see that S t f P A with S t f  “  f  for each f P A and t P r´1, 1s, and so the algebra A is homogeneous on the circle. We define en psq “ eiπns ps P r´1, 1sq for n P Z, so that lin {en : n P Z} is the space of trigonometric polynomials. Suppose that f P Cpr´1, 1sq with ω f ptq “ Optγ q as t Ñ 0 for some γ > α ´ 1. Then f P A. In particular, suppose that f “ en , where n P Z. Then ω f ptq ∼ t as t Ñ 0, and so f P A because α < 2. It follows that A contains all the trigonometric polynomials, and hence A is uniformly dense in CpTq. Our first claim is that pA,  · q is a unital Banach function algebra on T. Indeed, take f, g P A and t P r´1, 1s. For each s P r´1, 1s, we have       p f g ´ S t p f gqqpsq ď | f |T pg ´ S t gqpsq + |g|T p f ´ S t f qpsq , and so Ω f g ptq ď | f |T Ωg ptq + |g|T Ω f ptq. It follows that Ip f gq ď | f |T Ipgq + |g|T Ip f q and hence

 f g ď  f  g .

Further, 1T  “ 1. The claim follows.  Our  second claim is that A is natural on T. Indeed, take a function f P A with  f ptq ě δ > 0 pt P r´1, 1sq. Then we see easily that Ip1/ f q ď δ2 Ip f q < 8, and so 1/ f P A. Thus A is natural. Our third claim is that A is not equivalent to a uniform algebra. Indeed, for each n P N, we have

3.5 Pointwise approximate identities

 Ωen ptq “

1 ´1

and so  Ipen q “ 2

207

 iπns    e ´ eiπnps´tq  ds “ 2 1 ´ eiπnt 

1

  1 ´ eiπnt 

´1

|t|α

α´1

π

  1 ´ eiu 

π/2

|u|α



dt ě 2pπnq

pt P r´1, 1sq ,

du “ Cnα´1

for a constant C > 0. Thus en  ě Cnα´1 Ñ 8 as n Ñ 8 because α > 1, whereas |en |T “ 1 for each n P N, and so the claim follows. Our fourth claim is that the Banach function algebra A is contractive. Since A is homogeneous on T, it suffices for this to show that the particular maximal ideal M :“ { f P A : f p0q “ 0} has a contractive approximate identity. For this, define En psq “ max {1 ´ n |s| , 0}

ps P r´1, 1s, n P Nq .   Take n P N. Suppose first that |t| ď 1/n. Then En psq ´ En ps ´ tq ď n |t| for |s| ď 2/n and En psq “ En ps ´ tq “ 0 for |s| ě 2/n, and so  2/n ΩEn ptq ď 2 n |t| ds “ 4 |t| . 0

Second, suppose that |t| ě 1/n. Then  ΩEn ptq ď 2

1 ´1

En psq ds “

Hence 

1/n

IpEn q ď 8

t 0

1´α

2 dt + n



1

t

´α

1/n

 dt “ O

2 . n

1 n2´α

 Ñ 0 as

nÑ8

because α < 2, and so 1 ´ En  “ En  “ 1 + op1q as n Ñ 8. In fact, we may suppose that IpEn q ď 1 and En  ď 2 for all n P N. Finally, we claim that p1´En : n P Nq is an approximate identity for the maximal ideal M. Certainly 1 ´ En P M pn P Nq. Now write  δ  1 Ω f ptq Ω f ptq dt , J p f q “ 2 dt p f P Aq Iδ p f q “ δ α |t| |t|α ´δ δ for each δ > 0, so that Ip f q “ Iδ p f q + Jδ p f q p f P Aq. Fix f P M and ε > 0, and then choose δ > 0 such that Iδ p f q < ε and | f |r´δ,δs < ε. Then Iδ p f En q ď |En |T Iδ p f q + | f |r´δ,δs IpEn q < 2ε . Next choose n0 P N such that n0 δ > 1 and

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3 Banach function algebras



1/n0 ´1/n0

   f ptq dt < εδα .

Take n ě n0 . Then | f En |r´1,1s ď | f |r´δ,δs < ε and  Ω f En ptq ď 2

1/n ´1/n

   f psq ds < 2εδα

and so Jδ p f En q < 4εδ

α

 δ

1

pt P r´1, 1sq ,

dt < 4ε . tα

Hence  f En  ď ε + 2ε + 4ε “ 7ε pn ě n0 q. Thus f · p1 ´ En q Ñ f in pA,  · q as n Ñ 8, so that p1 ´ En : n P Nq is indeed an approximate identity for M. We conclude that pp1 ´ En q/ 1 ´ En  : n P Nq is a contractive approximate identity in M, and so A is contractive. By using suitable linear combinations of translates of the functions En , we see that the Banach function algebra A is also strongly pointwise contractive.

3.6 Uniform algebras and Gleason parts The class of uniform algebras was defined in §3.1: a uniform algebra on a nonempty, locally compact space K is a Banach function algebra on K that is a closed subalgebra of pC b pKq, | · |K q. We recall that A is natural on K when every character on A is given by evaluation at a point of K, and then A Ă C 0 pKq. We shall now give some theorems about uniform algebras, and we shall give a number of examples (mostly without proofs) of uniform algebras with diverse properties. We introduced the notion of a peak point and of a strong boundary point for Banach function algebras in §3.1; we shall use these to define various boundaries for a uniform algebra. A natural uniform algebra A on a compact space K is a Cole algebra if each point of K is a strong boundary point for A. Thus CpKq is a Cole algebra, but there are other examples. We shall show that A is a Cole algebra if and only if it is contractive if and only if it has the separating ball property; later, in §5.3, we shall show that non-trivial Cole algebras are not BSE algebras. Let A be a unital uniform algebra. We shall define the Gleason parts of ΦA , and we shall show in Theorem 3.6.32 that A is pointwise contractive if and only if each singleton in ΦA is a one-point Gleason part if and only if A has the weak separating ball property. Thus there are uniform algebras that are pointwise contractive, but not contractive.

3.6 Uniform algebras and Gleason parts

209

For studies of uniform algebras, see the classic texts [28, 124, 127, 277, 295, 296], and also [50, §4.3]. Note that, in most of these texts, a uniform algebra is unital by definition, but that we do not always require this. Let A be a uniform algebra on a locally compact space K. The set of strong boundary points for A is called the Choquet boundary of A, and is denoted by Γ0 pAq. S of K is a boundary for A if, for each f P A, there exists x P S with A subset   f pxq “ | f |K ; for example, Γ0 pAq is a boundary for A [50, Proposition 4.3.4]. A closed subspace L of K is a closed boundary for A if | f |L “ | f |K p f P Aq; the intersection of all the closed boundaries for A is a closed boundary, called the Šilov boundary, ΓpAq [50, Definition 4.3.1(iv)]. Suppose that K is compact and that A is a uniform algebra on K. Then, by [50, Corollary 4.3.7(i)], ΓpAq “ Γ0 pAq. The three clauses in the next proposition are Lemma 7.22, Corollary 7.33, and Theorem 27.3 of [295], respectively. Proposition 3.6.1 Let A be a uniform algebra on a non-empty, compact space K. (i) Suppose that L Ă K is a peak set for A. Then, for each f P A, there exists g P A such that g | L “ f | L and |g|K “ | f |L . (ii) Either Γ0 pAq “ K or Γ0 pAq is uncountable. (iii) Suppose that A is normal. Then A is natural. Definition 3.6.2 Let A be a uniform algebra on a compact space K. A non-empty, closed subspace L of K is an interpolation set for A if the restriction map R : f Þ→ f | L ,

A Ñ CpLq ,

is a surjection. The set L is a peak interpolation set for A if it is an interpolation set that is also a peak set. A countable set S in K is an interpolation sequence if the restriction map R : f Þ→ f | S , A Ñ 8 pS q, is a surjection. For example, by [295, Theorem 20.2], a closed subspace L of the circle T is a peak interpolation set for the disc algebra ApDq (see Example 3.6.10) if and only if L has Lebesgue measure zero. In the case where A has an interpolation sequence, there is a continuous surjection R : A Ñ Cpβ Nq. A remark on interpolation sequences for the algebra H 8 pDq will be made in Example 3.6.38. We shall use the following theorem of Pełczy´nski, given in [295, Theorem 20.17]. Theorem 3.6.3 Let A be a uniform algebra on an uncountable, compact metric space K. Then K contains a peak interpolation set that is homeomorphic to the Cantor set Δ. On the other hand, we have the following results of Rudin from [275] and [295, Corollary 12.8].

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Proposition 3.6.4 Let K be a non-empty, compact space. (i) Suppose that K is scattered. Then the only uniform algebra on K is CpKq. (ii) Suppose that K contains a copy of the Cantor set. Then CpKq contains a proper uniform algebra. Example 3.6.5 Let K be a non-empty, locally compact space. Then the commutative C ∗ -algebra C 0 pKq is Arens regular. Clearly Γ0 pC 0 pKqq “ K. As before, the dual space, C 0 pKq , of C 0 pKq is identified with MpKq. Further, as in Example 3.1.6, the bidual space C 0 pKq “ MpKq of C 0 pKq is a commutative, unital C ∗ r for a certain compact space K r “ pK, r σq, called algebra, and hence C 0 pKq “ CpKq r see [51], the hyper-Stonean envelope of K. For an extensive study of the space K, r r where various ‘constructions’ of K are given. Since CpKq is a von Neumann algebra, r is hyper-Stonean and, as in Theorem 2.2.15, the unique Banach-algebra predual K r is NpKq, r which is linearly homeomorphic to MpKq. As before, we shall of CpKq r and we shall regard points y P K r as elements of identify K with a subset of K, r MpKq by identifying them with the point mass δy at y. r σq, and so, for each In this example, K is exactly the set of isolated points of pK, r We x P K, the unique element of TIMpxq is the characteristic function of x in K. shall see this in a more general setting in Corollary 3.6.18. We recall that MpKq “ Md pKq ‘1 Mc pKq and that we identify Md pKq with 1 pKq. Thus r “ 8 pKq ‘8 Mc pKq “ Cpβ Kd q ‘8 Mc pKq . CpKq r and set K rc “ K r \ β Kd , also a clopen We regard β Kd as a clopen subspace of K, r subspace of K, so that rc q . r “ Cpβ Kd q ‘8 CpK CpKq r onto K8 is denoted by πK . Take a point The natural continuous projection from K x P K8 . Then r K{x} “ π´1 K p{x}q “ {y P K : πK pyq “ x} is the fibre of K at x, as in Equation (2.3.7). Each fibre K{x} is a closed subspace of r and clearly we have K, " r“ K {K{x} : x P K8 } . However, a fibre is not necessarily open. For example, take K “ N. Then clearly K{8} “ N∗ , which is not open in β N. Now suppose for convenience that K is compact, and regard CpKq as canonically r Take x P K and y P K{x} . For each f P CpKq, we embedded in CpKq “ CpKq. have f pyq “ εy p f q “ f pπK pyqq “ f pxq, and so f is constant on each fibre K{x} . r is constant on each fibre K{x} . Then we claim Conversely, suppose that F P CpKq that F | K P CpKq. Indeed, suppose that xν Ñ x0 in K. Then we may suppose r σq, say limν xν “ y0 P K. r Since πK is that pxν q converges in the compact space pK,

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continuous, limν xν “ πK py0 q P K, and so limν Fpxν q “ FpπK py0 qq “ Fpx0 q, and r as the algebra of functions the claim follows. Thus we can identify CpKq in CpKq r that are constant on each fibre in K. r in CpKq

r in terms of ultrafilters, and we We shall later need to understand the space K give the basic definitions here; for details and further results, see the memoir of Dales, Lau, and Strauss [59, Chapter 2]. Note that a positive measure μ on a locally compact space K is not necessarily a member of the Banach space MpKq because it may be that μpKq “ 8. Definition 3.6.6 Let K be a non-empty, locally compact space, and let μ be a positive measure on K. Then the character space of the commutative, unital C ∗ -algebra L8 pK, μq is a compact space that is denoted by Φμ . Thus L8 pK, μq is C ∗ -isomorphic to CpΦμ q; the map that implements this isomorphism is the Gel’fand transform Gμ : L8 pK, μq Ñ CpΦμ q . In the case where L1 pK, μq “ L8 pK, μq, the bidual L1 pK, μq of L1 pK, μq is the dual r space CpΦμ q “ MpΦμ q, and we can regard Φμ as a clopen subspace of K. Let F be a maximal family of pairwise-singular, continuous measures in MpKq+ . Then the compact spaces Φμ for μ P F are pairwise-disjoint, clopen subspaces of rc , and we have the identification K rc q “ Mc pKq “ 8 {L8 pK, μq : μ P F } “ 8 {CpΦμ q : μ P F } . CpK We define

UK “

"

{Φμ : μ P F } .

(3.6.1) (3.6.2)

rc , and that K rc can be identified with β U K . It is clear that U K is dense in K Let μ be a positive measure on K, and take B P BK . Then χB (or, more precisely, the equivalence class rχB s) is an idempotent in L8 pK, μq, and so Gμ pχB q is an idempotent in CpΦμ q; we set KB ∩ Φμ “ {ϕ P Φμ : Gμ pχB qpϕq “ 1} ,

(3.6.3)

a clopen subspace of Φμ . In particular, suppose that B P BK and μpBq “ 0. Then KB ∩ Φμ “ ∅. Clearly the Stone space S tpBμ q of Bμ is homeomorphic to the space Φμ . Indeed, first let p be an ultrafilter in Bμ . Then  {KB ∩ Φμ : B P p} is a singleton in Φμ , and so we can regard p as a point of Φμ . Conversely, each character ϕ P Φμ defines the ultrafilter in BK which is the equivalence class corresponding to the family

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{B P BK : ϕpχB q “ 1} . This family is directed by reverse inclusion, and so defines a net; we write ‘limBÑϕ ’ for convergence along this net. Thus we see that the corresponding net & % μ| B :BÑϕ μB “ μpBq in L8 pK, μqr1s converges weak-∗ to δϕ in MpΦμ q. This net is called the canonical net that converges to δϕ . Specifically, for each λ P L8 pK, μq, we have  1 lim xλ, μB y “ lim λ dμ “ Gμ pλqpϕq . (3.6.4) BÑϕ BÑϕ μpBq B Proposition 3.6.7 Let K be a non-empty, locally compact space, and take x P K r is in the weak-∗ closure of the set and V P N x . Then δψ P MpKq {μC : μ P PrpKq, C Ă Bμ , C Ă V} for each ψ P π´1 K pVq. r Proof Take ψ P π´1 K pVq. Since the subset U K is dense in K, it suffices to suppose ´1 that ψ P πK pVq ∩ U K , and hence that ψ P Φμ for some μ P PrpKq. Thus the result follows from Equation (3.6.4). We shall use the following result. We say that an element λ P L8 pK, μq is continuous at x P K if the equivalence class of λ contains a function which is continuous at x. Proposition 3.6.8 Let K be a non-empty, locally compact space, and take x P K and μ P PrpKq. Suppose that there exists V P N x such that μpUq > 0 for each non-empty, open subspace U of K with U Ă V. Take λ P L8 pK, μq, and suppose that Gμ pλq is constant on Φμ ∩ K{x} . Then λ is continuous at x. Proof We note that the set Φμ ∩ K{x} is not empty because it contains each weak-∗ accumulation point of the net {μB : B P N x }. We may suppose that λ is real-valued. Assume towards a contradiction that λ is not continuous at x. Then there exist α, β P R with α < β such that, setting A “ {x P V : λpxq < α} ,

B “ {x P V : λpxq > β} ,

we have A ∩ B “ ∅ and both A and B meet each neighbourhood of x in a non-empty, open set; by hypothesis, each such intersection has strictly positive μ-measure, and so {A} ∪ N x and {B} ∪ N x are contained in ultrafilters ϕ, ψ P Φμ ∩ K{x} , respectively, with ϕ  ψ. We have Gμ pλqpϕq ď α and Gμ pλqpψq ě β, a contradiction of the fact that Gμ pλq is constant on Φμ ∩ K{x} . Thus λ is continuous at x.

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We shall now give some standard examples of uniform algebras. Example 3.6.9 Let K be a non-empty, compact subspace of C n , where n P N. First consider the algebra consisting of the restrictions to K of all polynomials; the uniform closure in CpKq of this algebra is denoted by PpKq. Next consider the algebra of restrictions to K of all the rational functions defined on a neighbourhood of K, so that these functions have the form p/q, where p and q are polynomials and ZK pqq “ ∅; the uniform closure in CpKq of this algebra is denoted by RpKq. Finally, the algebra consisting of all functions in CpKq that are analytic on int K is denoted by ApKq. Thus PpKq, RpKq, and ApKq are unital uniform algebras on K, and PpKq Ă RpKq Ă ApKq Ă CpKq . The character spaces of PpKq and RpKq are identified with the polynomial and rational convex hulls of K, respectively. In the case where n “ 1, the space ΦPpKq is the union of K and the bounded components of C \ K, and ΦRpKq “ ΦApKq “ K, so that RpKq and ApKq are natural uniform algebras; the fact that ApKq is natural is a theorem of Arens given as [50, Theorem 4.3.14].

Example 3.6.10 We denote by ApDq the disc algebra of all functions f in CpDq such that f | D is analytic. Thus ApDq is a natural uniform algebra on D and ApDq “ PpDq. Set M “ { f P ApDq : f p0q “ 0}, so that Φ M “ D \ {0}. Then, for each z P C with 0 < |z| ď 1, we have εz  “ |z|, and so M does not have norm-one characters. We shall use the function Bα pζq “

p1 ´ αqpζ ´ αq p1 ´ αqp1 ´ αζq

pζ P Dq ,

(3.6.5)

    where  α P D. Note that Bα pζq “ Bα p1q “ 1 when |ζ| “ 1 and that Bα pαq “ 0, and so  Bα pζq < 1 when |ζ| < 1. Let A be a unital uniform algebra on a non-empty, compact space K with  |K| ě 2, and take x, y P K. Suppose that there exists f P Ar1s with f pxq “ 1 and  f pyq < 1. Then there exists g P Ar1s with gpxq “ 1 and gpyq “ 0. Indeed, set α “ f pyq; clearly g :“ Bα ◦ f P Ar1s with gpxq “ 1 and gpyq “ 0. Consider the semigroup algebra A “ p 1 pZ+ q, q. The character space of A is  n identified with D, and elements of A correspond to functions f “ 8 n“0 αn Z in 8 ApDq such that  f 1 “ n“0 |αn | < 8. Thus we can regard A as being the algebra A+ pDq of absolutely convergent Taylor series on D [50, Example 2.1.13(ii)]. Of course, A+ pDq is a natural Banach function algebra on D that is a dense subalgebra of ApDq. The maximal ideal { f P A+ pDq : f p0q “ 0} in A+ pDq corresponds to the semigroup algebra p 1 pNq, q.

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Theorem 3.6.11 Let A be a unital uniform algebra on a compact space K with |K| ě 2. Then A has the separating ball property at x if and only if x is a strong boundary point for A. Proof Suppose that A has the separating ball property at x. Then, by Proposition 3.4.4, x is a strong boundary point for A. Conversely, suppose that x is a strong boundary point for A,  and take y P K with y  x. Then there exists f P Ar1s with f pxq “ 1 and  f pyq < 1. As in Example 3.6.10, we may suppose that f pyq “ 0, and so A has the separating ball property at x. Proposition 3.6.12 Let A be a natural, regular uniform algebra on a locally compact space K, and take x P Γ0 pKq. Then A has the strong separating ball property at x. Proof Let U be an open neighbourhood of x. Since A is regular, there exists g P A with gpxq “ 1 and g | pK \ Uq “ 0. Take n P N, and set   Vn “ {y P K : gpyq < 1 + 1/2n } . Since x is a strong boundary point, there exists hn P A such that hn pxq “ |hn |K “ 1 and |hn g|K\Vn < 1/2n . We define f “

8  1 h g, n n 2 n“1

  so that f P A, f pxq “ 1, and supp f Ă U. We claim that  f pyq ď 1 py P Kq. Indeed, take y P Vk \ Vk+1 , where k P N. Then       k 8    1 1 1 1 1 1 1  f pyq ď · n “ 1´ k 1+ k + < 1, 1+ k + n n 2 2 2 2 2 2 3 · 4k n“1 n“k+1  and the claim is immediate if y P Vn or if y P K \ V1 . Thus the claim holds, and hence | f |K “ 1. This shows that A has the strong separating ball property at x. Let A and x P K satisfy the conditions of the above proposition. It seems to be an open question whether it follows that A is necessarily strongly regular at x. Let A be a natural uniform algebra on a compact space K. Then A is identified r r but in general A is not a uniform algebra on K with a closed subalgebra of CpKq, r In fact, take y, z P K, r and set y ≈ z if because it may not separate the points of K. r and the quotient Fpyq “ Fpzq pF P A q. Then ≈ is an equivalence relation on K, r ≈ is a compact space. Clearly we can regard A as a uniform algebra space L :“ K/ on L. In general, the character space, ΦA , of A is larger than L. We also regard A as a uniform algebra on ΦA and L as a subspace of ΦA ; as in Equation (2.3.6), we have a continuous projection π : ΦA Ñ K. Take x P K, with corresponding maximal ideal M x in A. Then M x is the maximal ideal {F P A : Fpxq “ 0} in A .

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r now, we also consider the Take x P K. Earlier, we defined the fibre K{x} in K; fibre Φ{x} “ π´1 p{x}q in ΦA , as in Equation (2.3.7). For a study of the uniform algebra A for a uniform algebra A, see [44], where the following result is included. Proposition 3.6.13 Let A be a natural uniform algebra on a non-empty, compact space K. Then: (i) A consists of the functions in A that are constant on each fibre in ΦA ; r (ii) A + CpKq is a closed linear subspace of CpKq; (iii) πpΓpA qq “ ΓpAq. Proof (i) Let f P A. Then, as in Example 3.6.5, f P A is constant on each fibre. Now suppose that F P A and that F is constant on each fibre in ΦA . As an r the function F is constant on each fibre in K, r and so, as in Example element of CpKq, r 3.6.5, F P CpKq, regarded as a closed subalgebra of CpKq. By Equation (1.3.4) (with E “ CpKq and F “ A), we have CpKq ∩ A “ A, and so F P A. (ii) This follows from Proposition 1.3.20. (iii) Set Γ “ ΓpAq. The embedding of A into CpΓq is an isometry, and so the r is an isometry. Take ϕ P ΓpA q. Then ϕ embedding of A into CpΓq “ CpΓq r and ψ gives a character on CpΓq. r corresponds to evaluation at a point, say ψ, in Γ, Then ψ | CpΓq is given by evaluation at a point, say x, in Γ. Thus ϕ | A “ ε x , and so πpϕq “ x P Γ. Thus πpΓpA qq Ă ΓpAq. On the other hand, take x P Γ0 pAq. For each U P N x , there exists ϕU P Γ0 pA q with πpϕU q P U. The set {ϕU : U P N x } has an accumulation point, say ϕ, in ΓpA q, and clearly πpϕq “ x, and so x P πpΓpA qq. Thus ΓpAq “ Γ0 pAq Ă πpΓpA qq. r to be a subConditions on A for the closed linear subspace A + CpKq of CpKq r are given in [44, Theorem 4.3]. algebra of CpKq Let A be a unital uniform algebra on a compact space K. Then we defined in Definition 2.1.16 the state space of A as KA “ {λ P A : λ “ x1K , λy “ 1} ; the elements of ex KA are the pure states. By the Hahn–Banach theorem and the Riesz representation theorem, for each λ P KA there exists a probability measure μ on K such that  f dμ p f P Aq . x f, λy “ K

Such a measure μ is a representing measure for λ on K. In particular, for each x P K, the element ε x has the representing measure δ x ; in general, ε x also has other representing measures. Proposition 3.6.14 Let A be a unital uniform algebra on a non-empty, compact space K.

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(i) Take λ P ex KA . Then there exists x P K such that δ x is the unique representing measure for λ, and λ “ ε x . (ii) Take x P K such that δ x is the unique representing measure for ε x . Then ε x P ex KA . (iii) Take x P Γ0 pKq. Then ε x P ex KA . Proof (i) Let μ be a representing measure for λ on K. For each Borel set B P BK such that μpBq  {0, 1}, we have λ “ μpBqλ1 + μpK \ Bqλ2 , where   1 1 x f, λ1 y “ f dμ and x f, λ2 y “ f dμ μpBq B μpK \ Bq K\B for f P A. Since λ1 , λ2 P KA and λ P ex KA , it follows that λ “ λ1 “ λ2 , and hence   f dμ “ μpBq f dμ p f P Aq . B

K

This implies that each f P A is constant almost everywhere with respect to μ, and this can only happen if μ “ δ x for some x P K, so that δ x is the unique representing measure for λ. It follows that λ “ ε x . (ii) Suppose that ε x “ pλ1 +λ2 q/2 for λ1 , λ2 P KA . Take representing measures μ1 and μ2 on K for λ1 and λ2 , respectively. Then 2δ x “ μ1 + μ2 , and so μ1 “ μ2 “ δ x . This shows that ε x P ex KA . (iii) Let μ be a representing measure for ε x . Take U P N x and η > 0. Then there exists f P A with f pxq “ | f |K “ 1 and | f |K\U < η. We have   f dμ ď 1 “ f pxq “ | f | dμ ď μpUq + η , K

K

and so μpUq “ 1. It follows that μ “ δ x , and so ε x P ex KA by (ii). Corollary 3.6.15 Let K be a non-empty, compact space. Then ex KCpKq “ K. Proof Take λ P ex KCpKq . Then λ “ ε x for some x P K, by (i), above; by (iii), each ε x belongs to ex KCpKq . Theorem 3.6.16 Let A be a natural uniform algebra on a compact space K with |K| ě 2, and take x P K. Then the following conditions on x are equivalent: (a) ε x P ex KA ; (b) x P Γ0 pAq ; (c) A has the separating ball property at x; (d) M x has a bounded approximate identity; (e) M x has an identity; (f) M x has a contractive approximate identity; (g) x is an isolated point of ΦA .

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Proof Clause (b) implies (a) by Proposition 3.6.14(iii), and the converse follows by [50, Theorem 4.3.5, (a) ⇒ (d)]. Clauses (b) and (c) are equivalent by Theorem 3.6.11. Suppose that (b) holds. For U P N x , there exists fU P A with fU pxq “ | fU |K “ 1 and | fU |K\U < 1, and so the set {1K ´ fUn : U P N x , n P N} is contained in pM x qr2s . Take h P M x and ε > 0. There exists U P N x with |h|U < ε, and there exists n P N with  fUn hK < ε. By Proposition 2.1.22(ii), M x has a BAI, and so (d) follows. Also (d) implies (b) by [50, Theorem 4.3.5, (e) ⇒ (d)]. Clauses (d) and (e) are equivalent by Proposition 2.3.66(ii), where we note that a mixed identity in M x is an identity because M x is Arens regular. Trivially (f) implies (d). Now suppose that (e) holds, say E is the identity of M x . Since E is an idempotent in A , necessarily |E|ΦA “ 1. By Proposition 2.3.66(ii) again, M x has a CAI, giving (f). Certainly (e) implies (g), and the converse follows from Šilov’s idempotent theorem for the algebra A . Thus, by the implication (b) ⇒ (d) of the above theorem and Theorem 2.1.46, M x factors whenever x is a strong boundary point for a unital uniform algebra A. We shall see in Example 3.6.38 that the converse to this is false, but it seems to be an open question whether the converse is false when the uniform algebra A is separable. For a study of factorization in commutative Banach algebras, see [55]. Corollary 3.6.17 Let A be a natural uniform algebra on a compact space K. Then A is weakly sequentially complete if and only if K is finite. Proof Certainly A is finite dimensional, and hence weakly sequentially complete, when K is finite. Suppose that A is weakly sequentially complete, and assume towards a contradiction that K is infinite, so that A is infinite dimensional. By passing to a closed subalgebra, we may suppose that A is separable. Take x P Γ0 pAq. Then M x is Arens regular, and, by Theorem 3.6.16, M x has a BAI. By Theorem 2.3.46, M x has an identity. So x is an isolated point of K, and the characteristic function, χ x , of {x} belongs   to A by Šilov’s idempotent theorem. Since χ x ´ χy K “ 1 when x, y P Γ0 pAq with x  y and A is separable, it follows that Γ0 pAq is countable. By Proposition 3.6.1(ii), K “ Γ0 pAq, and so every point of K is isolated in K. Thus K is finite, and so A is finite dimensional, a contradiction. Corollary 3.6.18 Let A be a natural uniform algebra on a compact space K such that |K| ě 2, and take x P K. Then   TIMpxq “ 1 px P Γ0 pAqq and TIMpxq “ ∅ px P K \ Γ0 pAqq . Proof Suppose that x P Γ0 pAq, so that M x has a BAI. By Proposition 3.4.15(ii), TIMpxq   ∅. Since A is Arens regular, it follows from Proposition 3.4.14(i) that TIMpxq “ 1.

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Now suppose that TIMpxq  ∅. Then essentially as in Proposition 3.4.15(i), M x has a BAI, and so x P Γ0 pAq by Theorem 3.6.16, (d) ⇒ (b). Definition 3.6.19 Let A be a natural uniform algebra on a non-empty, compact space K. Then A is a Cole algebra if every point of K is a strong boundary point, so that Γ0 pAq “ K. The reason for this terminology is the following. In the case where K is compact and metrizable, a natural uniform algebra on K is a Cole algebra if and only if every point of K is a peak point. It was a long-standing conjecture, called the ‘peakpoint conjecture’, that CpKq is the only Cole algebra on a compact, metrizable space K. The first counter-example was due to Cole [43], and is described in [295, §19]; an example of Basener [14], also described in [295, Example 19.8], gives a compact space K in C 2 such that the uniform algebra RpKq is a Cole algebra, but RpKq  CpKq. Also there is an example of a compact, polynomially convex space K in C 2 of Cole, Ghosh, and Izzo [45, Theorem 1.2] such that PpKq is a Cole algebra, but PpKq  CpKq. For a survey and history of the peak-point conjecture, see the article by Izzo [183]. Let A be a natural uniform algebra on the closed unit interval I. Then it is a very famous and formidable question of Gel’fand whether A is necessarily equal to CpIq. It is a consequence of Rossi’s local maximum modulus theorem [295, Corollary 9.14] that Γ0 pAq is dense in I. However, it seems to be unknown whether every such algebra is necessarily a Cole algebra, and also unknown whether every Cole algebra on I is necessarily trivial. It is a result of Wilken [323] that strongly regular uniform algebras on I are trivial. This is discussed in [296]. The following theorem is immediate from Theorem 3.6.16. Theorem 3.6.20 Let A be a natural uniform algebra on a compact space K such that |K| ě 2. Then the following conditions are equivalent: (a) A is a Cole algebra; (b) A has the separating ball property; (c) A is contractive; (d) each point of K is an isolated point of ΦA . Thus there are non-trivial, contractive uniform algebras. Definition 3.6.21 Let A be a natural uniform algebra on a non-empty, compact space K, and take x, y P K. Then dA px, yq < 2 ,   where dA is the Gleason metric given by dA px, yq “ ε x ´ εy  px, y P Kq. x∼y

if

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Then, as we shall see in Corollary 3.6.24, ∼ is an equivalence relation on K; the equivalence classes with respect to this relation are the Gleason parts for A. These parts form a partition of K, and each part is a completely regular and σ-compact topological space with respect to the Gel’fand topology; by a theorem of Garnett, these are the only topological restrictions on Gleason parts. For a discussion of Gleason parts, see [124, Chapter VI] and [295, §18], where Garnett’s theorem is proved as Theorem 18.5. Clearly {x} is a one-point Gleason part whenever x is a strong boundary point, but the converse fails, as we shall see in Examples 3.6.37, 3.6.38, and 3.6.42. Example 3.6.22 Consider the disc algebra ApDq. Each point z P T is a peak point for ApDq, and so {z} is a one-point part and r2s Mz “ Mz . Now take z P D. Then r2s

Mz “ Mz2 “ Mz2 “ { f P Mz : f  pzq “ 0}  Mz , and so Mz has no approximate identity. By the Schwarz–Pick theorem,   |w ´ z|  f pwq ď |f| |1 ´ z w| D

pw P Dq

(3.6.6)

for f P Mz , and so w ∼ z. It follows that the open disc D is a single Gleason part for ApDq.

The next proposition is standard [295, Lemma 16.1]. Proposition 3.6.23 Let A be a natural uniform algebra on a non-empty, compact space K, and take x, y P K. Then the following are equivalent:   (a) ε x ´ εy  “ 2 ; (b) there exists F P Ar1s with Fpxq “ 1 and Fpyq “ 0 ;   (c) for each ε > 0, there exists f P pM q with  f pxq > 1 ´ ε. y r1s

  Proof (a) ⇒ (b) There exists G P Ar1s with xG, ε x ´ εy y “ 2, and we may suppose that Gpxq “ 1, so that Gpyq “ ´1. Set F “ p1 +Gq/2. Then F P Ar1s with Fpxq “ 1 and Fpyq “ 0, and so (b) holds. (b) ⇒ (a) Take F as specified, and take n P N. Set αn “ 1 ´ 1/n, and then define Fn :“ Bαn ◦ F, where Bα was defined in Equation (3.6.5), so that Fn P Ar1s ,     ε x ´ εy  ě Fn pxq ´ Fn pyq “ 2 ´ 1/n. ´1 + 1/n. Thus Fn pxq “ 1, and Fn pyq “  It follows that ε x ´ εy  “ 2, giving (a). (b) ⇔ (c) This is immediate.

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Let A be a natural uniform algebra on a non-empty, compact space K. It follows that a one-point part in K is an isolated point with respect to the Gleason metric, but we shall see in Example 3.6.41 that the converse is not necessarily true. Corollary 3.6.24 Let A be a natural uniform algebra on a non-empty, compact space K. Then ∼ is an equivalence relation on K. Take x P K. Then A has the weak separating ball property at x if and only if {x} is a one-point part. Proof Clearly x ∼ x for x P K and y ∼ x whenever x ∼ y for x, y P K. Now take x, y, z P K with x ∼ y and y ∼ z, and assume that x  z. By Proposition there  3.6.23,  is F P Ar1s with Fpxq “ 1 and Fpzq “ 0. Since y ∼ z, necessarily Fpyq < 1. As in Example 3.6.10, we may suppose that Fpyq “ 0, and so x  y, a contradiction. Thus x ∼ z. We have shown that ∼ is an equivalence relation. Take x P K. It follows from Proposition 3.6.23, (a) ⇔ (c), that A has the weak separating ball property at x if and only if {x} is a one-point part. The following result is based on [28, Theorem 1.6.2]. Here π x is the bounded p M x Ñ M x such that π x p f ⊗ gq “ f g p f, g P M x q. linear map π x : M x ⊗ Theorem 3.6.25 Let A be a natural uniform algebra on a non-empty, compact space K, and take x P K. Suppose that the map π x is a surjection. Then x is an isolated point with respect to the Gleason metric on K. Proof We may suppose that |K| ě 2. p M x Ñ M x is a surjection, and so, by the open mapping The map π x : M x ⊗ theorem, Theorem 1.3.5(ii), there exists m > 0 such that, for each f P pM x qr1s , there exist f j , g j P pM x qr1s and α j > 0 for j P N such that f ´ f pxq “

8  j“1

α j f jg j “

8 

α j p f j ´ f j pxqqpg j ´ g j pxqq

(3.6.7)

j“1

 and 8 j“1 α j ď m. Now  take y P K \ {x} andε > 0. Then there exists a function f P pM x qr1s such that  f pyq ´ f pxq > εy ´ ε x  ´ ε. Using Equation (3.6.7), we have  2   εy ´ ε x  ´ ε ď m εy ´ ε x  ,   and so εy ´ ε x  ě 1/m. This shows that x is an isolated point with respect to the Gleason metric on K. We now give a result for uniform algebras that is close to Theorem 3.4.5; it is taken from [64, Theorem 4.17]. In the theorem, χP is the characteristic function of a part P. Theorem 3.6.26 Let A be a natural, uniform algebra on a non-empty, compact space K.

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(i) Let P be a Gleason part for A, and take x0 P P. Then there is an idempotent element E P Ar1s such that E | K “ χP and such that F E “ E whenever F P Ar1s with Fpx0 q “ 1. (ii) For each Gleason part P, there is an idempotent E P P Ar1s with E P | K “ χP and such that E P E Q “ 0 in A whenever P and Q are distinct parts. Proof (i) As in Example 3.1.6, the uniform algebra A is a dual Banach  function  algebra with Banach-algebra predual A . Further, ε x0 belongs to A with ε x0  “ 1, and so we can apply Theorem 3.4.10 with A replaced by A and F by A to conclude that there exists an idempotent element E P A such that Epx0 q “ 1 and F E “ E when F P Ar1s with Fpx0 q “ 1. Since E is an idempotent element in a uniform algebra, necessarily E P Ar1s . Suppose that y P K \ P. By Proposition 3.6.23, there exists F P Ar1s such that Fpx0 q “ 1 and Fpyq “ 0, and so Epyq “ 0. Suppose that y P K and Epyq “ 0. By Proposition 3.6.23, x0  y, and so y P K \ P. Thus E | K “ χP . (ii) Choose xP P P for each part P, so that there exists an idempotent element E P P Ar1s such that E P | K “ χP and such that F E P “ E P whenever F P Ar1s with FpxP q “ 1. Take distinct parts P and Q, and set F “ 1´E P . Then F is an idempotent in A , and so F P Ar1s . Also FpyQ q “ 1, and so p1 ´ E P q E Q “ FE Q “ E Q , and hence E P E Q “ 0. It the following corollary, we again regard K as a subset of ΦA , with closure K. Corollary 3.6.27 Let A be a natural uniform algebra on a non-empty, compact space K, and take x P K such that {x} is a one-point part. Then x is an isolated point of K. Suppose that each point of K is a one-point part. Then K is discrete with respect to the relative weak topology. Proof By Theorem 3.6.26(i), there exists E x P A such that E x | K is the characteristic function of {x} on K. Clearly E x | K is the characteristic function of {x} on K, and so x is an isolated point of K. It follows that K is weakly discrete when each point of K is a one-point part. Let A be as above, and suppose that there exists x P K such that {x} is a onepoint part, but x  Γ0 pAq. Then the corollary shows that x is an isolated point of K. However, by Theorem 3.6.16, (g) ⇒ (b), x is not an isolated point of ΦA , and so K is not an open subspace of ΦA . Let E x be the corresponding idempotent in A , and set T x “ {ϕ P ΦA : E x pϕq “ 1} . Since T x is a peak set for A , there exists ϕ P T x ∩Γ0 pA q. By Proposition 3.6.13(iii), πpϕq P ΓpAq. The part with respect to A that contains x is contained in T x ∩ Φ{x} , where Φ{x} is the fibre in ΦA above x in ΦA ; the description of this part seems to be unclear.

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Corollary 3.6.28 Let A be a natural uniform algebra on a non-empty, compact space K, and choose ζP P D for each part P of ΦA . Then there exists F P Ar1s such that F | P takes the constant value ζP for each part P. Proof Let S “ {P1 , . . . , Pn } be a finite set of pairwise-disjoint parts. Then, by Theorem 3.6.26, for each j P Nn , there is an idempotent E j P A with E j | K “ χP j and such that Ei E j “ 0 for i, j P Nn with i  j. Set FS “

n 

ζP j E j .

j“1

Then FS P Ar1s and FS | P “ ζP χP pP P Sq. The sets of the form S form a directed set with respect to inclusion, and so pFS q is net in Ar1s ; take F P Ar1s to be a weak-∗ accumulation point of this net. Clearly F has the required properties. Theorem 3.6.29 Let A be a natural uniform algebra on a compact space K such that |K| ě 2, and take x P K. Then the following are equivalent: (a) the singleton {x} is a one-point Gleason part; (b) M x has norm-one characters; (c) M x has a contractive pointwise approximate identity; (d) A has the weak separating ball property at x; (e) A contains an idempotent E x such that E x pyq “ δ x,y py P Kq. Proof Clearly (a) and (b) are equivalent and (c) implies (a). The equivalence of (a) and (d) is given in Corollary 3.6.24, and (a) implies (e) by Theorem 3.6.26(i). Now suppose that (e) holds, and set F x “ 1 ´ E x . Then F x is an idempotent in M x with |F x |ΦA “ 1, and so there is a net p fα q in pM x qr1s that converges pointwise on K to F x , giving (c). Corollary 3.6.30 Let A be a uniform algebra that has the weak separating ball property. Then A also has the weak separating ball property. Proof Take ϕ P ΦA . Then certainly A has the WSBP at ϕ, and so each point of ΦA is a one-point part in ΦA . Thus 8 is a one-point part in ΦA , and so, by Theorem 3.6.29, (a) ⇒ (d), A has the WSBP at 8. Hence A also has the WSBP. Theorem 3.6.31 Let A be a natural, pointwise contractive uniform algebra on a non-empty, locally compact space K. Then, for each f P 8 pKqr1s , there exists F P Ar1s such that Fpxq “ f pxq px P Kq, and so A is strongly pointwise contractive.

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223

Proof Set B “ A , so that B is a natural, unital uniform algebra on K8 ; we may suppose that f P 8 pK8 qr1s with f p8q “ 0. Clearly each singleton in K8 is a one-point part, and so the existence of F P Br1s such that F | K8 “ f follows from Corollary 3.6.28. Then we have F P Ar1s with F | K “ f . Take n P N, ε > 0, distinct points x1 , . . . , xn P K, and ζ1 , . . . , ζn P D. Then there is F P Ar1s with Fpx j q “ ζ j p j P Nn q, and so there is f P Ar1s such that    f px j q ´ ζ j  < ε p j P Nn q. This shows that A is strongly pointwise contractive. The following immediate consequence of Theorems 3.6.29 and 3.6.31 is a main result of this section. Theorem 3.6.32 Let A be a natural uniform algebra on a compact space K such that |K| ě 2. Then the following conditions on A are equivalent: (a) A is pointwise contractive; (b) A is strongly pointwise contractive; (c) A has the weak separating ball property; (d) M x has norm-one characters for each x P K; (e) each singleton in K is a one-point Gleason part. Example 3.6.33 We give an example of a contractive Banach function algebra that is equivalent to a uniform algebra, but which is not strongly pointwise contractive. Indeed, consider the space C 2 , and define B to be the absolutely convex hull of the three points p1, 0q, p0, 1q, and p1, 1q in C 2 . Then B is the unit ball of a norm,  · , on C 2 . Take z “ α1 p1, 0q + α2 p0, 1q + α3 p1, 1q and w “ β1 p1, 0q + β2 p0, 1q + β3 p1, 1q in   B, where 3i“1 |αi | ď 1 and 3i“1 |βi | ď 1. Then zw “ γ1 p1, 0q + γ2 p0, 1q + γ3 p1, 1q , where γ1 “ α1 β1 + α1 β3 + α3 β1 , γ2 “ α2 β2 + α2 β3 + α3 β2 and γ2 “ α3 β3 . Thus |γ1 | + |γ2 | + |γ3 | ď p|α1 | + |α2 | + |α3 |qp|β1 | + |β2 | + |β3 |q ď 1 , and so zw P B. This shows that A :“ pC 2 ,  · q is a Banach sequence algebra on a two-point set, S . Of course, the norm is equivalent to the uniform norm. Indeed, it is clear that |z|S ď z ď 2 |z|S pz P Aq. For the identity p1,1q of A andthe identities p1, 0q and p0, 1q of the two maximal p1, 1q “ p1, 0q “ p0, 1q “ 1, and so A is contractive. ideals of A, we have   However, p1, ´1q “ 2, and so A is not strongly pointwise contractive.

We now present various uniform algebras; we shall note in particular when their maximal modular ideals have contractive, or bounded, approximate identities.

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Example 3.6.34 Let A “ ApDq be the disc algebra, as in Example 3.6.22, and take z P D. It follows from Theorems 3.6.16 and 3.6.29 that Mz has a bounded pointwise approximate identity if and only if Mz has a contractive pointwise approximate identity if and only if z is a peak point for A if and only if Mz has a contractive approximate identity if and only if z P T. Let A+ pDq – p 1 pZ+ q, q be as in Example 3.6.10. Then the maximal ideal { f P A+ pDq : f p0q “ 0} clearly does not have a bounded pointwise approximate identity.

Example 3.6.35 Let A be a natural uniform algebra on a compact set K, and take x P K. It is also possible to have x P ΓpAq, but such that M x does not have a bounded pointwise approximate identity. Indeed, set K “ D × I, and take A to be the tomato can algebra [295, Example 7.8], so that A is the uniform algebra of all continuous functions f on D × I such that the function z Þ→ f pz, 1q, D Ñ C, belongs to ApDq. Then Γ0 pAq “ {pz, tq P K : 0 ď t < 1} ∪ {pz, 1q P K : z P T} and ΓpAq “ K. The set K \ Γ0 pAq “ {pz, 1q : z P D} is a Gleason part with respect to A, and again we see that, for x P K, the maximal ideal M x has a bounded pointwise approximate identity if and only if M x has a contractive pointwise approximate identity if and only if x is a peak point; if x P K \ Γ0 pAq, then M x2 “ M x2  M x , and so M x does not have an approximate identity.

Example 3.6.36 Let K be a non-empty, compact plane set, and consider the natural uniform algebra RpKq on K. For example, K can be a Swiss cheese [50, p. 455]; this is a compact set K in C formed by deleting a sequence pDn q of open discs with  pairwise-disjoint closures in D such that 8 n“1 rn < 1, where rn is the radius of Dn , and such that int K “ ∅. In this case, RpKq  ApKq “ CpKq. Take x P K. Then {x} is a one-point Gleason part if and only if x is a peak point [295, Corollary 26.14]. By [295, Corollary 26.15], RpKq “ CpKq if and only if each singleton in K is a one-point Gleason part. By [295, Corollary 26.13] each Gleason part for RpKq that is not a singleton has positive plane area. It follows that the following conditions on x P K with respect to the uniform algebra RpKq are equivalent: (a) x is a peak point; (b) {x} is a one-point part; (c) x is an isolated point of K with respect to the Gleason metric; (d) M x has a bounded approximate identity; (e) M x has a contractive approximate identity; (f) M x has a contractive pointwise approximate identity; (g) M x factors; (h) M x factors weakly. Here the implication (h) ⇒ (b) follows from Theorem 3.6.25. It follows that RpKq “ CpKq if and only if RpKq is pointwise contractive. A famous example of McKissick [237], given as [295, Theorem 27.4], shows that there are compact plane sets K such that RpKq is normal, but non-trivial; for a ‘classical Swiss cheese’ with this property, see [106]. An easier example is given by

3.6 Uniform algebras and Gleason parts

225

Körner in [193]. An example of Izzo gives a Swiss cheese K for which RpKq has only one non-trivial Gleason part [184, Theorem 1.5]. An example of Wang [321] shows that there is a Swiss cheese K and x P K such that x is not a peak point, but such that RpKq is strongly regular at x. The question whether RpKq is strongly regular at x whenever RpKq is regular and x is a peak point seems to be open.

Example 3.6.37 There are natural, separable uniform algebras on a compact space K that have one-point parts {x} for some x P K\ΓpAq. For such points x, the maximal ideal M x has a contractive pointwise approximate identity, but no bounded approximate identity. For example, the uniform algebra Aα that is the big disc algebra, described in [295, Theorem 18.1], has this property; see also [55, Proposition 2.5]. For substantial generalizations of this example, see [153]. It was shown by Cole that there is a natural, separable uniform algebra on a compact metric space K such that each point of K is a one-point Gleason part, so that A is pointwise contractive, but ΓpAq  K. Thus A is not a Cole algebra, and hence A is not contractive. It follows that some maximal ideals of A are pointwise contractive, but without a bounded approximate identity. It was proved in [54, Theorem 2.3] that there is such an example such that the invertibles are dense in the uniform algebra. In the paper of Cole, Ghosh, and Izzo [45, Theorem 1.1] it was shown that the uniform algebra can be taken to be PpKq for a certain compact set K Ă C 3 . Additional examples of uniform algebras such that the invertibles are dense but having large Gleason parts were given by Izzo in [184].

Let A be a natural uniform algebra on a non-empty, compact space K. A subset S of K is an analytic disc if there is a continuous bijection η : D Ñ S such that f ◦ η P OpDq for each f P A, and, in this case, a function f on S is analytic on S (with respect to A) if f ◦ η P OpDq. It is not required that η be a homeomorphism, and there are natural examples where it cannot be one; see [127, Chapter X]. Clearly each analytic disc is contained in a single Gleason part for A. For an extension of Garnett’s theorem, see an article of Izzo and Papathanasiou [185, Theorem 1.2], which shows that each completely regular and σ-compact space occurs in a character space that contains no analytic discs. Example 3.6.38 Let H 8 be the uniform algebra of all bounded analytic functions on D, so that H 8 is non-separable. The (large) character space of H 8 is denoted by Φ; it is studied in [173, Chapter 10] and [127, Chapters V and X]. The famous Carleson’s corona theorem [127, Chapter VIII, §2] states that D is dense in Φ. Each point of Γ “ ΓpH 8 q is a strong boundary point, and hence a one-point Gleason part. In fact, each Gleason part for H 8 is either a one-point part or an analytic disc [127, Theorem X.1.7] and there are one-point Gleason parts that are not in Γ. Suppose

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3 Banach function algebras

that x P Φ and {x} is a one-point part. Then M x factors [127, Chapter X, Theorem 2.4]. It is easy to see that the disc D contains interpolation sequences pzn q for H 8 such that limnÑ8 zn “ 1; a characterization of sequences in D that are interpolation sequences for H 8 is given in [127, Chapter VII, §1] and in [173, Chapter 10]. The space L8 pTq consists of the essentially bounded, measurable functions on T, so that L8 pTq is a unital, commutative C ∗ -algebra, and hence has the form CpXq for a certain compact space X; see [173, Chapter 10]. By Fatou’s lemma, H 8 is identified with a closed subalgebra of L8 pTq. The space X can be regarded as a closed subspace of Φ, and indeed X is identified with Γ [127, Chapter V, Theorem 1.7], so that H 8 can be regarded as a uniform algebra on X.

Example 3.6.39 We denote by A “ H 8 + C the set of functions on the above space X “ ΓpH 8 q of the form f + g, where f P H 8 and g P CpTq Ă L8 pTq – CpXq . The somewhat surprising fact that A is a closed subalgebra of CpXq is due to Sarason [289], and it is also proved in [127, Theorem IX.2.2]. More general uniform algebras, called Douglas algebras, are also considered in [127, Chapter IX]. The character space of A is the compact space ΦA “ Φ \ D. Take ζ P T. Then the fibre of Φ above the point ζ is denoted by Φζ . We claim that the Gleason parts for A are the same as the parts of H 8 in ΦA . First note that two points in distinct fibres are in different parts for H 8 , and that two points in ΦA that are in different parts for H 8 are certainly in different parts for A. Now suppose that ϕ and ψ are two points in the same fibre, say Φζ , that are in different parts with respect to A. Then, for each ε > 0, there exists h P Ar1s   with hpϕq ´ hpψq > 2 ´ ε, say h “ f + g, where f P H 8 and g P CpXq. Since g is constant on Φζ , we may replace g by this constant, and so obtain a function h1 P H 8 such that h1 | Φζ “ h | Φζ . Since Φζ is a peak set for H 8 , it follows 8 from Proposition 3.6.1(i) that there exists  h2 P H with  h2 | Φζ “ h1 | Φζ and 8 |h2 |Φ “ |h1 |Φζ ď 1. Thus h2 P Hr1s and h2 pϕq ´ h2 pψq > 2 ´ ε. This shows that ϕ and ψ are in distinct parts with respect to H 8 , giving the claim. For a further remark on this uniform algebra, see Example 5.3.11.

Example 3.6.40 Let A be a uniform algebra that is a Cole algebra. In the case where A is also regular, it follows from Proposition 3.6.12 that A has the strong separating ball property. An example of Feinstein [100, Theorem 3.6] shows that there is a nontrivial Cole algebra A on a compact metric space K (so that A is separable) and such that A is strongly regular on K, and so this example also has the strong separating ball property. Further, Feinstein [103, Corollary 8] gave an example of a non-trivial

3.6 Uniform algebras and Gleason parts

227

Cole algebra on a compact metric space that is not regular, and so is not strongly regular.

Example 3.6.41 In [101, Theorem 2.1], Feinstein constructed a separable, regular, natural uniform algebra A on a compact space K such that there is a two-point part, say P “ {x, y} with d :“ dpx, yq P p0, 2q, and such that all other points of K are one-point Gleason parts. Thus each point of K is an isolated point with respect to the Gleason metric.   There exists f x P Ar1s with  f x pxq ´ f x pyq > d/2. Define hx “

f x ´ f x pyq . f x pxq ´ f x pyq

Then h x P Arms , where m “ 4/d, h x pxq “ 1, and h x pyq “ 0. By Corollary 3.6.28, there exists F P Ar1s with F | K “ χP , and then h x F P Arms with h x F | K “ χ x . Thus 1K ´ h x F is the identity of M x , and so M x has a bounded pointwise approximate identity (with bound m + 1). Similarly, My has a bounded pointwise approximate identity, and each other maximal ideal of A has a contractive pointwise approximate identity. Thus, in this example, each maximal ideal has a bounded pointwise approximate identity (with a uniform bound), but the algebra is not pointwise contractive. This example will be mentioned again in Examples 5.1.20 and 5.3.15.

Example 3.6.42 In Examples 5.13 and 5.16 of [294], Sidney constructed natural uniform algebras A on compact spaces K and points x P K \ ΓpAq such that {x} is a one-point Gleason part, so that M x has a contractive pointwise approximate identity, but such that M x2 is not even dense in M x ; in particular, M x does not factor and M x does not have an approximate identity. In [135], Ghosh and Izzo show that there is a normal (and hence natural), separable uniform algebra A on a compact space K and x P K such that each point of K is a one-point Gleason part, and hence A is pointwise contractive, but again M x2 is not dense in M x , and, in particular, M x does not have approximate units. It is also shown in [135] that there is a natural, strongly regular, normal uniform algebra with a maximal ideal M x at a point x such that {x} is not a one-point Gleason part, but such that each maximal ideal factors.

Chapter 4

Banach algebras on locally compact groups

In this chapter, we shall consider the classical Banach algebras of harmonic analysis that are associated with a locally compact group G. The class of these algebras includes the group algebra L1 pGq, the measure algebra MpGq, the related algebras L p pGq (which are Banach algebras when G is compact and 1 ď p ď 8), and Beurling algebras on semigroups and locally compact groups. The product in each of these algebras is given by convolution, denoted by  . For example, pMpGq,  q is an isometric dual Banach algebra, with Banach-algebra predual C 0 pGq, for each locally compact group G. The definitions and some basic properties of these algebras will be recalled in §4.1. The case where G is abelian, and so L1 pGq and MpGq are identified by using the Fourier and Fourier–Stieltjes transforms with Banach function algebras ApΓq and BpΓq, respectively, on the dual group Γ of G, will be covered in §4.2. In §4.2, we shall also introduce Segal algebras with respect to L1 pGq and discuss Beurling algebras, particularly those of the form  1 pZ, ωq for a weight ω on Z. Now let Γ be a locally compact group. Then there are several classical Banach function algebras (with pointwise product) defined on Γ: these include the Fourier algebra ApΓq and the Fourier–Stieltjes algebra BpΓq, which will be introduced in §4.3, and their generalizations, the Figà-Talamanca–Herz algebras A p pΓq and B p pΓq, defined for p with 1 < p < 8, which will be introduced in §4.4. Many of the theorems on the algebras ApΓq and BpΓq flow from the seminal and very influential thesis of Eymard in 1964 [97]; a recent monograph on these algebras is that of Kaniuth and Lau [197]. The foundations of the theory of the algebras A p pΓq and B p pΓq were laid down by Herz in [164–166]; for an account, see the text of Derighetti [79]. As we indicated, the properties of these Banach algebras that we wish to think about are the following. When does the algebra have a bounded or contractive approximate identity? When is the algebra an ideal in its bidual? When is the algebra a dual Banach algebra? Most of these questions are classical and well-understood, but we do include some new results and proofs. Later, in Chapter 5, we shall consider when Banach function algebras associated with a locally compact group have a BSE norm and when they are BSE algebras. For such a Banach function algebra © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 H. G. Dales and A. Ülger, Banach Function Algebras, Arens Regularity, and BSE Norms, CMS/CAIMS Books in Mathematics 12, https://doi.org/10.1007/978-3-031-44532-3_4

229

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A, what are the corresponding algebras QpAq and C BS E pAq, to be defined in Chapter 5? In Chapter 6, we shall consider when such Banach function algebras are Arens regular or strongly Arens irregular (or neither).

4.1 Group and measure algebras We shall first recall some standard notation and properties concerning the group algebra L1 pGq and the measure algebra MpGq and their relatives on a locally compact group G. We shall prove Wendel’s theorem that identifies the multiplier algebra of L1 pGq with MpGq, we shall recall the definition of the almost periodic and weakly almost periodic functions on G, and we shall note when L1 pGq is a dual Banach algebra and when our algebras are ideals in their biduals. We shall also make some preliminary remarks on when our algebras are Arens regular, and we shall recall the concept of amenability for locally compact groups; we shall describe the spaces LUCpGq and RUCpGq that are defined on G, and we shall conclude the section by briefly mentioning Beurling algebras on locally compact groups and on semigroups. Standard texts on harmonic analysis include [119, 145, 168, 169, 201, 230, 270, 276]; see also, [50, §§3.3, 4.5]. Let G be a group, with identity eG ; the identity of an abelian group G is often denoted by 0G or 0. For t P G and a non-empty subset S of G, we set tS “ {ts : s P S } ,

S t “ {st : s P S } ,

S ´1 “ {s´1 : s P S } ;

for non-empty subsets S and T of G, we set S T “ {st : s P S , t P T }, as on page 4. A subset S is symmetric if S ´1 “ S . For a function f on G and s, t P G, we set pt f qpsq “ f ptsq, prt f qpsq “ f pstq, fqpsq “ f ps´1 q, frpsq “ f ps´1 q .

(4.1.1)

A space F of functions on G is left-translation-invariant (respectively, right-translation invariant) if t f P F (respectively, rt f P F) whenever t P G and f P F; F is translation-invariant if it is both left- and right-translation-invariant. We recall the following basic definition from page 12. Definition 4.1.1 A group that is also a locally compact topological space is a locally compact group whenever the group operations are continuous. Let G be a locally compact group. The identity eG of G has a neighbourhood base consisting of symmetric, open, relatively compact subspaces. Note that every open subgroup of G is also closed. The group G with the discrete topology is Gd . Each locally compact group G has a left Haar measure, called mG . This means that mG is a non-zero, positive, regular Borel measure on G such that mG ptBq “ mG pBq pt P G, B P BG q .

4.1 Group and measure algebras

231

It follows that mG pUq > 0 for each non-empty, open subspace U of G. The measure mG is uniquely defined up to a positive multiple by these properties. See [42, §9.2] or  [168, p. 194], for example. We shall sometimes write dt for dmG ptq and f for f ptq dmG ptq. G In the case where the group G is compact, we suppose that mG pGq “ 1; in the case where G is discrete, we suppose that mG is the counting measure on G. For each s P G, the function B Þ→ mG pBsq defined on BG is also a left Haar measure, and so there is ΔG psq > 0 such that mG pBsq “ ΔG psqmG pBq pB P BG q. The function ΔG : s Þ→ ΔG psq, G Ñ R+• , is the modular function of G; this function ΔG is clearly continuous and is such that ΔG pstq “ ΔG psqΔG ptq ps, t P Gq. The group G is unimodular if ΔG psq “ 1 ps P Gq. Each locally compact group that is either abelian or discrete or compact is certainly unimodular, but there are locally compact groups that are not unimodular; see Example 4.3.36. The following theorem of Kakutani and Kodaira is proved in [168, Theorem (8.7)]; it will allow us to make some reductions later. Theorem 4.1.2 Let G be a σ-compact, locally compact group. For each sequence pUn q of neighbourhoods of eG , there is a compact, normal subgroup N of G such  that N Ă {Un : n P N} and G/N is second countable, and so metrizable. Let G be any group. Then the coset ring of G is the Boolean algebra RpGq generated by the family of left cosets of the subgroups of G. For a locally compact group G, the closed coset ring is the Boolean algebra Rc pGq consisting of the closed sets in RpGq. For locally compact abelian groups, the structure of Rc pGq was completely described in [137] and [292], and the same description was verified for an arbitrary locally compact group by Forrest [121]; for an account, see [197, Appendix A.1]. Thus each set in Rc pGq for a locally compact group G is of the form ⎛ ⎞ ni n ⎜   ⎟⎟⎟ ⎜⎜⎜ ⎜⎜⎝ xi Hi \ yi, j Ki, j ⎟⎟⎟⎠ , i“1

j“1

where n P N, ni P N pi P Nn q, x1 , . . . , xn P G, H1 , . . . , Hn are a closed subgroups of G, and yi, j P G and the set Ki, j is either empty or an open subgroup of Hi for each i P Nn and j P Nni . The ring Rc pGq plays an important role in the study of homomorphisms between various Banach algebras associated to locally compact groups, in the study of the closed ideals in these algebras, and in the investigation of several other notions; see, for example, [41, 123, 197]. Let G be a locally compact group, and take p with 1 ď p ď 8. The Banach space L p pGq “ pL p pG, mG q,  ·  p q has been defined in Chapter 1. In particular, L8 pGq is the Banach space of equivalence classes of the space B b pGq of bounded Borel functions. We also use the notation of equation (4.1.1) when f P L p pGq. Recall also from Theorem 1.2.39(ii) that the spaces L p pGq are weakly sequentially complete for 1 ď p < 8.

232

4 Banach algebras on locally compact groups

For each f P L1 pGq, we have   ´1 ´1 f pt qΔG pt q dmG ptq “ f ptq dmG ptq G

(4.1.2)

G

and 

 f pstq dmG ptq “ ΔG psq

G

 f ptsq dmG ptq “

G

f ptq dmG ptq ps P Gq .

(4.1.3)

G

Now we can identify L1 pGq with L8 pGq for each locally compact group G; here the duality is given by  x f, λy “ f ptqλptq dmG ptq p f P L1 pGq, λ P L8 pGqq . G

We denote by 1G the function that is constantly equal to 1 on G, regarded as an element of L8 pGq. Note that pL8 pGq, · q is a commutative von Neumann algebra with respect to the pointwise product, and so, by Theorem 2.4.10, it is a dual Banach function algebra and, by Theorem 2.3.49, it is Arens regular; the Banach space L8 pGq has property pVq and is a Grothendieck space. Definition 4.1.3 Let G be a locally compact group, and let f and g be measurable functions on product of f and g at t is defined for those t P G  G. The convolution for which G f psqgps´1 tq dmG psq < 8 by the formula  f psqgps´1 tq dmG psq . p f  gqptq “ G

In the case where p f  gqptq is defined, the following formulae follow:   ´1 p f  gqptq “ f psqgps tq dmG psq “ f ptsqgps´1 q dmG psq G G  “ f pts´1 qgpsqΔG ps´1 q dmG psq pt P Gq . (4.1.4) G

For example, take f, g P L1 pGq. Then the product p f  gqptq is defined for almost all t P G, and f  g P L1 pGq with  f  g1 ď  f 1 g1 . Now suppose that f and g take values in R+ . Then also f  g takes values in R+ , and so the value of the norm  f  g1 is

    ´1 p f  gqptq dmG ptq “ f psq gps tq dmG ptq dmG psq , G

and so  f  g1 “

G



G

 p f  gqptq dmG ptq “

G

 f psq dmG psq

G

G

gptq dmG ptq “  f 1 g1 .

4.1 Group and measure algebras

233

In particular, suppose that K and L are compact subspaces of G. Then χK  χL 1 “ χK 1 χL 1 “ mG pKq mG pLq . It follows from equation (4.1.3) that   x f  g, λy “ f psqgptqλpstq dmG psq dmG ptq pλ P L8 pGqq . G

(4.1.5)

(4.1.6)

G

Hence the dual module actions of L1 pGq on L8 pGq are given by   pλ · f qptq “ f psqλpstq dmG psq , p f · λqptq “ f psqλptsq dmG psq (4.1.7) G

G

for t P G, f P L1 pGq, and λ P L8 pGq. The following is [50, Proposition 3.3.14]; see also [168, (20.19)]. Proposition 4.1.4 Let G be a locally compact group, and take p with 1 ď p < 8. q  g belongs to Then, for f P L p pGq and g P L pGq, where q “ p , the function f  q C 0 pGq and f  q g G ď  f  p gq . Definition 4.1.5 Let G be a locally compact group. The group algebra on G is L1 pGq “ pL1 pG, mG q,  · 1 ,  q . Of course, in the special case where G is discrete, we recover the group algebra p 1 pGq,  · 1 ,  q of Example 2.1.13(iv). We now consider pMpGq,  · q, the Banach space of all (complex-valued, regular, Borel) measures on a locally compact group G. Here each measure in MpGq is finite and the norm of μ P MpGq is the total variation norm: μ “ |μ| pGq. The measure that is the point mass at t P G is denoted by δt . By the Riesz representation theorem, Theorem 1.2.12, we identify MpGq with C0 pGq as Banach spaces The convolution product μ  ν of μ, ν P MpGq is defined by setting   x f, μ  νy “ f pstq dμpsq dνptq p f P C 0 pGqq , (4.1.8) G

G

so that μ  ν P MpGq with μ  ν ď μ ν, and hence pMpGq,  ·  ,  q is a Banach algebra. Thus, for each Borel set B in G, the two functions s Þ→ νps´1 Bq and t Þ→ μpBt´1 q are measurable on G, and   pμ  νqpBq “ νps´1 Bq dμpsq “ μpBt´1 q dνptq . G

More generally, it follows from (1.2.7) that   xκpλq, μ  νy “ λpstq dμpsq dνptq G

G

pλ P B b pGqq ,

G

where we note that λ P L1 pG, |μ|  |ν|q for λ P B b pGq.

(4.1.9)

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4 Banach algebras on locally compact groups

Definition 4.1.6 Let G be a locally compact group, and take μ P MpGq. Set μ∗ pBq “ μpB´1 q for each B P BG . Then the map ∗ : μ Þ→ μ∗ is a linear involution on the Banach space MpGq. Definition 4.1.7 Let G be a locally compact group. Then the measure algebra on G is the Banach space MpGq, with the convolution product  and the involution ∗. Theorem 4.1.8 Let G be a locally compact group. Then pMpGq,  ·  ,  q is an unital, semisimple, Banach ∗-algebra with identity δeG that is a Dedekind complete Banach lattice algebra. Let G be a locally compact group. As noted in Example 1.3.25(ii), the space C 0 pGq is an isometric concrete predual of the Banach space MpGq; we shall show that MpGq is a dual Banach algebra with isometric Banach-algebra predual C 0 pGq. This was also proved in [50, Theorem 3.3.15] and [282, Example 5.1.3]. We note that it follows from equation (4.1.8) that   x f, μ  νy “ p f · μqptq dνptq “ pν · f qptq dμptq p f P C 0 pGqq, μ, ν P MpGqq , G

G

where the module actions are given by   p f · μqpsq “ f ptsq dμptq , pμ · f qpsq “ f ptsq dμptq G

ps P Gq

G

for f P C 0 pGq and μ P MpGq. Theorem 4.1.9 Let G be a locally compact group. Then MpGq is a dual Banach algebra, with isometric Banach-algebra predual C 0 pGq. Proof Certainly C 0 pGq is a closed linear subspace of MpGq ; we must show that C 0 pGq is a submodule of MpGq . Fix f P C 0 pGq and μ P MpGq. Take s P G and ε > 0. There is a compact subspace K of G with |μ| pG\Kq < ε. Since f | K is uniformly continuous, there is a compact neighbourhood V of s such that sup{ f ptuq ´ f ptsq : t P K} < ε pu P Vq. It follows that   p f ptuq ´ f ptsqq dμptq < 2ε | f |G and p f ptuq ´ f ptsqq dμptq < ε μ . G\K K Hence p f · μqpuq ´ p f · μqpsq < εp2 | f |G + μq pu P Vq, showing that f · μ is continuous on G. Again take ε > 0 and a compact subspace K of G with |μ| pG\Kq < ε. There is a compact subspace L of G such that | f |G\L < ε. Then, for each s P G\K ´1 L, we have

4.1 Group and measure algebras

235

 p f · μqpsq ď

+

  f ptsq d |μ| ptq

G\K

K

ď | f |G |μ| pG\Kq + |μ | f |G\L < εp| f |G + μq . Thus f · μ P C 0 pGq. Similarly, we have μ · f P C 0 pGq, and so C 0 pGq is a submodule of MpGq , as required. It follows that C 0 pGq is introverted in MpGq , that C 0 pGqK is a closed ideal in MpGq , and that MpGq “ MpGq  C 0 pGqK , as in Theorem 2.4.10. It was shown by Daws, Pham, and White in [72] that C 0 pGq is the unique Banach-algebra predual of MpGq satisfying a certain extra condition. The spaces of discrete, continuous, absolutely continuous, and singular measures (with respect to Haar measure) in MpGq are denoted by Md pGq, Mc pGq, Mac pGq, and M s pGq, respectively. As before, we identify the closed subalgebra Md pGq of MpGq with p 1 pGq,  q, so that δ∗s “ δ s´1 ps P Gq, and we also identify Mac pGq with L1 pGq by using the Radon–Nikodým theorem: each f P L1 pGq corresponds to the measure dμ f “ f dmG in Mac pGq, and the convolution products are consistent. Take f P L1 pGq, and set f ∗ ptq “ frptqΔG pt´1 q pt P Gq . We claim that this definition is consistent with the previous definition of μ∗ when we identify L1 pGq as a closed linear subspace of MpGq. Indeed, for each Borel set B in G, we have   ´1 ´1 μ f ∗ pBq “ f pt qΔG pt q dmG ptq “ f ptq dmG ptq by p4.1.2q B

B´1

“ μ f pB´1 q “ μ∗f pBq , and so μ f ∗ “ μ∗f , giving the claim. This gives part of the next theorem. Theorem 4.1.10 Let G be a locally compact group. Then pL1 pGq,  q is a semisimple Banach algebra and a closed ∗-ideal in pMpGq,  q. Further, L1 pGq is a Dedekind complete Banach lattice algebra that is a projection band in MpGq. We note that t f “ δt´1  f ,

rt f “ f  δt´1

pt P Gq ,

and so a subset S of L1 pGq is translation-invariant if and only if δt  f P S and f  δt P S for each t P G and f P S . The space Mc pGq is also a closed ∗-ideal in MpGq and M s pGq is a closed linear ∗-subspace of MpGq; as in [50, §3.3] and [168, §9], in the case where G is nondiscrete, we have

236

4 Banach algebras on locally compact groups

MpGq “ Md pGq ‘1 Mc pGq “ Md pGq ‘1 Mac pGq ‘1 M s pGq “  1 pGq ‘1 L1 pGq ‘1 M s pGq , and, in particular, L1 pGq “ Mac pGq is 1-complemented in MpGq. Recall from Corollary 1.2.36 that the subalgebra  1 pGqr1s is σpMpGq, C 0 pGqq-dense in MpGqr1s . Each of the algebras L1 pGq and MpGq is commutative if and only if the group G is abelian. The following theorem is given in [53, §11] as a consequence of a more general theorem; further representation theorems are given in [53, Theorem 11.10]. Theorem 4.1.11 (i) Let G be a locally compact group. Then the left-regular representations MpGq Ñ Br pMpGqq and L1 pGq Ñ Br pL1 pGqq are isometric Banach lattice algebra homomorphisms. (ii) Let S be a cancellative semigroup, and let ω be a weight on S . Then the left-regular representation  1 pS , ωq Ñ Br p 1 pS , ωqq is an injective Banach lattice algebra homomorphism. (iii) Let ω be a continuous weight on R+ . Then the left-regular representation 1 L pR+ , ωq Ñ Br pL1 pR+ , ωqq is an injective Banach lattice algebra homomorphism. Let G be a locally compact group, and set A “ L1 pGq. As in Example 1.3.85, A and MpGq are weakly sequentially complete as Banach spaces, and so they are KB-spaces as Banach lattices. Thus A and MpGq are projection bands in A and MpGq , respectively. The space C b pGq “ Cpβ Gq is a closed submodule of the dual spaces MpGq and of  1 pGq “  8 pGq with xδ s , λy “ λpsq ps P G, λ P C b pGqq ,

(4.1.10)

but C b pGq is not left- or right-introverted in these spaces unless G is discrete. For example, take G “ pR, + q. Then, for each n P N, choose λn P C b pRq such that  λn pnq “ |λn | R “ 1 and supp λn Ă rn ´ 1/n, n + 1/ns, and set λ “ 8 n“1 λn , so that λ P C b pRq. Let M be a weak-∗ accumulation point of {δn : n P N} in  1 pRq . Then xδt , M · λy “ xM, λ · δt y “ lim λpn + tq pt P Rq , nÑ8

and so pM · λqp0q “ 1, whereas pM · λqptq “ 0 p0 < t < 1q, and so M · λ  C b pRq. Thus C b pRq is not left-introverted in either MpGq or  1 pGq . A small variation of this argument shows that C b pRq is not left-introverted in L1 pRq . The following identification of a projective tensor product is given in [50, Proposition 3.3.20]; the case where G and H are discrete was already given in Example 2.1.36. Theorem 4.1.12 Let G and H be locally compact groups. Then there is an isometric p L1 pHq Ñ L1 pG × Hq such that ∗-isomorphism θ : L1 pGq ⊗ θp f ⊗ gqps, tq “ f psqgptq

ps P G, t P H, f P C00 pGq, g P C00 pHqq .

(4.1.11)

4.1 Group and measure algebras

237

Proof The   map p f, gq Þ→ θp f ⊗gq, C00 pGq×C00 pHq Ñ C00 pG × Hq, is bilinear, and θp f ⊗ gq1 ď  f 1 g1 p f P C00 pGq, g P C00 pHqq, and so θ extends to a bilinear map L1 pGq × L1 pHq Ñ L1 pG × Hq. By the defining property of the projective tensor p L1 pHq Ñ L1 pG × Hq product, there is a unique continuous contraction θ : L1 pGq ⊗ satisfying equation (4.1.11). We see immediately that θ is a ∗-homomorphism. Take E to be the linear subspace of L1 pGq ⊗ L1 pHq that is spanned by elements of the form f ⊗ g, where f and g are characteristic functions of Borel subsets of G and p L1 pHq. Each H with finite Haar measure, respectively. Then E is dense in L1 pGq ⊗  element of E can be written as a finite sum of the form F “ i, j αi, j χS i ⊗ χT j , where and the sets S i and T j are Borel subsets with finite measure of G and H, respectively,  the rectangles S i × T j are pairwise-disjoint in G × H. It follows that θpFq1 ě Fπ , and so θ is an isometry. The range of θ contains the function χS ×T for each rectangle S × T that is a Borel set in G × H of finite measure, and it is easy to see that the linear span of these functions is dense in L1 pG × Hq. It follows that the map θ is a surjection, and hence an isometric ∗-isomorphism. Each Banach algebra L1 pGq has a contractive approximate identity. Indeed, let pVα q be a directed set of symmetric neighbourhoods of eG consisting of relatively compact sets and ordered by reverse inclusion that is a base for NeG . For each α, choose fα P C 00 pGq+ such that supp fα Ă Vα and G fα dmG “ 1, and then set eα “ p fα + fα∗ q/2, so that supp eα Ă Vα and  ∗ eα “ eqα “ eα , eα dmG “ eα 1 “ 1 . G

Then peα q is a net which is contractive approximate identity for L1 pGq [50, Theorem 3.3.23]. In particular, this implies that the algebra L1 pGq always factors. In the case where G is metrizable, so that {eG } is a Gδ -set, pL1 pGq,  q has a sequential contractive approximate identity. We shall also consider certain subalgebras of group algebras that we specify below; more general subalgebras are considered in the literature. Definition 4.1.13 Let G be a locally compact group, and let S be a closed subsemigroup of G such that eG P S and such that mG pV ∩ S q > 0 for each open neighbourhood V of eG . Then the semigroup algebra L1 pS q consists of the functions f in L1 pGq such that f “ 0 almost everywhere on G\S , regarded as functions on S . Let L1 pS q be a semigroup algebra, as in the definition. Then the product of two functions f, g P L1 pS q is given by  p f  gqptq “ f psqgps´1 tq dmG psq , S ∩tS ´1

and L1 pS q is indeed a closed subalgebra of L1 pGq that is also a Banach lattice algebra. Let peα q be the contractive approximate identity for L1 pGq described above.

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4 Banach algebras on locally compact groups

  Then the condition that we have imposed on S implies that peα | S / eα | S 1 q is a contractive approximate identity for L1 pS q. For example, L1 pR+ q is a semigroup algebra contained in the group algebra 1 L pRq; the product of two functions f, g P L1 pR+ q is given by  t p f  gqptq “ f psqgpt ´ sq ds pt P R+ q . 0

Proposition 4.1.14 Let G be a locally compact group. Then L1 pGq/I is weakly sequentially complete for each closed ideal I of L1 pGq that has a bounded right approximate identity. Proof The algebra L1 pGq has a CAI, and L1 pGq “ L8 pGq is WSC, and so this follows from Corollary 2.3.76. We shall now give a short proof that the Banach algebras L1 pGq and MpGq are not Arens regular for each infinite locally compact group G. This result was first proved by Young in [328]; stronger results will be proved in §6.4. Theorem 4.1.15 Let G be an infinite locally compact group. Then L1 pGq and MpGq are not Arens regular. Proof Assume towards a contradiction that L1 pGq is Arens regular. Since L1 pGq has a CAI and is weakly sequentially complete, it follows from Theorem 2.3.46 that L1 pGq has an identity. Hence G is discrete (and infinite), and so L1 pGq “  1 pGq is Arens regular, a contradiction of Example 2.3.52. Since L1 pGq is a closed ideal in MpGq, also MpGq is not Arens regular. Proposition 4.1.16 Let G be a locally compact group. Then L1 pGq is a dual Banach algebra if and only if L1 pGq “ MpGq, i.e., if and only if G is discrete. Proof Suppose that G is discrete. Then L1 pGq “  1 pGq “ c 0 pGq is a dual Banach algebra, as in Theorem 2.4.29. Suppose that L1 pGq is a dual Banach algebra. Since L1 pGq has a CAI, it follows from Proposition 2.4.7 that L1 pGq is unital; the identity of L1 pGq is δeG , and so G is discrete. In fact, suppose that L1 pGq is isometrically a dual Banach space. Then it follows from Theorem 1.2.35 that ex L1 pGqr1s  ∅. But each element of ex L1 pGqr1s is a point mass, and so δeG P L1 pGq. Hence G is discrete. The following identification is Wendel’s theorem; for more details and more general versions, see [50, Theorem 3.3.40], [169, Theorem 35.5], and [254, §1.9.13]. Theorem 4.1.17 Let G be a locally compact group. Then MpL1 pGqq is isometrically isomorphic to MpGq as a Banach algebra.

4.1 Group and measure algebras

239

Proof Take μ P MpGq, and set Lμ pf q “ μ  f and Rμ p f q “ f  μ for f P L1 pGq, so that pLμ , Rμ q P MpL1 pGqq with pLμ , Rμ qop ď μ. Conversely, let peα q be a CAI in L1 pGq, as above, and take T P M  pL1 pGqq, say with T op ď 1. Then pT eα q is a net in L1 pGqr1s Ă MpGqr1s , and we may suppose that limα T eα “ μ in σpMpGq, C 0 pGqq for some μ P MpGqr1s with μ ď T op . Take h P L1 pGq. Then T eα  h “ T peα  hq Ñ T h in L1 pGq, and so, for each λ P C 0 pGq, we have xT h, λy “ lim xT eα  h, λy “ lim xT eα , h · λy “ xμ, h · λy “ xμ  h, λy , α

α

whence T h “ μ  h. Thus T “ Lμ as an operator on L1 pGq and T op “ μ. A similar argument applies when T P M r pL1 pGqq. Further, f  Lμ g “ Rμ f  g p f, g P L1 pGqq, and so the map μ Þ→ pLμ , Rμ q ,

MpGq Ñ MpL1 pGqq ,

is an isometric Banach-algebra isomorphism. Let G be a locally compact group. We shall identify the dual module action of L1 pGq on L8 pGq “ L1 pGq in terms of the convolution product.   }  1 Indeed, first note that, by (4.1.2), we have Δ} G f P L pGq with ΔG f  “  f 1 for 1

8 each f P L1 pGq. Second, we note that, for each f P L1 pGq and λ P L pGq, we have f  λ, λ  fq P C b pGq with | f  λ|G ď  f 1 λ8 and λ  fq ď  f 1 λ8 . Then G we claim that the module products in L8 pGq are given by

f · λ “ λ  fq,

λ · f “ Δ} Gf  λ

p f P L1 pGq, λ P L8 pGqq .

(4.1.12)

The first of these is immediate from equation (4.1.6). To check the second part of (4.1.12) for f P L1 pGq and λ P L8 pGq, take g P L1 pGq. Then   xg, λ · f y “ x f  g, λy “ f pts´1 qgpsqΔG ps´1 qλptq dmG psq dmG ptq G

and xg, Δ} G f  λy “

  G

G

gpsqΔG pts´1 q f pts´1 qλptqΔG pt´1 q dmG psq dmG ptq ,

G

and so λ · f “ Δ} G f  λ. It follows that { f  λ : f P L1 pGqr1s } “ {λ · g : g P L1 pGqr1s } .

(4.1.13)

Since the group algebra L1 pGq is a closed ideal in the measure algebra MpGq, it follows that L8 pGq is a Banach MpGq-bimodule; the module operations are given by x f, μ · λy “ x f  μ, λy , x f, λ · μy “ xμ  f, λy p f P L1 pGqq

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4 Banach algebras on locally compact groups

for μ P MpGq and λ P L8 pGq. For example, we see that λ · δt “  t λ ,

δt · λ “ ΔG pt´1 qrt λ

pt P Gq

for each λ P L8 pGq, and we set λ · t “ λ · δt and t · λ “ δt · λ. Further, L1 pGq is a Banach MpGq-bimodule, and we set xt · Λ, λy “ xΛ, λ · ty ,

xΛ · t, λy “ xΛ, t · λy pt P G, λ P L8 pGq, Λ P L1 pGq q .

Proposition 4.1.18 Let G be a locally compact group. (i) Suppose that λ P L8 pGq and that the map s Þ→ λ · s, G Ñ L8 pGq, is continuous. Then λ P C b pGq. (ii) Suppose that ν P MpGq and that λ · ν P CpGq for each λ P L8 pGq. Then ν P L1 pGq. Proof (i) Set Fpsq “ λ · s ps P Gq. Let peα q be a CAI for L1 pGq as specified on page 237, say supp eα Ă V for a compact neighbourhood V of eG for each α. Take s, t P G, so that F is uniformly continuous on tV. We have  pλ  eα qpstq ´ pλ  eα qptq ď λpstuq ´ λptuq eα puq dmG puq V ď sup{ λpsxq ´ λpxq : x P tV} , and so the family {λ  eα : α} is equicontinuous on each compact subset of G. By Theorem 1.1.11, there exists μ P C b pGq such that limα λ  eα “ μ uniformly on compact subsets of G. It follows that wk∗ – limα λ  eα “ μ in L8 pGq. However wk∗ – limα λ  eα “ λ in L8 pGq. Thus λ “ μ locally almost everywhere (with respect to mG ), and so λ “ μ as elements of L8 pGq. (ii) This is [169, Theorem (35.13)]. Definition 4.1.19 Let G be a locally compact group. Then the functional ϕ 0 : μ Þ→ μpGq “ xμ, 1G y ,

MpGq Ñ C ,

is the augmentation character on MpGq, and its kernel M0 pGq is the augmentation ideal of MpGq. The restriction of ϕ 0 to L1 pGq is the augmentation character on L1 pGq, and its kernel is    1 1 1 M0 pGq ∩ L pGq “ L 0 pGq “ f P L pGq : f psq dmG psq “ 0 . G

Of course, the augmentation character (on L1 pGq) is a character on L1 pGq; for some groups G, it is the unique character on L1 pGq. The ideal L10 pGq is a maximal modular ideal in L1 pGq. It is immediately checked that

4.1 Group and measure algebras

241

f · 1G “ 1G · f “ ϕ 0 p f q1G

p f P L1 pGqq ,

M · 1G “ 1G · M “ xM, ϕ 0 y1G

pM P L1 pGq q .

(4.1.14)

Let F be a closed, translation-invariant linear subspace of L8 pGq, and take M P F  . Then δt · M and M · δt are defined on F for t P G by the formulae xδt · M, λy “ xM, λ · δt y ,

xM · δt , λy “ xM, δt · λy pλ P Fq .

We shall see that many properties of group algebras depend on whether the underlying group is amenable; we shall recall the definition of this notion. Definition 4.1.20 Let G be a locally compact group, and let F be a closed, translation-invariant linear subspace of L8 pGq with 1G P F. Take M P F  . Then M is a mean on F if M “ xM, 1G y “ 1 , and M is left-invariant (respectively, right-invariant) if δt · M “ M (respectively, M · δt “ M) for all t P G; a mean on F is invariant if it is left- and right-invariant. The group G is amenable if there is a left-invariant mean on L8 pGq. Suppose that M is a mean on L8 pGq. Then M is left-invariant if and only if f · M “ ϕ 0 p f q M p f P L1 pGqq . For an amenable group G, there is an invariant mean M on L8 pGq [50, Proposition 3.3.49]. Each compact group and each locally compact abelian group is amenable; a group G such that Gd is amenable is itself amenable (but not conversely); each closed subgroup and each quotient group of an amenable group is amenable. For accounts of amenable locally compact groups, see [256, 262, 282] and also [50, §3.3], for example. For a discussion of the size of the set of left-translationinvariant means on L1 pGq for an amenable group G, see [256, Chapter 7]. An account of amenable semigroups is given in [256]. The following characterization of amenable groups involving a bounded approximate identity is given in [282, Theorem 1.1.19]; see also [282, Corollary 2.3.11]. Proposition 4.1.21 Let G be a locally compact group. Then G is amenable if and only if L01 pGq has a bounded approximate identity. In this case, L01 pGq has a bounded approximate identity of bound 2. It follows from Theorem 2.1.46 that L01 pGq factors whenever the group G is amenable; in fact, by a result of Willis [50, Theorem 3.3.30], L01 pGq always factors weakly, and each element of L01 pGq is the sum of four products. Example 4.1.22 The standard example of a non-amenable group is F2 , the free group on two generators. It follows that the locally compact group S Lp2, Rq is not

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amenable because it contains a closed subgroup that is isomorphic to F2 [256, Example (0.7)]. In particular, S Lp2, Rq is not amenable as a discrete group. For several other, related groups that are not amenable, see [282, Example 1.2.10]. The special orthogonal group S Op3q is compact, and hence amenable, but it is not amenable as a discrete group [282, Theorem 0.1.4].

Let G be a locally compact group. We now consider various closed linear subspaces of the dual space L8 pGq “ L1 pGq . Definition 4.1.23 Let G be a locally compact group. The spaces LUCpGq and RUCpGq denote the closed linear subspaces of C b pGq Ă L8 pGq consisting of the bounded, left (respectively, right) uniformly continuous functions on G, so that LUCpGq “ {λ P C b pGq : t Þ→ t λ , G Ñ C b pGq, is continuous} , RUCpGq “ {λ P C b pGq : t Þ→ rt λ , G Ñ C b pGq, is continuous} , and then UCpGq “ LUCpGq ∩ RUCpGq. It follows from Proposition 4.1.18(i) that LUCpGq “ {λ P L8 pGq : t Þ→ t λ , G Ñ L8 pGq, is continuous} . In fact, LUCpGq, RUCpGq, and UCpGq are unital C ∗ -subalgebras of C b pGq and of L1 pGq “ L8 pGq, and C 0 pGq Ă UCpGq. The following result is contained in [169, (32.45)]. Theorem 4.1.24 Let G be a locally compact group. Then L8 pGq · L1 pGq “ LUCpGq · L1 pGq “ LUCpGq , and LUCpGq is a norm-closed Banach L1 pGq-bimodule in L8 pGq. Further, LUCpGq is left-introverted. Proof Set A “ L1 pGq and X “ LUCpGq, and let peα q be the CAI for A specified above. Take f P A and λ P A . For s, t P G, we have } pλ · f qptsq ´ pλ · f qpsq “ pΔ} G f  λqptsq ´ pΔG f  λqpsq ď λ8  f  δt ´ f 1 Ñ 0

as

t Ñ eG ,

and so λ · f P X. Thus A · A Ă X. In particular, X is a Banach right A-module. Now take λ P X. Given ε > 0, there exists V P NeG such that λpstq ´ λptq < ε ps P Vq .

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For each α such that supp eα Ă V and for each t P G, it follows that ´1 ~ pλ · eα qptq ´ λptq “ pΔ G eα qpsqλps tq ´ λptq   ´1 ´1 ´1 eα psqλptq dmG psq “ ΔG ps qeα ps qλps tq dmG psq ´ G G “ eα psqpλpstq ´ λptqq dmG psq G “ eα psqpλpstq ´ λptqq dmG psq < ε . V

Thus limα λ · eα “ λ in X, and so λ P XA. By the ‘right-handed’ form of Cohen’s factorization theorem, Theorem 2.1.46, X “ X · A. This shows that X Ă A · A, and so X “ A · A. Hence X is a norm-closed Banach A-bimodule in A . Since M · λ P A pM P A , λ P A q, it follows that X is left-introverted. Let G be a locally compact group, and set A “ L1 pGq. The above result says that LUCpGq “ A · A. Similarly, RUCpGq “ A · A . But [168, (19.28)] states that LUCpGq “ RUCpGq only if G is unimodular, so A · A  A · A when G is not unimodular. Let A be a Banach algebra. Then the spaces APpAq and WAPpAq were defined in Definition 2.3.27; we remarked that they are norm-closed Banach A-bimodules in A and that APpAq Ă WAPpAq. By Corollary 2.3.59, APpAq and WAPpAq are introverted submodules of A . We now consider these spaces when A “ L1 pGq and A “ MpGq for a locally compact group G. Proposition 4.1.25 Let G be a locally compact group. Then C 0 pGq Ă WAPpMpGqq “ WAPpL1 pGqq . Proof Set A “ L1 pGq, and take λ P C 0 pGq, so that λ P A ; we may regard λ as an element of MpGq . Since C 0 pGq is a Banach-algebra predual of MpGq, it follows from Theorem 2.4.10 that λ P WAPpMpGqq. By Proposition 2.3.33 (with θ taken to be the injection of L1 pGq into MpGq), necessarily λ P WAPpAq. This shows that C 0 pGq Ă WAPpMpGqq Ă WAPpAq. Now take λ P WAPpAq. The algebra A has a CAI, and so, by Proposition 2.3.66(ii), there is a mixed identity E in A with E “ 1. Since MpAq “ MpGq by Theorem 4.1.17, Proposition 2.3.65 applies to show that the map μ Þ→ λ · μ ,

MpGq Ñ MpGq ,

is weakly compact. By Corollary 2.3.30, it follows that λ P WAPpMpGqq. This implies that WAPpAq Ă WAPpMpGqq, giving the result.

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Definition 4.1.26 Let G be a locally compact group. For λ P L8 pGq, set LOpλq “ {t λ : t P G}

and

ROpλq “ {rt λ : t P G} ,

the left orbit and right orbit of λ. The functional λ is weakly almost periodic (respectively, almost periodic) if the set LOpλq is relatively compact in the weak (respectively, norm) topology on L8 pGq. The spaces of these functionals are denoted by WAPpGq and APpGq, respectively. Let G be a locally compact group. Then we shall see shortly that λ P L8 pGq is in WAPpGq (respectively, APpGq) if and only if ROpλq is relatively compact in 8 the weak  (respectively, norm)  topology on L pGq, and this holds if and only if 1 the set f  λ : f P L pGqr1s is relatively weakly compact (respectively, relatively compact) in L8 pGq. Clearly 1G P APpGq Ă WAPpGq Ă LUCpGq Ă C b pGq Ă L8 pGq .

(4.1.15)

The spaces APpGq and WAPpGq are unital C ∗ -subalgebras of C b pGq, and we have APpGq “ APpGd q ∩ C b pGq ,

WAPpGq “ WAPpGd q ∩ C b pGq .

(4.1.16)

In the case where G is compact, we have APpGq “ WAPpGq “ WAPpL1 pGqq “ LUCpGq “ CpGq . We shall see in Proposition 6.4.20 that WAPpGq “ LUCpGq if and only if G is compact. Definition 4.1.27 Let G be a locally compact group. Then the character space of the commutative C ∗ -algebra APpGq is the Bohr compactification of G, and is denoted by bG. Thus APpGq is C ∗ -isomorphic to CpbGq. There is an obvious continuous map of G into bG. However, this map is not always an injection, so that bG may not be a ‘compactification’ of the locally compact space G. (For example, bG “ {eG } when G is the locally compact group S Lp2, Rq.) The group G is defined to be maximally almost periodic when this map is an injection, equivalently, when APpGq separates the points of G; see [254, §3.2.17]. For example, abelian groups and compact groups are maximally almost periodic; maximally almost periodic groups are unimodular. In fact, bG is always a compact group; an elementary proof of this and of other properties of bG is given in [230, §41]; for a topological approach, see [172, Chapter 21]. We note that bG has the following universal property. Let H be a compact group. Then each continuous group homomorphism from G into H extends to a continuous homomorphism from bG into H.

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The following result of Ülger is taken from [306, Theorem 4]; see also [282, Example 5.1.12]. Theorem 4.1.28 Let G be a locally compact group. Then: (i) APpGq “ APpL1 pGqq; (ii) WAPpGq “ WAPpL1 pGqq “ WAPpMpGqq “ WAPp 1 pGqq ∩ C b pGq. Proof Fix a CAI peα q in L1 pGq such that e∗α “ eqα “ eα in L1 pGq for each α. (i) Take λ P APpL1 pGqq Ă C b pGq, so that the set {peα  δt q · λ : t P G, α} is relatively compact in pC b pGq, | · |G q. For each t P G, the net peα  δt q “ prt´1 eα q, which is bounded in C b pGq , is weak-∗ convergent to δt in C b pGq , and so, by Theorem 1.3.45, the net ppeα  δt q · λq is convergent in pC b pGq, | · |G q; its limit is δt · λ. This implies that the set {λ · δt : t P G} “ {t λ : t P G} is relatively compact in C b pGq, i.e., λ P APpGq. Conversely, take λ P APpGq. Then { f  λ : f P L1 pGqr1s } is relatively compact in C b pGq, and so, by equation (4.1.13), {λ · g : g P L1 pGqr1s } is relatively compact in C b pGq, i.e., λ P APpL1 pGqq. (ii) Take λ P WAPpL1 pGqq, so that {λ · f : f P L1 pGqr1s } is relatively weakly compact in L8 pGq. Then {λ · peα  δt q : t P G, α P A} is relatively weakly compact in L8 pGq. Since wk∗ – limα λ · peα  δt q “ λ · δt for each t P G, it follows from Proposition 1.2.11 that {λ · δt : t P G} is relatively weakly compact in L8 pGq, i.e., λ P WAPpGq. Conversely, take λ P WAPpGq. Then {δt  λ : t P G} is relatively weakly compact in L8 pGq. By the Krein–Šmulian theorem, Theorem 1.2.24(iv), the set { f  λ : f P  1 pGqr1s } is relatively weakly compact in L8 pGq. However, by Corollary 1.2.36,  1 pGqr1s is weak-∗ dense in MpGqr1s , and so, by Proposition 1.2.11, { f  λ : f P MpGqr1s } is relatively weakly compact in L8 pGq. Again by equation (4.1.13), λ P WAPpL1 pGqq. We have shown that WAPpGq “ WAPpL1 pGqq; by Proposition 4.1.25, we also have WAPpL1 pGqq “ WAPpMpGqq. It follows from equation (4.1.16) that WAPpGq “ WAPp 1 pGqq ∩ C b pGq. Corollary 4.1.29 Let G be a locally compact group. Then: (i) WAPpGq is a closed, introverted, translation-invariant linear subspace of L8 pGq; (ii) C 0 pGq Ă WAPpGq “ WAPpG op q; (iii) WAPpGq “ L1 pGq · WAPpGq “ WAPpGq · L1 pGq. Proof (i) This follows from Theorem 4.1.28(ii) and Corollary 2.3.59. (ii) This follows from Proposition 4.1.25, Theorem 4.1.28(ii), and Theorem 2.3.29. (iii) This is now a special case of Theorem 2.3.44(i).

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Corollary 4.1.30 Let G be a locally compact group, and take λ P C b pGq. Then λ P WAPpGq if and only if the function ps, tq Þ→ λpstq, G × G Ñ C, clusters on G × G. Proof By equation (4.1.16), λ P WAPpGq if and only if λ P WAPpGd q, and so it follows from Theorem 2.3.29 that λ P WAPpGq if and only if λ satisfies the iteratedlimit condition as an element of  1 pGq . We note that λpstq “ xδ st , λy ps, t P Gq; set r λ : ps, tq Þ→ λpstq, G × G Ñ C. Suppose that λ P WAPpGq. Then certainly r λ clusters on G × G. The converse follows because aco {δ s : s P G} is dense in  1 pGqr1s . Theorem 4.1.31 Let G be a locally compact group. Then there is a unique invariant mean on the space WAPpGq. We shall sketch the proof of the above theorem, giving some extra information about the mean; full details are given in [256, (2.36)]. Set A “ L1 pGq, and take λ P WAPpGq “ WAPpAq. Define K to be the weak closure of the convex hull of LOpλq. For t P G, define Lt : μ Þ→ μ · δt , K Ñ K. Clearly L s ◦ Lt “ L st ps, t P Gq, and so the family F “ {Lt : t P G} is a semigroup of weakly continuous, affine maps on K. Further, for each μ1 , μ2 P K with μ1  μ2 , we have Lt μ1 ´ Lt μ2  “ μ1 ´ μ2 , so that 0 is not in the norm-closure of the set {Lt μ1 ´Lt μ2 : t P G}. By the Ryll-Nardzewski theorem, Theorem 1.3.75, there exists μ0 P K with μ0 · δt “ μ0 pt P Gq. Clearly μ0 is a constant function on G. It follows from [256, (2.13)] that there is a unique left-invariant mean, say M, on WAPpGq. Take λ P WAPpGq, and let co LOpλq and co ROpλq be the closures in WAPpGq of the convex hulls of the left and right orbits, respectively, of λ. Then it is shown in [256, (2.36)] that each of these sets contains exactly one constant function, and these constant functions are the same; they are both equal to xM, λy 1G . Further, M is the unique invariant mean on WAPpGq. Theorem 4.1.32 Let G be a locally compact group. Then APpGq “ CpGq when G is compact and APpGq ∩ C 0 pGq “ {0} when G is not compact. Proof The case where G is compact is immediate. Now suppose that G is not compact. We first claim that xM, λy “ 0 pλ P C 0 pGqq, where M is the unique invariant mean on WAPpGq described above. Indeed, take λ P C 0 pGq. Since G is not compact, there is a net ptα q in G that is eventually outside each compact subset of G, and so the net ptα λq converges pointwise to 0 on G. By Corollary 4.1.29(ii), λ P WAPpGq, and so LOpλq is relatively weakly compact in C 0 pGq. Thus we may suppose that the net ptα λq converges weakly, and hence pointwise, in C 0 pGq, say to g P C 0 pGq. Clearly g “ 0, and so co LOpλq∩C1G “ {0}. The claim follows from the above remarks. Assume towards a contradiction that there exists λ P APpGq ∩ C 0 pGq with λ  0. By replacing λ by |λ|, we may suppose that λ ě 0. Choose s0 P G with λps0 q > 0, and set ε “ λps0 q. Since LOpλq is totally bounded, there exist n P N and t1 , . . . , tn P G

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 such that LOpλq Ă {B ε/2 pti λq : i P Nn }. Thus, for each s P G, there exists i P Nn with  s0 s´1 λ ´ ti λ G < ε/2, and so, in particular, λpti sq > ε/2. Consider the function n 1 μ“ t λ , n i“1 i so that μ P C 0 pGq+ . However, for each s P G, we have μpsq ě ε/2n, and this implies that xM, μy ě ε/2n, a contradiction of the fact that xM, μy “ 0. This shows that C 0 pGq ∩ APpGq “ {0}. Let A be a Banach algebra with dual module A . We recall that the notion of a closed submodule F of A that is faithful and left-introverted was given in Definition 2.3.57, and that it was shown in Theorem 2.3.60 that in this case F K is a weak∗-closed ideal in pA , l q and pF  , l q is a quotient Banach algebra of pA , l q; the quotient map is denoted by qF : A Ñ F  . As in Theorem 2.3.81, there is a canonical Banach-algebra embedding θ : M r pAq Ñ pF  , l q whenever A has a mixed identity and F “ FA. We now consider this situation in the case where A “ L1 pGq for a locally compact group G and X is a faithful, left-introverted A-submodule of A “ L 8 pGq with C 0 pGq Ă X. In this case, equation (2.3.15) shows that xθpμq, λ · f yX  ,X “ x f  μ, λyA,A

p f P A, λ P X, μ P MpGqq .

Now suppose that X Ă C b pGq, and define ηpμq for μ P MpGq by  λpsq dμpsq pλ P Xq . ηpμqpλq “

(4.1.17)

G

Take μ P MpGq. Then ηpμq P X  and ηpμq | C 0 pGq “ μ. Further, the map η : MpGq Ñ X  is a linear isometry. In particular, ηpδ s qpλq “ λpsq ps P G, λ P Xq. Take f P A, λ P X, and μ P MpGq. Then     f ptqλptsq dmG ptq dμpsq xηpμq, λ · f yX  ,X “ pλ · f qpsq dμpsq “ G G G    ´1 ´1 “ f pts qΔG ps q dμpsq λptq dmG ptq “ x f  μ, λyA,A , G

G

and so the maps θ and η are the same, at least when X “ XA. Hence, in this case, η is a Banach-algebra isometry; later, in equation (4.1.20), we shall identify MpGq with ηpMpGqq when X is also an unital C ∗ -subalgebra of C b pGq. In particular, it follows from equation (4.1.9) that   xμ l ν, λy “ xμ  ν, λy “ λpstq dμpsq dνptq pλ P X, μ, ν P MpGqq G

when X “ XA.

G

(4.1.18)

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Now suppose that X is an unital C ∗ -subalgebra of L8 pGq with C 0 pGq Ă X. The compact character space of X is denoted by ΦX . We can identify X with CpΦX q and X  with MpΦX q. In the case where X is left-introverted in L8 pGq, so that pX  , l q is a Banach algebra, this gives a Banach-algebra structure to MpΦX q, and it is denoted by pMpΦX q, l q. It follows from equation (4.1.15) and remarks within Example 3.1.6 that we have natural continuous surjections ΦL8 pGq Ñ ΦC b pGq “ β G Ñ ΦLUCpGq Ñ ΦWAPpGq .

(4.1.19)

Set Φ “ ΦL8 pGq . Then we can regard X “ CpΦX q as a C ∗ -subalgebra of CpΦq; there is a continuous Banach-space epimorphism from MpΦq onto MpΦX q, and this coincides with the quotient map qX : pL1 pGq , l q “ pMpΦq, l q Ñ pX  , l q “ pMpΦX q, l q . For a study of the above compactifications βG, ΦLUCpGq , and ΦWAPpGq of the locally compact group G, see [172, Chapter 21]. (The space ΦWAPpGq is denoted by WAPpGq in [172], and often by GWAP .) Again, let G be a locally compact group, and now suppose that X is an unital C ∗ -subalgebra of L8 pGq with C 0 pGq Ă X Ă C b pGq. Then X is a Banach function algebra on G, and there is a natural embedding of G into ΦX , so that we may regard G as a dense, open subspace of ΦX , and hence ΦX is a compactification of G. In the case where X is left-introverted, we regard ΦX as a compact subspace of pMpΦX q, l q, and so it is clear that pΦX , l q is a compact, right topological semigroup, and that the semigroup operation on ΦX extends that on G. Clearly the continuous surjections β G Ñ ΦLUCpGq Ñ ΦWAPpGq are semigroup morphisms. In fact, we shall usually denote the semigroup product in ΦX by juxtaposition. Thus we can write X  “ MpΦX q “ MpGq ‘1 MpΦX \Gq “ MpGq ‘1 C 0 pGqK

(4.1.20)

as a Banach space, where MpGq “ MpΦX q | G, MpΦX \Gq “ MpΦX q | pΦX \Gq, and C 0 pGqK is the annihilator of C 0 pGq in X  . We recall from Corollary 2.3.59 that a closed L1 pGq-submodule X of L 8 pGq with X Ă WAPpGq is introverted; in particular, C 0 pGq is introverted in L8 pGq, and so, by Proposition 2.3.62, C 0 pGqK is an ideal in pMpΦX q, l q. Further, the range of the mapping η : MpGq Ñ X  of equation (4.1.17) is exactly the space MpGq specified in equation (4.1.20) when X “ XA. Thus we obtain the following theorem, where we use the above notation. The theorem is related to [215, Lemma 4.1]; a version for Beurling algebras is given in [57, Theorem 7.25]. Theorem 4.1.33 Let G be a locally compact group, and let X be an unital C ∗ subalgebra of L8 pGq with C 0 pGq Ă X Ă C b pGq such that X is a faithful and

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249

left-introverted L1 pGq-submodule of L8 pGq and X “ XA. Then pX  , l q “ pMpΦX q, l q . Further, pMpGq,  q “ pMpΦX q | G, l q is a closed subalgebra of pMpΦX q, l q and pC 0 pGqK , l q “ pMpΦX \Gq, l q is a closed ideal in pMpΦX q, l q, and MpΦX q “ MpGq ‘1 C 0 pGqK “ MpGq  C 0 pGqK . In the case where G is compact, then necessarily X “ CpGq and so ΦX “ G and pMpΦX q, l q “ pMpGq,  q. For example, suppose that X “ WAPpGq or X “ LUCpGq or X “ C b pGq. Then X is an unital C ∗ -subalgebra of L8 pGq with C 0 pGq Ă X Ă C b pGq, using Corollary 4.1.29(ii), and so we have Banach algebras MpΦWAPpGq q, MpΦLUCpGq q, and Mpβ Gq, respectively; see §6.4 for further properties of these algebras. For X “ WAPpGq, the space X is faithful and introverted, and X “ X · A by Corollary 4.1.29(iii). For X “ LUCpGq, the space X is faithful and left-introverted, and X “ A · A “ X · A by Theorem 4.1.24. Thus these two examples satisfy the conditions of Theorem 4.1.33. We remark that, in the case where X “ L8 pGq (so that XA  X), it is shown in [216] that ΦX is a semigroup only when G is either discrete or compact. In the case where X “ C b pRq, so that ΦX “ β R, we have noted that X is not left-introverted; in fact, it is shown in [172, Theorem 21.47] that β R cannot be a compact, right topological semigroup such that the semigroup operation on β R extends addition on R. Let G be a locally compact group, set A “ L1 pGq and X “ LUCpGq, so that X “ A · A. Since A has a contractive approximate identity, the bidual A has a mixed identity E with E “ 1. It follows from Theorem 2.3.82 (and using the notation of that result) that A “ E l A  pI ´ Eq l A “ j E pX  q  X K ,

(4.1.21)   where X K Ă rad A . The element qX pEq is the identity of X  and qX pEq “ 1, and so pX  , l q is an unital Banach algebra (cf. [217, Theorem 2.3]). We remark that, by [57, Proposition 7.20], the space pLUCpGq, l q is the same whether LUCpGq be regarded as a left-introverted submodule of L8 pGq or of  8 pGq. We shall require the following facts about LUCpGq in §6.4; they are taken from [29]. Proposition 4.1.34 Let G be a locally compact group. (i) Take disjoint, non-empty, compact subsets K and L of ΦLUCpGq . Then there is a symmetric neighbourhood W of eG with K ∩ WL “ ∅. (ii) Take a symmetric, open neighbourhood W of eG in G and a subset S of G. Then there exists h P CpΦLUCpGq , Iq such that hpϕq “ 1

pϕ P WS q ,

hpϕq “ 0 pϕ P G\W 3 S q .

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(iii) Suppose that S and T are subsets of G such that S ∩UT “ ∅ for a symmetric neighbourhood U of eG and that W is a symmetric, open neighbourhood of eG with W 4 Ă U. Then WS ∩ WT “ ∅. In particular, S ∩ T “ ∅. Proof Set X “ LUCpGq. (i) Choose h P CpΦX , Iq with hpϕq “ 1 pϕ P Kq and hpϕq “ 0 pϕ P Lq. Since h | G P X, it follows that there is a symmetric neighbourhood W of eG such that hpwsq ´ hpsq < 1/2 ps P G, w P Wq. For each w P W and ϕ P ΦX , we have hpwϕq ´ hpϕq ď 1/2, and so hpwϕq ď 1/2 pw P W, ϕ P Lq. Thus K ∩ WL “ ∅. (ii) Set fW “ χW /mG pWq, so that fW P L1 pGq with  fW 1 “ 1. Now take a non-empty subset U of G, so that WU is open in G and χWU P L8 pGq. By Theorem 4.1.24, the function χWU · fW belongs to X, and, by equation (4.1.7), we have  pχWU · fW qptq “ fW psqχWU pstq dmG psq pt P Gq . G

Thus pχWU · fW qptq “ 1 pt P Uq and pχWU · fW qptq “ 0 pt P G\W 2 Uq, and also pχWU · fW qptq P I pt P Gq. We obtain the required function h by taking U “ WS . (iii) Since WT Ă G\W 3 S , this follows from (ii). Let G be a compact group, so that mG pGq “ 1. Since G is unimodular, it follows from equation (4.1.2) that   ´1 f ps q dmG psq “ f psq dmG psq p f P L1 pGqq . G

G

Take p with 1 ď p ď 8. Then the Banach spaces pL p pGq,  ·  p q and pCpGq, | · |G q are Banach algebras with respect to the convolution product  , and CpGq Ă L8 pGq Ă L p pGq Ă L1 pGq . (Incidentally, it is a famous result of Saeki [285] that, for a locally compact group G, the Banach space L p pGq is closed under convolution for some p with 1 < p < 8 only when G is compact.) The space CpGq is a dense subalgebra of L p pGq whenever 1 ď p < 8. Example 4.1.35 Let G be a compact group, and take p with 1 < p < 8. Then pL p pGq,  q is a reflexive Banach algebra, and so L p pGq is a dual Banach algebra that is an ideal in its bidual and it is Arens regular. Suppose that L p pGq has a bounded approximate identity. Since the Banach space L p pGq is weakly sequentially complete, it follows from Theorem 2.3.46 that L p pGq has an identity, and so G is discrete, and hence finite.

Theorem 4.1.36 Let G be a compact group, and take p with 1 ď p ď 8. Then pL p pGq,  q is a compact algebra.

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251

Proof First, suppose that 1 ď p < 8. Take f P CpGq, and regard CpGq as a linear subspace of Lq pGq, where q “ p . Set B “ L p pGqr1s . We claim that f  B is an equicontinuous subset of CpGq. Indeed, take s, t P G and g P B. Then  p f  gqpsq ´ p f  gqptq ď  s f puq ´ t f puq gpu´1 q dmG puq G

ď  s f ´ t f q g p ď  s f ´ t f q “  st´1 f ´ f q using Hölder’s inequality. Since u f ´ f q Ñ 0 as u Ñ eG , the claim follows. By the Ascoli–Arzelà theorem, Theorem 1.1.11, the set f  B is relatively compact in CpGq. Since the embedding of CpGq into L p pGq is continuous, f  B is also relatively compact in L p pGq, and so L f P KpL p pGqq. Since CpGq is dense in L p pGq, it follows that L f P KpL p pGqq for each f P L p pGq. Similarly, R f P KpL p pGqq for each f P L p pGq. Hence pL p pGq,  q is a compact algebra. Now consider the case of L8 pGq. Take f P L8 pGq, so that f P L1 pGq. By (4.1.12), there exists g P L1 pGq such that L f pλq “ λ · g pλ P L8 pGqq, and so L f “ Lg , where Lg P BpL1 pGqq. Since Lg P KpL1 pGqq, also L f P KpL8 pGqq. Similarly, R f P KpL8 pGqq. Thus pL8 pGq,  q is also a compact algebra. Applications of the above results that give information about the Arens regularity p B will be given in Examples 6.2.8(ii). of tensor products L p pGq ⊗ Let G be a locally compact group. Then it was proved by Wong [327] that L1 pGq is an ideal in its bidual whenever G is compact, and the converse has been proved in several different ways, including [310, Proposition 4.8]. Theorem 4.1.37 Let G be a locally compact group. Then pL1 pGq,  q is an ideal in its bidual if and only if G is compact. Proof Set A “ pL1 pGq,  q. Suppose that G is compact. Then it follows from Theorems 4.1.36 and 2.3.7 that A is an ideal in A . Conversely, suppose that A is an ideal in A . Since A factors and satisfies the Dunford–Pettis property, it follows from Corollary 2.3.11 that R f is compact for each f P A, and so the map g Þ→ λ · p f  gq, A Ñ A , is compact for each f P A and λ P A . Thus A A Ă APpGq. However C 0 pGq Ă LUCpGq “ A A by Theorem 4.1.24, and so C 0 pGq Ă APpGq. By Theorem 4.1.32, G is compact. We shall now list some properties of the algebras pL8 pGq,  q in the case where G is compact. In this case, L8 pGq is a dense ideal in pL1 pGq,  q, and λ  f 8 ď λ8  f 1 ,

 f  λ8 ď λ8  f 1

pλ P L8 pGq, f P L1 pGqq ,

and so L8 pGq is a Banach L1 pGq-bimodule. We recall that f · λ “ λ  fq and λ · f “ fq  λ for f P L1 pGq and λ P L8 pGq.

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Theorem 4.1.38 Let G be a compact group. Then: (i) L1 pGq · L8 pGq “ L8 pGq · L1 pGq “ APpGq “ WAPpGq “ CpGq; (ii) L8 pGq  L8 pGq Ă CpGq, and so pCpGq,  q is a closed ideal in pL8 pGq,  q; (iii) pL8 pGq,  q is a dual Banach algebra, with Banach-algebra predual L1 pGq; (iv) pL8 pGq,  q is Arens regular; (v) pL8 pGq,  q is an ideal in its bidual; (vi) L8 pGq “ L8 pGq ‘ L1 pGqK and p f, g P L8 pGq, R, S P L1 pGqK q .

p f + Rq l pg + Sq “ f  g

(4.1.22)

Proof Set A “ pL1 pGq,  q and I “ pL8 pGq,  q, so that I is a dense ideal in A that is a Banach A-bimodule. (i) By Theorem 4.1.36, A is a compact algebra, and so we can apply Theorem 2.3.44(iv) and Theorem 4.1.28 to obtain the result. (ii) Since CpGq Ă A, this follows from (i). (iii) For this, we shall apply Proposition 2.4.3. The map T : I Ñ A is given by  x f, T λy “ f psqλpsq ds p f P A , λ P Iq . G

We must show that T is an A-bimodule isomorphism. For this, take f, g P A and λ P I. Then    f psqpq g  λqpsq ds “ f psqgpus´1 qλpuq du ds x f, T pλ · gqy “ G

and

G

G

 

 pg  f qpuqλpuq du “

x f, T pλq · gy “ G

G

gpus´1 q f psqλpuq ds du ,

G



and hence T pλ · gq “ T pλq · g P A . Thus T is a left A-module isomorphism. Similarly, T is a right A-module isomorphism, and hence an A-bimodule isomorphism, as required. (iv) Since pI, · q is a C ∗ -algebra, it follows from Theorem 2.3.49 that pI,  q is Arens regular. (v) By Theorem 4.1.36, I is a compact algebra, and so it is an ideal in its bidual. (vi) This follows from Theorem 2.4.17. Let ω be a continuous weight on a locally compact group G. Then the Banach space L1 pG, ωq is defined to be    1 f psq ωpsq dmG psq < 8 . L pG, ωq “ f measurable :  f ω “ G

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253

The spaces L1 pG, ωq are Banach algebras with respect to convolution multiplication,  . In the case where G is discrete, we obtain the Beurling algebras  1 pG, ωq of page 80. Definition 4.1.39 Let ω be a continuous weight on a locally compact group G. Then the Banach algebras L1 pG, ωq are Beurling algebras. For studies of general Beurling algebras, see [50, §4.6], [57], and [129]; in [57], many properties of L1 pG, ωq that are analogous to those given earlier in this section for the group algebras L1 pGq are established. For example, let G be a locally compact group, and let ω be a continuous weight function on G with ωpsq ě 1 ps P Gq, so that L1 pG, ωq is a subalgebra of L1 pGq. In the case where G is arbitrary and ω is symmetric and in the case where G is maximally almost periodic and ω is arbitrary, it is shown in [57, Theorem 7.13] that L1 pG, ωq is semisimple, but it seems to be unknown whether this is the case for every G and every such weight ω.

4.2 Locally compact abelian groups In this section, we shall discuss group and measure algebras and some related Banach algebras that are defined on locally compact abelian groups. In particular, we shall recall the definitions of the Fourier and Fourier–Stieltjes transforms that specifically identify these algebras as Banach function algebras. We shall also define Segal algebras on a group G, noting that they are then Segal algebras with respect to the group algebra L1 pGq in the sense of Definition 3.1.31; in particular, we shall show in Theorem 4.2.14 that some Segal algebras on the locally compact abelian group pR, + q have a contractive pointwise approximate identity. We shall consider certain Beurling algebras, and we shall conclude the section with a famous theorem of Varopoulos that ApΓq is a closed subalgebra of VpΓq for each compact abelian group Γ. Let pG, + q be a locally compact abelian (LCA) group. As on page 6, a character p on G is a group homomorphism from G onto the circle group T. The set Γ “ G of all continuous characters on G is an abelian group with respect to the pointwise operation given by: pγ1 + γ2 qpsq “ γ1 psqγ2 psq ps P G, γ1 , γ2 P Γq . The topology on Γ is that of uniform convergence on compact subsets of G; with this topology, Γ is also a LCA group, called the dual group to G. For each s P G, the map γ Þ→ γpsq “ xs, γy , Γ Ñ T , is a continuous character on Γ, and the famous Pontryagin duality theorem asserts that each continuous the character on Γ has this form and that the topology of

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4 Banach algebras on locally compact groups

uniform convergence on compact subsets of Γ coincides with the original topolp “ G. For example, see [119, Chapter 4] or [276, §1.7]. It is ogy on G, so that Γ standard that the dual group of a compact group is discrete and that the dual group p “ T, T p “ Z, and R p “ R. of a discrete group is compact. For example, Z In the case where G is abelian, the Bohr compactification bG of G is the dual group of Γd , Let pG, + q be a LCA group. Then the convolution product f  g of f, g P L1 pGq is given by  f psqgpt ´ sq ds

p f  gqptq “

pt P Gq ,

G

and the Fourier transform of f P L1 pGq is denoted by fp “ F f , so that  p f psq x´s, γy ds pγ P Γq , pF f qpγq “ f pγq “ G

and

  ApΓq “ fp : f P L1 pGq ,

  where  fp “  f 1 p f P L1 pGqq. In the case where G is a discrete abelian group, this definition coincides with that given in Example 2.1.19. The space ApΓq is a self-adjoint subalgebra of C 0 pΓq, called the Fourier algebra of Γ. The map F : f Þ→ fp,

pL1 pGq,  · 1 ,  q Ñ pApΓq,  ·  , · q ,

is an isometric ∗-isomorphism, and ApΓq is a natural, strongly regular Banach function algebra on the dual group Γ, and so each character on L1 pGq has the form f Þ→ fppγq for some γ P Γ; the Fourier transform F is identified with the Gel’fand transform G in this case. Thus ApΓq – L8 pGq. For properties of ApΓq, see Theorem 4.3.12; in particular, ApΓq has the strong separating ball property. Example 4.2.1 Let Δ be the Cantor set, defined on page 14. Then Δ is identified with {0, 1} ω , and so Δ is a compact abelian group, called the Cantor group. The Haar measure mΔ on Δ is the product measure of the measure on {0, 1} that gives the value 1/2 to each of the two points, so that mΔ pΔq “ 1. Later we shall consider the Fourier algebra ApΔq on Δ.

We recall the following standard facts about L1 pGq. Let G be a LCA group with dual group Γ, and take f P L1 pGq. Suppose that fp P L1 pΓq, and set  fppγqxt, γy dγ pt P Gq , gptq “ Γ

where dγ is a suitably normalized Haar measure on Γ. Then g “ f , and the corresponding map L1 pΓq Ñ ApGq is the inverse Fourier transform.

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255

Take f, g P L2 pGq. Then f g P L1 pGq and  f g1 ď  f 2 g2 . By Plancherel’s theorem, there is a linear isometry F : f Þ→ fp,

L2 pGq Ñ L2 pΓq ,

such that F agrees with the Fourier transform on L1 pGq ∩ L2 pGq. Further, we have x f g “ fp  p g p f, g P L2 pGqq, and     g : f, g P L2 pGq ApΓq “ fp  p with  fp  p (4.2.1) g ď  f 2 g2 . The Fourier–Stieltjes transform of μ P MpGq is denoted by p μ “ F μ, so that  pF μqpγq “ p μpγq “ x´s, γy dμpsq pγ P Γq , G

and BpΓq “ {p μ : μ P MpGq} ,   where p μ “ μ pμ P MpGqq. The space BpΓq is a self-adjoint subalgebra of C b pΓq, called the Fourier–Stieltjes algebra of Γ. The map F : μ Þ→ p μ,

pMpGq,  ·  ,  q Ñ pBpΓq,  ·  , · q ,

is an isometric ∗-isomorphism, and BpΓq is a Banach function algebra on the locally compact group Γ. The Fourier–Stieltjes transform extends the Fourier transform defined on L1 pGq, and so ApΓq is a closed subalgebra of BpΓq. For example, let G be a compact, abelian group, so that the dual group Γ is discrete. Then ApΓq and BpΓq are Banach sequence algebras on Γ. For another example, consider the LCA group R “ pR, + q. Then the dual group of R is identified with the imaginary axis iR, and the Fourier–Stieltjes transform of μ P MpRq is p μ, where  8 p μpiyq “ e´iyt dμptq py P Rq , ´8

so that L pRq and MpRq are identified with ApiRq and BpiRq, respectively. The corp to obtain the inverse Fourier transform is rect normalization of Haar measure on R dy/2π. We take A b pΠq and A 0 pΠq to be the algebras of functions in C b pΠq and C 0 pΠq, respectively, whose restrictions to Π are analytic, so that A 0 pΠq is a natural uniform algebra on Π. For μ P MpRq, we also define the Laplace transform, Lμ, of μ by  8 e´zt dμptq pz P DLμ q , pLμqpzq “ 1

´8

DLμ , the domain of Lμ, consists of the complex numbers z “ x + iy such that where 8 ´xt e d |μ| ptq < 8. Define ´8

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4 Banach algebras on locally compact groups

αpμq “ inf supp μ , with αp0q “ 8, so that αpμq P r´8, 8s. In the case where αpμq ě 0, equivalently when μ P MpR+ q, clearly DLμ ⊃ Π, and pMpR+ q,  q is identified as a subalgebra of A b pΠq and pL1 pR+ q,  q is identified as a subalgebra of A 0 pΠq that is a natural Banach function algebra on Π. As noted after Definition 4.1.13, L1 pR+ q is a semigroup algebra that is a closed subalgebra of the group algebra L1 pRq. For a related algebra, see Example 4.2.21. The famous Titchmarsh’s convolution theorem [50, Theorem 4.7.22] states that αpμ  νq “ αpμq + αpνq whenever μ, ν P L1 pRq and αpμq, αpνq > ´8. We shall use the following result, which is contained in [50, Proposition 4.7.19]. Proposition 4.2.2 Let μ P MpRq. Then the following are equivalent: (a) αpμq ě 0 (b) there exists G P A b pΠq such that Gpiyq “ p μpiyq py P Rq. We now mention two commutative, radical Banach algebras, so that they are not Banach function algebras; these algebras are related to the above-mentioned algebras and are discussed in [50, §4.7]. Example 4.2.3 Let L1 pIq be the Banach space of integrable functions on I, and, for f, g P L1 pIq, define  t f pt ´ sqgpsq ds pt P Iq , p f  gqptq “ 0

so that here  denotes the truncated convolution multiplication. Then pL1 pIq,  q is a commutative, radical Banach algebra, called the Volterra algebra, and is denoted by V. An element f P L1 pIq is nilpotent if αp f q :“ sup{a P r0, 1s : f | r0, as “ 0 a.e.} > 0 , and so the nilpotent elements are dense in V. The algebra V has a sequential contractive approximate identity and the Banach space L1 pIq is weakly sequentially complete, but V does not have an identity, and so, by Lemma 2.3.45, V is not Arens regular; we shall see in Example 6.1.21 that V is strongly Arens irregular. It is a theorem of Dixmier [50, Theorem 4.7.58(i)] that each closed ideal in V has the form { f P V : αp f q ě a} for some a P I. The argument of Theorem 4.1.36 shows that the map L f : g Þ→ f  g, V Ñ V, is compact; this is also immediate from [90, Corollary VI.8.11]. Thus V is a compact algebra. In particular, V is an ideal in its bidual.

Example 4.2.4 Let C pIq denote the Banach space CpIq, with the relative truncated convolution multiplication  given above. Then pC pIq,  q is also a commutative, radical Banach algebra.

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257

We claim that C pIq is also a compact algebra. Indeed, take f P CpIq. Given ε > 0, there exists δ > 0 such that f puq ´ f pvq < ε whenever u, v P I with |u ´ v| < δ. It follows that p f  gqptq ´ p f  gqpt0 q < ε whenever g P CpIqr1s and t, t0 P I with |t ´ t0 | < δ. In particular, the set f  CpIqr1s is pointwise bounded and equicontinuous. By the Ascoli–Arzelà theorem, Theorem 1.1.11, f  CpIqr1s is relatively compact in CpIq, and so L f is a compact operator, giving the claim. By Theorem 2.3.49, C pIq is Arens regular.

The following result will be considerably extended in Chapter 6. Proposition 4.2.5 Let Γ be an infinite, locally compact abelian group. Then ApΓq and BpΓq are not Arens regular. p it follows Proof Since ApΓq is Banach-algebra isomorphic to L1 pGq, where G “ Γ, from Theorem 4.1.15 that ApΓq is not Arens regular. Since ApΓq is a closed subalgebra of BpΓq, it further follows that BpΓq is not Arens regular. Example 4.2.6 Consider the algebra ApZq “ F pL1 pTq,  q, so that ApZq is a Tauberian Banach sequence algebra on Z; here the linear subspace c 00 of ApZq corresponds to the space of trigonometric polynomials on T. The analogue of the sequence pΔn q (of equation (1.1.4)) is pDn q, where Dn is defined for n P N to be the characteristic function of the subset {´n, ´n + 1, . . . , 0, . . . , n ´ 1, n} of Z. For f P L1 pTq and n P N, the function corresponding to Dn fp is S n f , the partial sum of the Fourier series, so that pS n f qpθq “

n 

fppkqeikθ

pθ P p´π, πsq .

k“´n

It is not true that limnÑ8 S n f “ f for each f P L1 pTq, and so pDn q is not an approximate identity for ApZq. However ApZq does have a contractive approximate identity.

Let A be a Banach function algebra with multiplier algebra MpAq. In Definition 3.1.24, we defined the algebras M 00 pAq and M 0 pAq. We now consider these algebras when A is a group algebra on a LCA group. Example 4.2.7 Let G be a non-discrete, locally compact abelian group with dual group Γ, and set A “ ApΓq, so that MpAq “ BpΓq. The character space Φ MpGq of MpGq contains Γ as an open subspace, but the Wiener–Pitt phenomenon shows that Γ is not dense in Φ MpGq . The Fourier–Stieltjes transform of μ P MpGq is the restriction to Γ of the Gel’fand transform of μ defined on Φ MpGq , and so the subset Γ of Φ MpGq is determining for BpΓq. For a study of

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4 Banach algebras on locally compact groups

Φ MpGq , see [145, Chapter 8]. In particular, it is shown that there exists an element μ P MpGq such that p μpγq ě 1 pγ P Γq, but μ is not invertible in MpGq. Further, BpΓq is far from being regular on ΦBpΓq ; we have ΓpBpΓqq  ΦBpΓq , where ΓpBpΓqq is the Šilov boundary of BpΓq. In fact, it is shown in [251] that the space Φ MpGq is not separable. We set B 00 pΓq “ M 00 pAq, so that B 00 pΓq corresponds to the measures μ P MpGq such that p μ | pΦBpΓq \Γq “ 0. It was shown by Hewitt and Zuckerman [171] μ | Γ. Then f 2 P ApΓq, that there exists μ P M s pGq with μ  μ P L1 pGq. Set f “ p and so f P B 00 pΓq. However f  ApΓq, and so ApΓq  B 00 pΓq. Now set B 0 pΓq “ M 0 pAq “ BpΓq ∩ C 0 pΓq , so that B 0 pΓq is the Rajchman algebra on Γ. Then B 0 pΓq is identified with a Banach function algebra on Γ. Then it follows by combining Theorems 6.1.1(i) and 6.8.1 of μ P C 0 pΓq, and such [145] that there exists μ P MpGq such that μ “ μ∗ , such that p μ, so that f P B 0 pΓq and f pΓq Ă R, which implies that that σpμq “ D. Set f “ p f pΓq  σp f q. This shows that B 0 pΓq is not natural on Γ and that B 00 pΓq  B 0 pΓq. In particular, take G “ T, so that B 0 pΓq “ B 0 pZq is a Banach sequence algebra on Z with B 0 pZq Ă c0 pZq such that B 0 pZq is not natural. For a definition of B 0 pΓq for a general locally compact group Γ, see Definition 4.3.43.

Definition 4.2.8 Let G be a LCA group. A Banach space pS ,  · q is a Segal algebra on G if S is a dense linear subspace of pL1 pGq,  · 1 q, if  f S ě  f 1 p f P S q, if S is invariant under all the translations a and a f S “  f S p f P S , a P Gq, and if the map a Þ→ a f, G Ñ S , is continuous for each f P S . Let Γ be the dual group of G, and suppose that S is a Segal algebra on G. Then it is shown in [50, p. 492] that the algebra F pS q of Fourier transforms of elements of S is a Segal algebra with respect to ApΓq in the sense of Definition 3.1.31; in particular, it is a natural Banach function algebra on Γ. Indeed, since L1 pGq has a contractive approximate identity, it follows from Proposition 3.1.29 that F pS q has a multiplier-bounded approximate identity with multiplier-bound 1. We shall give many examples of Segal algebras below and later in the text. For more on Segal algebras, see [50, p. 492] and [270, §6.2]. Clause (i) of the following result is standard [270, Proposition 6.2.8]; clause (ii) was also proved in [245]. Proposition 4.2.9 Let S be a Segal algebra on a LCA group G with dual group Γ. Then: (i) the algebra F pS q is a strongly regular Banach function algebra on Γ; (ii) S is an ideal in its bidual if and only if G is compact.

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259

Proof (i) As we shall note in Theorem 4.3.12(i), the Banach function algebra ApΓq is strongly regular, and so this follows from Corollary 3.1.33. (ii) This follows from Proposition 3.1.34 and Theorem 4.1.37. Let G be a LCA group, with dual group Γ, set A “ pL1 pGq,  q, let pI,  · I q be a weak Segal algebra with respect to A, and let F pIq be the algebra of Fourier transforms of elements in I, so that F pIq is a natural Banach function algebra on Γ. We set J8 pIq “ { f P I : fp P J8 pF pIqq} . By Proposition 3.1.28, F pIq is normal on Γ and J8 pIq “ J8 pAq. For each compact subspace K of Γ, there exists hK P J8 pF pIqq with hK | K “ χK , and so there is a constant C K such that  f I ď C K  f 1 for each f P I with supp fp Ă K; indeed, we can take C K “ hK . The following result of Cigler is given in [39]; we use the notation of the above paragraph. Proposition 4.2.10 Let G be a LCA group, and let pI,  · I q be a weak Segal algebra with respect to L1 pGq. Then I is a Segal algebra on G if and only if J8 pIq is dense in I. Proof By Proposition 4.2.9(i) J8 pIq is dense in I whenever I is a Segal algebra. g Ă K for a compact subset K of Γ. For the converse, take g P J8 pIq, say supp p We first claim that a gI “ gI pa P Gq. Indeed, take a P G. For each ε > 0, there exists f P Ar1s with  f  g ´ g1 < ε, and so     pa f q  g ´ a gI ď C K a p f  g ´ gq1 “ C K  f  g ´ g1 < C K ε . This implies that     a gI ď a g ´ pa f q  gI + pa f q  gI ď C K ε +  f 1 gI ď C K ε + gI , from which the claim follows. We also claim that, for each ε > 0, there is a neighbourhood U of eG such that a g ´ gI < ε pa P Uq. Indeed, for each a P G, we have supp p{ a g ´ gq Ă K, and so a g ´ gI ď C K a g ´ g1 , from which the second claim follows. Now suppose that J8 pIq is dense in I, and take f P I. Then there is a sequence pgn q in J8 pIq with limnÑ8  f ´ gn I “ 0. Take a P G. It follows from the first claim that pa gn q is Cauchy in I, and so converges to an element, say h, in I. We have a gn ´ a f 1 “ gn ´ f 1 ď gn ´ f I , and so h “ a f ; further, a f I “  f I . Finally, take ε > 0. Then there exists g P J8 pIq with  f ´ gI < ε and, by the second claim, there is a neighbourhood U of eG with a g ´ gI < ε pa P Uq, and hence such that a f ´ f I < 3ε pa P Uq. We have shown that I is a Segal algebra on G.

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4 Banach algebras on locally compact groups

Example 4.2.11 Let G be a compact abelian group, and take p and q such that 1 ď p ď q < 8. Then Lq pGq is an ideal in L p pGq that is a Segal algebra with respect to L p pGq.

Example 4.2.12 Let G be a LCA group, with dual group Γ, and take p such that 1 ď p < 8. Define   S p pGq “ f P L1 pGq : fp P L p pΓq , and define

   f S p pGq “  f 1 +  fp

p f P S p pGqq .

p

Then it is easily checked that pS p pGq,  · S p pGq q is a Segal algebra on G, and it is regarded as a strongly regular Banach function algebra on Γ. Indeed, for f P S p pGq and g P L1 pGq, we have   g Γ ď  f S p pGq g1 .  f  gS p pGq ď  f 1 g1 +  fp p p

p “ iR, and write S p for S p pRq. Take In particular, take G “ R and Γ “ R 1 p 2 p χ “ χr´π,πs as an element of L pRq ∩ L pRq, and set u “ χ  χ, so that u is the triangular function with up0q “ 1, with upiyq “ 0 p|y| ě 2πq, and with u linear on p Further, u1 “ 1 and u “ fp, where r´2π, 0s and on r0, 2πs in iR. Thus u P ApRq. f P L1 pRq ∩ C 0 pRq with  f 1 “ 1. For α > 0, set fα ptq “ α f pαtq pt P Rq, so that fα P L1 pRq with  fα 1 “ 1, and set uα “ fpα . Then uα piyq “ upiy/αq py P Rq, so that uα  “ α1/p u p . Clearly fα P S p with  fα S p “ 1 + α1/p u p . By taking α “ n P N, we see that the norm on S p is not equivalent to the norm on L1 pRq, and so S p isa proper subalgebra of L1 pRq. By taking α “ 1/n for n P N, we see that p f1/n /  f1/n S q is a sequence in pS p qr1s that is eventually in Miy for each p   y P R• and is such that u p0q/  f  Ñ 0 as n Ñ 8, and so S has the weak 1/n

1/n S p

p

p Hence S p has the weak separating ball property. separating ball property at 0 P R. For further results on the algebra S p , see Theorem 4.2.14 and Example 5.3.33.

We shall now show that the above Segal algebras on R have a contractive pointwise approximate identity; the constructions are based on those of Inoue and Takahasi in [178]. We require a lemma. Lemma 4.2.13 Let F be a non-empty, finite subset of R, and take ε > 0. Then there is a strictly increasing sequence pnk q in N such that 1 ´ eink y < ε pk P N, y P Fq .

4.2 Locally compact abelian groups

261

In the following theorem, we shall continue to use the notation of Example 4.2.12. Theorem 4.2.14 Take p with 1 ď p < 8. Then the Segal algebra S p has a contractive pointwise approximate identity. Proof Fix ε > 0 and a non-empty, finite subset F of R. First choose n0 P N with n0 > 4/ε and such that 1 ´ upiy/n0 q < ε py P Fq, and g “ χr´2πn0 ,2πn0 s , so that |g| R ď 2n0 . Next choose then take g P L1 pRq ∩ C 0 pRq with p L0 > 0 such that gptq < ε/2 p|t| ě L0 q. Set m “ n20 . By Lemma 4.2.13, there exist numbers N 1 , . . . , Nm P N such that N1 ě L0 and N j+1 > N j p j “ 1, . . . , m ´ 1q and such that 1 ´ e´iN j y < ε/4 p j “ 1, . . . , mq. We define a measure μ on R by setting 1  pδN + δ´N j q , 2m j“1 j m

μ“

so that μ “ 1, and then we set E “ fn0  μ  μ, so that E P L1 pRq with E1 “ 1 μ 2 . Note that 0 ď un0 ď χr´2πn0 ,2πn0 s and that and Ep “ un0 · p 1 cospN j yq m j“1 m

p μpiyq “

py P Rq ,

  so that p μpiyq P R and p μ 2 piyq P I for y P R, and so we can estimate Ep1 as follows: m m   1  ´iN j y 1  iN j y ´iNk y iNk y Ep1 ď pe + e qpe + e q “ gp±N j ± Nk q . 4m2 j,k“1 4m2 j,k“1

Suppose that j  k. Then each of the four numbers of the form ±N j ± Nk are such that ±N j ± Nk > 4L0 , and so gp±N j ± Nk q < ε/2. Thus their contribution to the sum is at most ε/2. Suppose that j “ k. Then we obtain terms gp´2N j q + gp2N j q + 2gp0q, and gp´2N j q + gp2N j q + 2gp0q < 2n0 + ε, and their contribution to the sum   is at most mp2n0 + εq/4m2 < 1/2n0 + ε/4 < ε/2. We conclude that Ep1 < ε. It   follows that Ep p < ε and that ES p < 1 + ε. Now take y P F. Then 

 iy iy p Epiyq p μpiyq2 ´ 1 ď 1 ´ u ´ 1 “ u + 2 1 ´ p μpiyq . n0 n0 We have 1 ´ upiy/n0 q < ε, and also m  ε 1   1 ´ e´iN j y + 1 ´ eiN j y < . 1 ´ p μpiyq ď 2m j“1 2

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4 Banach algebras on locally compact groups

p It follows that Epiyq ´ 1 < 2ε. The above estimates show that S p has a CPAI. Now suppose that G is a compact abelian group, so that the algebras pL p pGq,  q for 1 ď p ď 8 are commutative Banach algebras. By taking Fourier transforms, we can identify L p pGq with a Banach sequence algebra on the discrete dual group Γ. Definition 4.2.15 Let Γ be a discrete abelian group, and take p with 1 ď p ď 8. Then F p pΓq “ {F f : f P L p pGq} , p and F f F pΓq “  f  p p f P L p pGqq. where G “ Γ p Example 4.2.16 Let G be a compact abelian group, with discrete dual group Γ. For p with 1 < p < 8, the algebra S :“ pL p pGq,  q is a Segal algebra on G, and so pF p pΓq, · q is a natural, Tauberian Banach sequence algebra on Γ. The algebra S is a reflexive Banach algebra, and hence it is Arens regular. Further, suppose that G is infinite. Then F p pΓq does not have a bounded pointwise approximate identity, and, by Theorem 4.1.15, ApΓq, is not Arens regular. Thus the fact that a Segal algebra S with respect to a Banach function algebra A is Arens regular does not imply that A is Arens regular. Note that F2 pΓq “  2 pΓq and that, by the Hausdorff–Young theorem, the image of L p pGq under the Fourier transform is contained in  q pΓq whenever 1 < p ď 2, where q “ p . Thus, for each p with 1 < p < 8, we have F p pΓq Ă  r pΓq, where r “ max{2, p }, and so  f F p pΓq ě  f r

p f P F p pΓqq .

(4.2.2)

For a discussion of the multiplier algebra MpS q, see Chapter 4 of the book of Larsen [208] and also [209]; in particular, MpS q “  8 pΓq when 1 < p ď 2, and so MpL1 pGqq  MpS q. Consider F p pZq, and let pDn q and pS n f q, now for f P L p pTq, be as in Example 4.2.6. Then limnÑ8 S n f “ f for each f P L p pTq [201, §1.5], and so pDn q is a multiplier-bounded approximate identity for F p pZq. For further remarks on this algebra, see Example 5.5.16(i).

Example 4.2.17 Let G be a compact abelian group, with dual group Γ, and consider the dense linear subspace pL8 pGq,  · 8 q of pL1 pGq,  · 1 q. Take f P L1 pGq and g P L8 pGq. Then f  g P L8 pGq with  f  g8 ď  f 1 g8 , and so I :“ F8 pΓq is a weak Segal algebra with respect to ApΓq. However the Banach function algebra I is not Tauberian when G is infinite. Indeed, f  g P CpGq p f, g P L8 pGqq; since CpGq  L8 pGq, the ideal I does not have approximate units, and so I is not a Segal algebra. On the other hand, the algebra CpGq is a Segal algebra on G; this follows from Proposition 4.2.10.

4.2 Locally compact abelian groups

263

By Theorem 4.1.38, (iii), (iv) and (v), I is a dual Banach function algebra with Banach-algebra predual ApΓq, I is Arens regular, and I is an ideal in its bidual. By Corollaries 2.4.9, 2.4.18, and 2.4.19, pI  , l q is a dual Banach algebra,  I l I  Ă I, and pI  , l q is Arens regular and also an ideal in its bidual. For further remarks on this algebra, see Example 5.5.16(ii).

The next example, based on an example of Mirkil [242], exhibits a Tauberian Banach sequence algebra that does not have approximate units; further properties of the example are given in [50, Example 4.5.33]. Example 4.2.18 We start with the Banach space pL2 pTq,  · 2 q, and identify T with the closed interval r´π, πs. Write S “ r´π/2, π/2s, and set M “ { f P L2 pTq : f | S P CpS q} . For f P M, define 1  f  “  f 2 + | f |S “ √ 2π



π

´π

2 f pθq d θ

1/2 + | f |S .

We claim that pM,  ·  ,  q is a commutative Banach algebra (for an equivalent norm). Indeed, take f, g P M, and set h “ f  g. By Proposition 4.1.4, we have h P CpTq Ă L2 pTq with h2 ď |h|T ď  f 2 g2 , and so h ď 2  f 2 g2 ď 2  f  g , giving the claim. Further, the trigonometric polynomials are a dense subalgebra of M. Indeed, given f P M and ε > 0, there exists g P CpTq with g | S “ f | S and g ´ f 2 < ε, and there is a trigonometric polynomial h with |h ´ g|T < ε, and then h ´ f  < 3ε, giving the result. We now identify M with its algebra of Fourier transforms on Z, so that M is a Tauberian Banach sequence algebra on Z. Define g0 on T by setting g0 pθq “ 1 p|θ| ď π/2q so that g0 P M. We have 1 gp0 pkq “ π



π 0

g0 pθq “ ´1

pπ/2 < |θ| ď πq ,

 kπ 2 sin gpθq cos kθ dθ “ kπ 2

pk P Z• q ,

with gp0 p0q “ 0, and so gp0 P IpFq, where F “ 2Z. Define μ0 “ δπ/2 + δ´π/2 , so that μ0 P MpS q Ă M  . We have xZ k , μ0 y “ eikπ/2 + e´ikπ/2 “ 2 cospkπ/2q

pk P Zq ,

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4 Banach algebras on locally compact groups

and so xZ k , μ0 y “ 0 for k P Z\F, whence μ0 | JpFq “ 0. However xg0 , μ0 y “ g0 pπ/2q + g0 p´π/2q “ 2 , and so g0 P IpFq\JpFq. Since gp0 J8 pMq Ă JpFqand J8 pMq “ M, it follows that gp0 M Ă JpFq, and so gp0  gp0 M. In particular, M does not have approximate units. Set F1 “ 4Z ∪ {8} and F2 “ p4Z + 2q ∪ {8}. It is noted in [50, Example 4.5.33] that F1 and F2 are sets of synthesis for M  , but that F1 ∪ F2 is not a set of synthesis. On the product space E :“ L2 pTq × CpS q, define the norm   p f, gq “  f 2 + |g|S p f P L2 pTq, g P CpS qq . The map f Þ→ p f, f | S q, M Ñ E, is an isometric embedding. Since the dual space of E is E  “ L2 pTq × MpS q, each Λ P M  is the restriction of a functional pλ, μq, where λ P L2 pTq and μ P MpS q, and it acts on M as follows:  π  π/2 xΛ, f y M ,M “ f pxqλpxq dx + f pxq dμpxq . ´π

´π/2

Since L2 pTq is reflexive and each operator from CpS q to MpS q is weakly compact by Corollary 1.3.53, it follows that M is Arens regular.

Example 4.2.19 Again, take G to be a compact abelian group, with dual group Γ. Take p and q with 1 < p, q < 8, and set A “ pL p pGq,  q “ pF p pΓq, · q and B “ pLq pGq,  q “ pFq pΓq, · q, so that A and B are both reflexive, Tauberian Banach sequence algebras on Γ. The space p B “ L p pGq ⊗ p Lq pGq A “ A⊗ is (isometrically isomorphic to) a natural, Tauberian Banach sequence algebra on Γ × Γ, and hence is an ideal in its bidual. The algebra A has a multiplier-bounded approximate identity. The conjugate indices to p and q are p and q , respectively. Since L p pGq has the approximation property and the Radon–Nikodým property, it follows from Proposition 2.4.20 that the space 



q Lq pGq,  · ε q F :“ pL p pGq ⊗ is a Banach-algebra predual of A, and so A is a dual Banach algebra. We can again identify F as a Banach space with KpA, B q. By Theorem 2.4.17 and Corollary 2.4.19, A and A are Arens regular, and ΦA Ă F; as in Theorem 2.4.10, A “ A  F K .

4.2 Locally compact abelian groups

265

Again, by Theorem 1.4.18(iv), the Banach space A is reflexive if and only if F is reflexive if and only if KpA, B q “ BpA, B q. For example, in the case where G “ T, p Lq pTq is not reflexive for any p and q with 1 < p, q < 8 the Banach space L p pTq ⊗ p  q that was discussed [283, Corollary 2.26], in distinction to the situation for  p ⊗ in Example 3.3.6(i). For further remarks on this algebra, see Example 5.5.16(iii).

We have defined Beurling algebras on an arbitrary group in Example 2.1.13(v). We now consider some Beurling algebras that are defined on the particular abelian group pZ, + q. Example 4.2.20 Let ω be a weight on pZ, + q, and consider the Beurling algebra Bω “ p 1 pZ, ωq,  · ω ,  q. Set ρ1 “ inf{ω1/n n : n P N}

and

ρ2 “ sup{ω´1/n ´n : n P N} ,

so that 0 < ρ2 ď ρ1 < 8. Then set Xω “ {z P C : ρ2 ď |z| ď ρ1 } , so that Xω is an annulus or a circle depending on whether ρ2 < ρ1 or ρ2 “ ρ1 . Of course, when ω is the constant sequence 1, we recover the group algebra  1 pZq, regarded as a natural Banach function algebra on T. Take ϕ P ΦBω , and set ζ “ ϕpδ1 q. Then |ζ| ď ρ1 . Also 1 “ ϕpδ0 q “ ϕpδ´1 qϕpδ1 q, and so |ζ| ě ρ2 . Hence ζ P Xω . For each ζ P Xω , the evaluation map 8  ´8

αn δn Þ→

8 

αn ζ n ,

Bω Ñ C ,

´8

is a character on Bω , and so ΦBω “ Xω . Thus Bω is identified via the Fourier transform as a natural Banach function algebra on Xω ; clearly, Bω is dense in the uniform algebra ApXω q that was defined in Example 3.6.9. We consider when the algebras Bω have the standard properties that we are considering. First, as in Proposition 2.4.28, Bω is an isometric dual Banach function algebra, with Banach-algebra predual c 0 pZ, 1/ωq. Second, the algebras Bω are unital, and so they are not ideals in their biduals. Third, in contrast to the situation for the case where ω “ 1 (see Example 2.3.52), the algebras Bω are often Arens regular; we shall discuss this in §6.3. Again let ω be a weight on pZ+ , + q, and consider the closely related semigroup algebra p 1 pZ+ , ωq,  · ω ,  q . This algebra can be regarded as a subalgebra of the algebra CrrXss of all formal power series on Z+ that contains the algebra of polynomials CrXs, and so it is a

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4 Banach algebras on locally compact groups

Banach algebra of power series; see [50, §4.6] for a discussion of these algebras. 1 + Suppose that inf{ω1/n n : n P N} “ 1. Then  pZ , ωq is identified as a natural Banach function algebra on D and is a dense subalgebra of the disc algebra ApDq. Suppose : n P N} “ 0. Then { f P  1 pZ+ , ωq : f p0q “ 0} is a radical Banach that inf{ω1/n n algebra. These algebras will be considered further in Examples 5.1.34 and 5.3.2(i).

Example 4.2.21 A closely related example of a Beurling algebra is the following. Let ω be a continuous weight on R+ . Then    8 1 + f ptq ωptq dt < 8 . L pR , ωq “ f measurable :  f ω “ 0

Then L1 pR+ , ωq is a commutative Banach algebra with respect to the truncated convolution product that is given by  t f psqgpt ´ sq ds pt P R+ q p f  gqptq “ 0

for f, g P L pR , ωq. Set ρ “ limtÑ8 ωptq1/t . Then L1 pR+ , ωq is semisimple when ρ > 0 and radical when ρ “ 0. For example, the weights ωptq “ p1 + tqα pt P R+ q 2 for α ě 0 give Banach function algebras and the weight ωptq “ e´t pt P R+ q gives a radical Banach algebra. Again set αp f q :“ sup{a P r0, 1s : f | r0, as “ 0 a.e.} for f P L1 pR+ , ωq, and define Ma “ { f P L1 pR+ , ωq : αp f q ě a} 1

+

for a P R+ , so that Ma is a closed ideal, called a standard closed ideal, in L1 pR+ , ωq. In the case where ρ > 0, take σ P R with e´σ “ ρ, set Πσ “ {z P C : z > σ}, and define the Laplace transform L f of f P L1 pR+ , ωq by  8   f ptqe´zt dt z P Πσ . pL f qpzq “ 0

Then L f is analytic on Πσ and L f belongs to C 0 pΠσ q, and L1 pR+ , ωq can be identified as a natural Banach function algebra on Πσ . The algebra L1 pR+ , ωq has a sequential bounded approximate identity and is weakly sequentially complete as a Banach space, but it does not have an identity. It follows from Theorem 2.3.46 that L1 pR+ , ωq is not Arens regular. Take a P R+ . Then the weight ω is regulated at a if ωps + tq “ 0 pt > aq , sÑ8 ωpsq lim

as in [11, Definition 1.3]; in the case where ω is regulated at some a P R+ , set

4.2 Locally compact abelian groups

267

αω “ inf{a P R+ : ω is regulated at a} . (There are radical weights on R+ that are not regulated at any a P R+ .) It is shown in [11, Theorem 2.7] that, when ω is regulated at some a P R+ , the standard ideal Mαω consists of the elements f P L1 pR+ , ωq such that L f is a compact operator on L1 pR+ , ωq; in this case, Mαω is an ideal in its bidual. Thus L1 pR+ , ωq is a compact algebra if and only if the weight ω is regulated at 0; for example, this is the case 2 when ωptq “ e´t pt P R+ q, as above. We shall see in Example 6.1.22 that L1 pR+ , ωq is strongly Arens irregular whenever ω is regulated at some a P R+ . In particular, we can take ω “ 1R+ to obtain the Banach function algebra L1 pR+ q, a closed subalgebra of the group algebra L1 pRq that was mentioned on page 256. For more on this example, see Proposition 5.3.29 and Corollary 6.4.22.

The following example originates with Grabiner in [142, Theorem 6.7(D)]. Example 4.2.22 Fix p with 1 < p < 8, set q “ p , the conjugate index to p, and take a weight sequence ω “ pωn : n P Z+ q with ωn ě 1 pn P Z+ q and ω0 “ 1. Consider the Banach space ⎧ ⎫ ⎛8 ⎞1/p ⎪ ⎪ ⎜⎜⎜ ⎟⎟ ⎪ ⎪ ⎨ ⎬ p + + p p⎟ ⎜ ⎟ : n P Z q : :“ ω < 8  pZ , ωq “ ⎪ α “ pα . α |α | ⎜ ⎟ ⎪ n n n p,ω ⎝ ⎠ ⎪ ⎪ ⎩ ⎭ n“0

+

We claim that the Banach space  pZ , ωq is a Banach algebra with respect to convolution multiplication whenever there exists a constant C > 0 such that p

n  k“0

ωqn ďC ωqk ωqn´k

pn P Z+ q .

(4.2.3)

Indeed, take elements α “ pαn : n P Z+ q and β “ pβn : n P Z+ q in  p pZ+ , ωq, and  set γn “ nk“0 αk βn´k pn P Z+ q. Then it follows from (4.2.3) by Hölder’s inequality that ⎛ n ⎞1/p n  ⎜⎜ ⎟ ωn 1/q ⎜ p p p p ⎟ ⎜ ď C ⎜⎝ |αk | ωk |βn´k | ωn´k ⎟⎟⎟⎠ , |αk | ωk |βn´k | ωn´k |γn | ωn ď ωk ωn´k k“0 k“0 for all n P Z+ , and so 8 

p |γn | p ωn ď C p/q

n“0

8  n 

p

p

|αk | p ωk |βn´k | p ωn´k

n“0 k“0

ď C p/q

8  k“0

p

|αk | p ωk

8  m“0

p p p |βm | p ωm “ C p/q α p,ω β p,ω .

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4 Banach algebras on locally compact groups

Hence α  β P  p pZ+ , ωq with α  β p,ω ď C 1/q α p,ω β p,ω . To obtain a commutative Banach algebra that satisfies our precise definition, we must replace the given norm by an equivalent norm. We also note that it follows from Hölder’s inequality that  p pZ+ , ωq Ă  1 pZ+ q  q whenever 8 n“0 1/ωn < 8, and so, in this case, we can define the Fourier transform p + of elements in  pZ , ωq. For example, set ωr pnq “ p1 + nqr pn P Z+ q , where r > pp ´ 1q/p, so that rq > 1. For each n P Z+ , we see (by considering separately the cases where |n ´ k| ď n/2 and |n ´ k| ě n/2) that equation (4.2.3) is satisfied with 8  1 < 8. C“2 p1 + jqrq j“0 With this choice of weight, the algebra  p pZ+ , ωr q is isometrically algebraically isomorphic, via the Fourier transform, to a natural, unital Banach function algebra on the closed unit disc D that is contained in A+ pDq. This algebra is a reflexive Banach algebra, and so is Arens regular and an ideal in its bidual.

Example 4.2.23 Take p and q with 1 < p, q < 8, and then take r and s such that r > pp ´ 1q/p “ p and s > pq ´ 1q/q “ q . Define A “ p p pZ+ , ωr q,  q and B “ p q pZ+ , ω s q,  q in the notation of Example 4.2.22. Now set p B, A “ A⊗ so that A is a natural, unital Banach function algebra on ΦA “ D × D.   Set F “  p pZ+ , 1/ωr q and G “  q pZ+ , 1/ω s q, so that F and G are Banachq G is a Banachalgebra preduals of A and B, respectively. By Proposition 2.4.20, F ⊗ algebra predual of A. p  q as a Banach space, A is reflexive if and only if Since A is isomorphic to  p ⊗ pq > p + q. When this condition fails, A is not an ideal in its bidual because the identity map IA “ LeA on A is not weakly compact. For more on this algebra, see Example 5.3.2(iii).

We conclude this section with a seminal theorem of Varopoulos; the proof is taken from [145, Theorem 11.1.1], where references to the original sources are given. See also [254, Theorem 1.10.15]. Let Γ be a compact abelian group. In the following theorem, we define the map M : CpΓq Ñ CpΓ × Γq by setting pM f qpx, yq “ f px + yq

px, y P Γ, f P CpΓqq,

4.2 Locally compact abelian groups

269

so that M is an isometric algebra homomorphism, and we also define the map P : CpΓ × Γq Ñ CpΓq by setting  pPFqpyq “ Fpy ´ x, xq dmΓ pxq py P Γ, F P CpΓ × Γqq , Γ

so that P is a linear contraction. Further, Ppg ⊗ hq “ g  h pg, h P CpΓqq and pP ◦ Mqp f q “ f p f P CpΓqq. As before, the Varopoulos algebra on Γ is VpΓq. Theorem 4.2.24 Let Γ be a compact abelian group. Then M : ApΓq Ñ VpΓq is an isometric Banach-algebra embedding whose image is a closed, complemented subalgebra of VpΓq. p “ Γ and pApΓq, · q is identified Proof Take G to be the dual group of Γ, so that G with p 1 pGq,  q. Let f P ApΓq, so that f pxq “

8 

αk xsk , xy

px P Γq ,

k“1

where sk P G and αk P C for k P N and  f  “ pM f qpx, yq “

8 

αk xsk , x + yy “

k“1

8 

8

k“1

|αk |. Then

αk xsk , xy xsk , yy px, y P Γq ,

k“1

 8 and so M f “ 8 k“1 αk sk ⊗ sk P VpΓq with M f π ď k“1 |αk | “  f . Clearly M : ApΓq Ñ VpΓq is a contractive algebra homomorphism. Now let F P VpΓq, and take ε > 0. Then there exist gk , hk P CpΓq for k P N such that 8  gk pxqhk pyq px, y P Γq Fpx, yq “ 8

k“1

and k“1 |gk |Γ |hk |Γ < Fπ + ε. Thus PF “ g, h P CpΓq, we have g  h P ApΓq with

8

k“1

gk  hk in CpΓq. But, for each

g  hApΓq ď g2 h2 ď |g|Γ |h|Γ using equation (4.2.1), and so PF P ApΓq with PFApΓq ď Fπ + ε. This holds for each ε > 0, and so P : VpΓq Ñ ApΓq is a linear contraction. Since pP ◦ Mqp f q “ f p f P ApΓqq, the map M : ApΓq Ñ VpΓq is necessarily an isometry, and this establishes the theorem.

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4 Banach algebras on locally compact groups

4.3 Fourier and Fourier–Stieltjes algebras In this section, our aim is to introduce the Fourier algebra ApΓq and the Fourier– Stieltjes algebra BpΓq that are associated with an arbitrary locally compact group Γ, and then to recall, mostly without proofs, the important properties of these algebras; they often possess some of the abstract properties of Banach function algebras that were introduced in Chapter 3. These algebras are Banach function algebras on Γ, and ApΓq is always natural. Of course, there are several other Banach algebras on Γ related to these algebras that we shall mention; these include the group C ∗ -algebra C ∗ pΓq, the reduced group C ∗ -algebra Cρ∗ pΓq, the group von Neumann algebra V NpΓq, and the Rajchman algebra B 0 pΓq. The seminal study of the Fourier and Fourier–Stieltjes algebras is given in the memoir [97] of Eymard; as we stated, a recent text is [197]. See also [282, §3.2]. Let Γ be a locally compact group. For p with 1 ď p ď 8, we continue to denote the Lebesgue space of p-integrable functions on Γ by L p pΓq, the space being defined with respect to the left Haar measure mΓ . The space L2 pΓq is a Hilbert space with respect to the inner product r · , · s that is defined by  f ptqgptq dmΓ ptq p f, g P L2 pΓqq . r f, gs “ Γ

Let π be a representation of the group algebra pL1 pΓq,  q, so that, as in Theorem 2.2.5, the map π : L1 pΓq Ñ BpHπ q is a contractive ∗-homomorphism for some Hilbert space Hπ . For f P L1 pΓq, define   ||| f ||| “ sup{πp f q : π is a representation of L1 pΓq} , so that ||| f ||| ď  f 1 . Then ||| · ||| is an algebra norm on pL1 pΓq,  q such that ||| f ∗  f ||| “ ||| f |||2

p f P L1 pΓqq ,

and the completion of pL1 pΓq, ||| · |||q is a C ∗ -algebra. Definition 4.3.1 Let Γ be a locally compact group. The completion of the normed algebra pL1 pΓq, ||| · ||| ,  q is the group C ∗ -algebra of Γ; it is denoted by C ∗ pΓq or pC ∗ pΓq, ||| · |||q. Thus the natural embedding of pL1 pΓq,  q into C ∗ pΓq is a contractive ∗-monop is abelian, C ∗ pΓq is identified with C 0 pGq. morphism. In the case where Γ “ G We now define the Fourier–Stieltjes algebra, BpΓq, on Γ as follows. Definition 4.3.2 Let Γ be a locally compact group. Then the Fourier–Stieltjes algebra BpΓq on Γ consists of the functions u P C b pΓq such that    (4.3.1) u “ sup xu, f y “ f u dmΓ : f P pL1 pΓq, ||| · |||qr1s < 8 . Γ

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The following basic result was first given by Eymard in [97, (2.1)]; see also the comments after [197, Lemma 2.1.4]. Theorem 4.3.3 Let Γ be a locally compact group. Then C ∗ pΓq “ BpΓq. Thus the space pBpΓq,  · q is isometrically the dual Banach space of the group C ∗ -algebra C ∗ pΓq, with the duality specified by  f u dmΓ p f P L1 pΓq, u P BpΓqq , (4.3.2) xu, f y “ Γ

and clearly u ě |u|Γ pu P BpΓqq, and hence BpΓq is a dual Banach function algebra, with Banach-algebra predual the group C ∗ -algebra C ∗ pΓq. In the case where p is abelian, this says that C 0 pGq “ MpGq, and so the new definition of BpΓq Γ“G coincides with the one given in §4.2 in this case. Take u, v P BpΓq. It follows from equation (4.3.2) that xuv, f y “ xu, v · f y p f P L1 pΓqq , and so v · f , for v P BpΓq and f P L1 pΓq, is just the pointwise product of the two functions v and f . Since v is bounded on Γ, the function v · f belongs to L1 pΓq; since L1 pΓq is a dense linear subspace of C ∗ pΓq, it follows that v · f P C ∗ pΓq whenever v P BpΓq and f P C ∗ pΓq. Thus xuv, f y ď u |||v · f ||| ď u v ||| f ||| p f P L1 pΓqq , and so uv ď u v. Hence BpΓq is a Banach function algebra on Γ, and C ∗ pΓq is a closed submodule of BpΓq . Proposition 4.3.4 Let Γ be a locally compact group. Then pBpΓq,  ·  , · q is a Banach function algebra on Γ. Theorem 4.3.5 Let Γ be a locally compact group. Then C ∗ pΓq “ BpΓq Ă WAPpΓq “ WAPpL1 pΓqq .

(4.3.3)

Proof We first recall that, by Theorem 4.1.28(ii), WAPpΓq “ WAPpL1 pΓqq. Let ι : L1 pΓq Ñ C ∗ pΓq be the canonical embedding, so that ι is a contractive Banach-algebra ∗-monomorphism. Since the algebra C ∗ pΓq is Arens regular, it follows from Theorem 2.3.29 that WAPpC ∗ pΓqq “ C ∗ pΓq “ BpΓq. Hence, by Proposition 2.3.33, BpΓq Ă WAPpΓq. The following theorem is clear; that BpΓq is weakly sequentially complete follows from Theorem 2.2.16(ii). Theorem 4.3.6 Let Γ be a locally compact group. Then BpΓq is an unital, selfadjoint, translation-invariant, isometric dual Banach function algebra on Γ, with Banach-algebra predual C ∗ pΓq, and BpΓq is weakly sequentially complete.

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In fact, the predual of BpΓq is unique when Γ is discrete. The following result of Pham is [261, Corollary 3]; in this paper, more general results are proved. Proposition 4.3.7 Let Γ be a discrete group. Then C ∗ pΓq is the unique Banachalgebra predual of BpΓq such that BpΓq is a dual Banach algebra. Proof For temporary convenience, we write χ s and δ s for the point mass at s P Γ, where these functions are regarded as elements of BpΓq and of  1 pΓq Ă C ∗ pΓq, respectively. Take M P BpΓq “ C ∗ pΓq and s P Γ. Then χ s · M is such that xχ s · M, uy “ xM, upsqχ s y “ xM, χ s y xu, δ s y pu P BpΓqq , and so χ s · M “ xM, χ s y δ s . Let F Ă BpΓq be a Banach-algebra predual of BpΓq. Then C ∗ pΓq is the closed linear span of {χ s · M : s P Γ, M P F} in BpΓq , and so C ∗ pΓq Ă F. By Proposition 1.3.28, F “ C ∗ pΓq, which gives the result. Corollary 4.3.8 Let G be a compact abelian group. Then CpGq is the unique Banach-algebra predual of pMpGq,  q. There is an alternative approach to the definition of BpΓq that we mention briefly. Definition 4.3.9 Let Γ be a locally compact group. A function f P CpΓq is positivedefinite if, for each n P N, each t1 , . . . , tn P G, and each α1 , . . . , αn P C, necessarily n 

αi α j f pti´1 t j q ě 0 .

i, j“1

The space of positive-definite functions on Γ is denoted by PDpΓq. Let t P Γ and f P PDpΓq. By taking n “ 2, t1 “ t, and t2 “ eG in the above definition, we see that the matrix ! f peΓ q f ptq f pt´1 q f peΓ q 2 is positive and hermitian, and so f pt´1 q “ f ptq and f peΓ q2 ´ f ptq ě 0. Thus each positive-definite function on Γ is bounded, and | f |Γ “ f peΓ q p f P PDpΓqq. Let f, g P PDpΓq, and take α P R+ . Then f + g and α f belong to PDpΓq, and so PDpΓq is a cone in C b pΓq+ . The space PDpΓq is discussed in [85, 13.4.4] and [197, §1.4]. The proof of the following theorem that characterizes the functions in PDpΓq and the Banach function algebra BpΓq can be found in [85, Theorem 13.4.5] and [197,

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§2.1]. A unitary representation of Γ on a Hilbert space H is a group homomorphism π from Γ into the unitary group UpHq of BpHq such that π is continuous when UpHq has the weak operator topology. Theorem 4.3.10 Let Γ be a locally compact group. A continuous function f : Γ Ñ C is positive definite if and only if there is a unitary representation π of Γ on a Hilbert space H and a cyclic vector ξ P H such that f ptq “ rπptqξ, ξs

pt P Γq .

Further, BpΓq “ lin PDpΓq. For each f P BpΓq, there are a unitary representation π of Γ on a Hilbert space H and ξ, η P H such that f ptq “ rπptqξ, ηs

pt P Γq ;

the norm of f in BpΓq is the infimum of ξ η over all such representations of f , and the infimum is attained. We now turn to the definition of the Fourier algebra, ApΓq. Definition 4.3.11 Let Γ be a locally compact group. Then ApΓq is the closure in BpΓq of the ideal BpΓq ∩ C00 pΓq; ApΓq is the Fourier algebra of Γ. Clearly ApΓq is a closed ideal in BpΓq, and so BpΓq Ă MpApΓqq; ApΓq is the closed linear span in BpΓq of PDpΓq ∩ C 00 pΓq. The following basic result is contained in [97] and [197, §§2.3, 2.4]; clause (v) is a special case of Proposition 4.4.4, to be proved below, and clause (vi) follows because ApΓq is a closed linear subspace of BpΓq. Theorem 4.3.12 Let Γ be a locally compact group. Then: (i) ApΓq is a natural, self-adjoint, translation-invariant, strongly regular Banach function algebra on Γ, and ApΓq is dense in C 0 pΓq; (ii) the algebra ApΓq consists precisely of the functions of the form f  r g, where f, g P L2 pΓq, and g, f, g P L2 pΓq} u “ inf{ f 2 g2 : u “ f  r

pu P ApΓqq ,

the infimum being attained; (iii) ApΓq is unital if and only if ApΓq “ BpΓq if and only if Γ is compact; (iv) ΦBpΓq “ Γ ∪ H and Γ ∩ H “ ∅, where H is the hull of ApΓq as an ideal in BpΓq, and Γ is determining for BpΓq; (v) ApΓq has the strong separating ball property; (vi) ApΓq is weakly sequentially complete.

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p Then, in the above identification Again let G be an LCA group, and set Γ “ G. of BpΓq with MpGq, the ideal ApΓq corresponds to L1 pGq, and so the new definition of ApΓq coincides with the one given in §4.2 in this case. Now suppose that Γ is a discrete group. Then δeΓ P PDpΓq. Since ApΓq is translation-invariant, it follows that c 00 pΓq Ă ApΓq. Since c 00 pΓq is dense in ApΓq, the Fourier algebra ApΓq is a Tauberian Banach sequence algebra on Γ, and it is clear that (4.3.4)  1 pΓq Ă ApΓq Ă c 0 pΓq . The algebra ApΓq is an ideal in its bidual when Γ is discrete; conversely, it will be shown in Proposition 4.4.5 that Γ is discrete whenever ApΓq is an ideal in its bidual. We discussed A-invariant linear subspaces of F when pA, Fq is a dual Banach algebra in Definition 2.4.14 and Proposition 2.4.15. Now suppose that pA, Fq is a dual Banach algebra, where F is Banach function algebra on a locally compact space Ω. Take x P Ω. Then ε x · f and f · ε x are defined in F as in Definition 2.4.14. For example, xa, ε x · f y “ xaε x , f y pa P A, f P Fq, where aε x is the product in A. Lemma 4.3.13 Let V be a closed linear subspace of F, and suppose that ε x · f P V and f · ε x P V for each x P Ω and F P V. Then V is A-invariant. Proof This follows because lin {ε x : x P Ω} is weak-∗ dense in F  “ A and V K is weak-∗ closed, and so V K is an ideal in A. Hence V is A-invariant. Now suppose that Ω is a semigroup and the product of ε x and εy in A is such that ε x εy “ ε xy px, y P Ωq. This implies that ε x · f “ r x f and f · ε x “  x f for f P V and x P Ω. Thus in this case V is A-invariant if and only if r x f and  x f belong to V whenever x P Ω and f P V, so that V is translation-invariant by elements of Ω. Hence we obtain the following result as a consequence of Theorem 2.4.15. Theorem 4.3.14 Let pA, Fq be a dual Banach algebra, where F is Banach function algebra on a locally compact space Ω. Suppose that Ω is a semigroup such that ε x εy “ ε xy px, y P Ωq, and suppose that V is a closed linear subspace of F that is translation-invariant by elements of Ω and is such that V K has an identity. Then V is complemented in F. Corollary 4.3.15 Let Γ be a locally compact group, and let V be a closed linear subspace of BpΓq that is invariant under left and right translations by the elements of Γ. Then V is complemented in BpΓq. In particular, ApΓq is complemented in BpΓq. Proof Let V be a closed linear subspace of BpΓq as specified, and set A “ BpΓq , so that A “ C ∗ pΓq and ε x εy “ ε xy px, y P Γq in A. Then A is von Neumann algebra, and V is A-invariant, as above. Since V K is a weak-∗-closed ideal in A, it follows that V K has an identity, and so the result follows from Theorem 4.3.14. Let Γ be a locally compact group, and take H to be the Hilbert space L2 pΓq. For t P Γ and h P H, define

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275

λt phqpsq “ hpt´1 sq ps P Γq , so that λt phq “ t´1 h and λt P BpHq. Then t Þ→ λt , Γ Ñ BpHq, is the left-regular representation of Γ in BpHq. We recall from page 96 that the weak-∗ (or ultra-weak) topology on BpHq comes p H  q . In this duality, an operator T P BpHq acts on from the duality BpHq – pH ⊗ f ⊗ g, where f, g P H, by the formula x f ⊗ g, T y “ rT f, gs, and so  f ptsqgpsq dmΓ psq pt P Γq , x f ⊗ g, λt´1 y “ rλt´1 f, gs “ Γ

whereas pf  r gqptq “



 Γ

f psqgpt´1 sq dmΓ psq “

Γ

f ptsqgpsq dmΓ psq

pt P Γq ,

and hence pf  r gqptq “ x f ⊗ g, λt´1 y pt P Γq .

(4.3.5)

This shows that λt acts as a character on ApΓq for each t P Γ. Since ApΓq is natural, each character on ApΓq has this form. See also [197, Theorem 2.3.8]. Thus the character εt for t P Γ corresponds to the functional λt´1 . Definition 4.3.16 Let Γ be a locally compact group. Then the weak-∗ closure of the linear space lin {λt : t P Γ} in BpL2 pΓqq is the group von Neumann algebra of Γ; it is denoted by V NpΓq. p is Thus V NpΓq is indeed a von Neumann algebra. In the case where Γ “ G 8 abelian, V NpΓq is identified with L pGq. For t P Γ, the character εt corresponds to λt´1 P BpHq, and so we can embed Γ in V NpΓq as εt Þ→ λt´1 , and this map is a group monomorphism taking εeΓ to 1H . Thus lin {εt : t P Γ}, where the closure is taken in V NpΓq, is a C ∗ -subalgebra of V NpΓq. Definition 4.3.17 Let Γ be a locally compact group. Then Cδ∗ pΓq “ lin {εt : t P Γ}. Thus Cδ∗ pΓq is a C ∗ -subalgebra of V NpΓq. For more on this C ∗ -algebra, see [15], [35], [36], and [211, §4]. The following theorem was proved by Eymard in [97, Théorème (3.10)], and it is given with extra information in [197, Theorem 2.3.9]. In the theorem, H “ L2 pΓq. Theorem 4.3.18 Let Γ be a locally compact group. For each λ P ApΓq , there is a unique operator T λ P V NpΓq such that rT λ p f q, gs “ xg  fq, λy p f, g P Hq . The mapping

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4 Banach algebras on locally compact groups

λ Þ→ T λ ,

ApΓq Ñ V NpΓq ,

is an isometric linear isomorphism, and the weak-∗ topology, σpApΓq , ApΓqq, corp H  q, on V NpΓq, and so ApΓq responds to the relative weak-∗ topology, σpBpHq, H ⊗ is the unique isometric Banach-algebra predual of V NpΓq. Corollary 4.3.19 Let Γ be a locally compact group. Then ApΓq is weakly sequentially complete. Proof Since ApΓq “ V NpΓq, it follows from Theorem 2.2.16(ii) that the bidual space ApΓq “ V NpΓq is weakly sequentially complete. The linear subspace WAPpApΓqq of V NpΓq is discussed by Granirer in [147]. Let Γ1 and Γ2 be two locally compact groups. Then there is a natural contraction p ApΓ2 q Ñ ApΓ1 × Γ2 q θ : ApΓ1 q ⊗ such that θp f1 ⊗ f2 qpγ1 , γ2 q “ f1 pγ1 q f2 pγ2 q pγ1 P G1 , γ2 P G2 q for f1 P ApΓ1 q and f2 P ApΓ2 q. We record the following theorem of Losert [233] that is related to Theorem 4.1.12; see also [197, Theorem 3.6.2]. It seems to be an open question whether θ is always an injection. p ApΓ2 qq Theorem 4.3.20 Let Γ1 and Γ2 be locally compact groups. Then θpApΓ1 q ⊗ is a dense subalgebra of ApΓ1 × Γ2 q, with   p ApΓ2 qq . θpFqApΓ ×Γ q ď FApΓ1 q ⊗p ApΓ2 q pF P ApΓ1 q ⊗ 1

2

p ApΓ2 qq “ ApΓ1 × Γ2 q if and only if either of the groups Γ1 or Further, θpApΓ1 q ⊗ Γ2 has an abelian subgroup of finite index, and, in this case, the Banach function p ApΓ2 q and ApΓ1 × Γ2 q are isomorphic as Banach algebras. algebras ApΓ1 q ⊗ We now mention two other algebras that are closely related to C ∗ pΓq and BpΓq. They are the reduced group C ∗ -algebra Cρ∗ pΓq and the reduced Fourier–Stieltjes algebra Bρ pΓq, respectively. Let Γ be a locally compact group, and again set H “ L2 pΓq. For f P L1 pΓq, define λ f : g Þ→ f  g , H Ñ H , so that λ f P BpHq. Then λ : f Þ→ λ f , L1 pΓq Ñ BpHq, is the left-regular representation of L1 pΓq on H; this map is a representation of L1 pΓq. Definition 4.3.21 Let Γ be a locally compact group. Then the closure of the set {λ f : f P L1 pΓq} in BpHq is the reduced group C ∗ -algebra of Γ; it is denoted by Cρ∗ pΓq.

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277

Clearly Cρ∗ pΓq is a C ∗ -subalgebra of V NpΓq. For f P L1 pΓq, we have   λ f op “ sup { f  g : g P L2 pΓqr1s } .   Then λ f op ď ||| f ||| p f P L1 pΓqq, and so the map λ : L1 pΓq Ñ Cρ∗ pΓq extends to a contractive C ∗ -epimorphism from C ∗ pΓq onto Cρ∗ pΓq. Hence Cρ∗ pΓq is C ∗ isomorphic to a quotient algebra of the group C ∗ -algebra pC ∗ pΓq, ||| · |||q. It follows that the dual space Cρ∗ pΓq of Cρ∗ pΓq is isometrically a weak-∗-closed ideal in the algebra BpΓq. Definition 4.3.22 Let Γ be a locally compact group. Then the dual space of Cρ∗ pΓq is the reduced Fourier–Stieltjes algebra of Γ; it is denoted by Bρ pΓq. Since ApΓq “ { f  r g : f, g P H}, it is clear that each function in ApΓq acts on Cρ∗ pΓq as a bounded linear functional, and so ApΓq Ă Bρ pΓq Ă BpΓq Ă MpApΓqq ,

(4.3.6)

where ApΓq and Bρ pΓq are closed ideals in BpΓq. For each u P Bρ pΓq, we have    1 u “ sup uptq f ptq dmΓ ptq : f P pL pΓq,  · op qr1s , Γ

  and so xu, λ f y ď u λ f op pu P Bρ pΓq, f P L1 pΓqq. For each u P Bρ pΓq and each f P L1 pΓq, as before the element u · f is just the pointwise product of the functions u and f , and so the algebra Bρ pΓq is a dual Banach function algebra, with Banach-algebra predual Cρ∗ pΓq. Theorem 4.3.23 Let Γ be a locally compact group. Then u “ sup { xu, λy : λ P Cδ∗ pΓqr1s } pu P Bρ pΓqq .

(4.3.7)

Proof We have Cδ∗ pΓq Ă Bρ pΓq “ Cρ pΓq . Since Γ is a determining set for the Banach function algebra Bρ pΓq, the space Cδ∗ pΓq is weak-∗ dense in the von Neumann algebra Bρ pΓq . So, by Kaplansky’s density theorem, Theorem 2.2.14, Cδ∗ pΓqr1s is weak-∗ dense in Bρ pΓqr1s . It follows from Corollary 1.2.14 that equation (4.3.7) holds. The following theorem of Granirer and Leinert is taken from [152, Theorem B2 ]; a related result is given as [197, Proposition 2.7.1]. Theorem 4.3.24 Let Γ be a locally compact group. Take u P BpΓq, and suppose that puα q is a net in BpΓq that converges to u in the weak-∗ topology, σpBpΓq, C ∗ pΓqq, and is such that limα uα  “ u. Then lim uα v ´ uv “ 0 pv P ApΓqq . α

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The next result will be used when we determine the BSE norm of ApΓq; see Theorem 5.3.18(ii). Clauses (ii) and (iii) in the theorem are similar to Propositions 2.19 and 2.20 of the thesis of Arsac [10]. Theorem 4.3.25 Let Γ be a locally compact group. Then: (i) ApΓqr1s is σpBρ pΓq, Cρ∗ pΓqq-dense in Bρ pΓqr1s ; (ii) the closed unit ball Bρ pΓqr1s is closed in C b pΓq in the compact-open topology on Γ; (iii) for each u P Bρ pΓqr1s , there is a net puα q in ApΓqr1s that converges to u in the compact-open topology on Γ. Proof (i) Set E “ Bρ pΓq, F “ ApΓq, and G “ Cρ∗ pΓq, so that F is closed in E, G is closed in F  , and G “ E. Then we are in the setting of Proposition 1.2.23, and so (i) follows from that result. (ii) Take a net puα q in Bρ pΓqr1s that converges to a function u P C b pΓq in the compact-open topology. Since C00 pΓq is dense in pL1 pΓq,  · 1 q, it follows that   uptq f ptq dmΓ ptq p f P L1 pΓqq , lim uα ptq f ptq dmΓ ptq “ α

Γ

Γ

and this implies that   uptq f ptq dm ptq “ lim u ptq f ptq dm ptq ď  f  ∗ Γ α Γ Cρ pΓq Γ

α

Γ

p f P L1 pΓqq .

Since L1 pΓq is dense in Cρ∗ pΓq, we conclude that uBρ pΓq ď 1, and hence that u P Bρ pΓqr1s , giving the result. (iii) Take u P Bρ pΓq with u “ 1. By (i), there is a net puα q in ApΓqr1s that converges to u in the topology σpBρ pΓq, Cρ∗ pΓqq. Since 1 “ u ď lim inf uα  ď lim sup uα  ď 1 , α

α

where we are using inequality (1.2.6), we have limα uα  “ u. Take g P L1 pΓq. Then, in both the pBpΓq, C ∗ pΓqq and pBρ pΓq, Cρ∗ pΓqq dualities, we have  gptquα ptq dmΓ ptq “ xλg , uα y . xg, uα y “ Γ

Since the net puα q is bounded in ApΓq and L1 pΓq is dense in C ∗ pΓq, it follows that uα Ñ u in the topology σpBpΓq, C ∗ pΓqq. By Theorem 4.3.24, limα uα v ´ uv “ 0 for each v P ApΓq. Since ApΓq is regular, this implies that limα uα “ u in the compact-open topology. Corollary 4.3.26 Let Γ be a locally compact group, and take u P C b pΓq. Then u P Bρ pΓq if and only if there is a bounded net puα q in ApΓq that converges to u in the compact-open topology on Γ.

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279

Proof This follows by combining clauses (ii) and (iii) of Theorem 4.3.25. Corollary 4.3.27 Let Γ be a discrete group. Then Bρ pΓqr1s is closed in p 8 pΓq, τ p q. Proof This follows from Theorem 4.3.25(ii) because, in the case where Γ is discrete, the compact-open topology is the same as τ p . Clauses (i) and (ii) of the following theorem are given in [197, Corollaries 2.2.2 and 2.2.3], respectively, following Eymard [97]; the result is a consequence of a more general theorem given in [197]. Theorem 4.3.28 Let Γ be a locally compact group. Then: (i) BpΓq “ BpΓd q ∩ C b pΓq and uBpΓq “ uBpΓd q pu P BpΓqq ; (ii) the closed unit ball BpΓqr1s is closed in pC b pΓq, τ p q. p we have already noted that In the case where G is a LCA group and Γ “ G, ∗ 1 C pΓq “ Cρ pΓq “ C 0 pGq, that ApΓq “ L pGq, that APpGq Ă L8 pGq “ L1 pGq , and that BpΓq “ Bρ pΓq “ MpGq “ MpApΓqq. There are very many properties of the Fourier algebra ApΓq that depend on whether the locally compact group Γ is amenable; we shall collect some of these below. In the following theorem, the fact that ApΓq has a bounded approximate identity if and only if Γ is amenable was first proved by Leptin [225]; for a proof, see [197, Theorem 2.7.2] or [282, Theorem 1.4.1]. The equivalence of (a) and (j) is a major theorem of Losert that generalizes Wendel’s theorem; for a proof of the equivalence of (a), (h), (i), and (j), see [197, Theorem 5.1.10]. A more general version of the fact that (g) implies (a) will be noted in Theorem 4.4.11; a proof of this implication will also be given in Proposition 5.3.22. ∗

Theorem 4.3.29 Let Γ be a locally compact group. Then the following conditions are equivalent: (a) the group Γ is amenable; (b) Cρ∗ pΓq “ C ∗ pΓq ; (c) Bρ pΓq “ BpΓq ; (d) 1Γ P Bρ pΓq ; (e) ApΓq has a bounded approximate identity; (f) ApΓq has a contractive approximate identity; (g) ApΓq has a contractive pointwise approximate identity; (h) pBpΓq,  · BpΓq q “ pMpApΓqq,  · op q ; (i) ApΓq is closed in pMpApΓqq,  · op q; (j)  f op “  f  p f P ApΓqq.

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It follows that, in the case where Γ is an amenable locally compact group, we have (4.3.8) ApΓq Ă Bρ pΓq “ BpΓq “ MpApΓqq . Corollary 4.3.30 Let Γ be an amenable locally compact group. Then Bρ pΓqr1s is closed in C b pΓ, τ p q. Proof This follows from Theorem 4.3.28(ii) because Bρ pΓq “ BpΓq in the case where Γ is amenable. We do not know whether Bρ pΓqr1s is closed in C b pΓ, τ p q for each locally compact group Γ; this is true when Γ is discrete by Corollary 4.3.27 and when Γ is amenable by Corollary 4.3.30. Let Γ be a locally compact group, and consider the set of pairs pK, εq, where K is a compact subspace of Γ and ε > 0 ; for two such pairs, pK1 , ε1 q and pK2 , ε2 q, we set pK1 , ε1 q ď pK2 , ε2 q if K1 Ă K2 and ε1 ě ε2 , so obtaining a directed set. Now suppose that Γ is amenable. Then, as in [197, Theorem 2.7.2], we can suppose that the contractive approximate identity in ApΓq consists of functions upK,εq ,  of the form where upK,εq P J8 pApΓqq, where upK,εq ´ 1K K < ε, and where Γ\K upK,εq < ε. It seems to be unknown whether ApΓq has approximate units for every locally compact group Γ. Some identifications of elements of MpApΓqq\BpΓq in the case where Γ is not amenable are given in [197, Chapter 5]. Example 4.3.31 Let Γ be a compact group. Then ApΓq “ BpΓq “ C ∗ pΓq , and so ApΓq is a dual Banach algebra, with Banach-algebra predual C ∗ pΓq. Now let Γ be a locally compact, amenable group, and suppose that ApΓq is a dual Banach algebra. Since ApΓq has a bounded approximate identity, it follows from Proposition 2.4.7 that ApΓq has an identity, and so Γ is compact. It appears to be an open question whether there is a group Γ that is not amenable, but is such that ApΓq is a dual Banach algebra; see [72]. In the case where Γ is abelian, but not compact, ApΓq is not isometrically a dual space. However there are locally compact groups Γ that are not compact such that ApΓq is isometrically a dual space; see the paper of Taylor [302, Theorem 4.1] for a discussion of this and related geometric properties of ApΓq. The following theorem is given within [197, Theorem 3.1.4, Proposition 6.5.7, and Theorem 6.5.11], and is based on [123]. Recall that Rc pΓq is the closed coset ring of Γ; clause (i) is Host’s idempotent theorem. Theorem 4.3.32 (i) Let Γ be a locally compact group. Then the non-zero idempotents in BpΓq are the functions χE , where E is a finite union of translates of open subgroups of Γ. (ii) Let Γ be an amenable locally compact group. Then each set E in Rc pΓq is a set of synthesis for ApΓq.

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(iii) Let Γ be an amenable locally compact group, and let I be a non-zero, closed ideal in ApΓq. Then I has a bounded approximate identity if and only if I “ IpEq for some set E P Rc pΓq. Corollary 4.3.33 Let Γ be an amenable locally compact group. Then ApΓq/IpEq is weakly sequentially complete for each set E P Rc pΓq. Proof By Theorem 4.3.29, the algebra ApΓq has a CAI, by Corollary 4.3.19, ApΓq is weakly sequentially complete, and, by Theorem 4.3.32(iii), IpEq has a BAI when E P Rc pΓq. Thus this follows from Corollary 2.3.76. Corollary 4.3.34 Let Γ be an amenable locally compact group. Then int E is closed in Γ for each E P Rc pΓq. Proof Take E P Rc pΓq, and set U “ int E; we may suppose that U  ∅. By Theorem 4.3.32(iii), the ideal IpEq in ApΓq has a BAI, say peα q. We may suppose that limα eα “ u in pBpΓq, σpBpΓq, C ∗ pΓqq. For each f P IpEq, we have limα f eα “ f in ApΓq and limα f eα “ f u in BpΓq, and so f “ f u. In particular, eα “ eα u for each α, and so u “ u2 . Clearly upγq “ 1 pγ P Γ\Eq, and hence upγq “ 1 pγ P Γ\Uq because u P C b pGq. Now let K be a compact neighbourhood of a point in U, so that χK P L1 pΓq Ă C ∗ pΓq, and   u dmΓ “ upsqχK psq dmΓ psq “ xu, χk y “ lim xeα , χK y “ 0 , K

Γ

α

and so u | K “ 0. It follows that u “ χΓ\U , and hence U is closed in Γ. Corollary 4.3.35 Let Γ be an amenable locally compact group. Suppose that I and J are two closed ideals in ApΓq such that I ‘ J “ ApΓq. Then there is a clopen set E in Rc pΓq such that I “ IpEq and J “ IpΓ\Eq. Proof Since Γ is amenable, the algebra ApΓq has a CAI; since I ‘ J “ ApΓq, both I and J have a BAI, and so, by Theorem 4.3.32(iii), there exist E, F P Rc pΓq such that I “ IpEq and J “ IpFq. Clearly E ∪ F “ Γ and E ∩ F “ ∅, and so F “ Γ\E and both E and F are clopen in Γ. Example 4.3.36 A notable example of an amenable group is the so-called ‘ax + bgroup’, also called S 2 . This group can be considered as the subgroup of S Lp2, Rq consisting of matrices of the form

 ab , 01 where a P R+• and b P R, so that S 2 is a non-compact, locally compact group; the group S 2 is not unimodular. As in [256, Example (0.5)], S 2 is amenable, and

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so, by Theorem 4.3.29, ApS 2 q has a (sequential) contractive approximate identity. The algebra ApS 2 q is not unital, and so it follows from Proposition 2.4.7 that ApS 2 q is not a dual Banach algebra. However, it is shown by Khalil in [203, Proposition 3] that ApS 2 q is isometrically the dual Banach space of a certain C ∗ -subalgebra of C ∗ pS 2 q.

Example 4.3.37 For a study of multiplier-bounded approximate identities in the Fourier algebra ApΓq, see [197, §5.6]. Let F2 be the free group on two generators. Then F2 is not amenable, and so ApF2 q does not have a bounded pointwise approximate identity. However, Haagerup in [158] proved that Cρ∗ pF2 q has the ‘metric approximation property’. As an application, he proved that ApF2 q has a multiplier-bounded approximate identity (with multiplier-bound 1). For a proof of this result, see [197, Theorem 5.7.6]; see also the seminal paper of de Cannière and Haagerup [33] and the book [34]. It is shown in [197, Corollary 5.7.7] that the Fourier algebra ApS Lp2, Rqq has a multiplier-bounded approximate identity (with multiplier-bound 1). An example of a locally compact group Γ such that ApΓq does not have a multiplier-bounded approximate identity is given by Dorofaeff in [86].

Proposition 4.3.38 Let Γ be a locally compact group. Then Γ is amenable if and only if C ∗ pΓq is a closed subalgebra of V NpΓq. Proof Suppose that Γ is amenable. By Theorem 4.3.29, (a) ⇒ (b), we have Cρ∗ pΓq “ C ∗ pΓq, and always Cρ∗ pΓq is a closed subalgebra of V NpΓq. Conversely, suppose that C ∗ pΓq is a closed subalgebra of V NpΓq. We first claim that ApΓqr1s is weak-∗ dense in BpΓqr1s . Indeed, define the Banach spaces E “ BpΓq, F “ ApΓq, and G “ C ∗ pΓq. Then F is closed in E and G “ E, and G is closed in F  by the hypothesis, and so the claim follows from Proposition 1.2.23. The constant function 1Γ belongs to BpΓqr1s , and so there is a net in ApΓqr1s that converges weak-∗ to 1Γ . But Bρ pΓq is a weak-∗-closed ideal in BpΓq, and so 1Γ P Bρ pΓq. By Theorem 4.3.29, (d) ⇒ (a), Γ is amenable. In §3.4, we discussed the space TIMpϕq of topologically invariant means at a point ϕ P ΦA for an arbitrary Banach function algebra A; see Definition 3.4.12. We first expand this notation in the special case that A “ pL1 pGq,  q for a locally p to be the point eΓ . Here the correscompact abelian group G, taking ϕ P Γ “ G ponding character on A is the augmentation character ϕ 0 of Definition 4.1.19; an element M P L8 pGq belongs to the space TIMpeΓ q if xM, ϕ 0 y “ 1 and

f · M “ ϕ 0p f q M

p f P L1 pGqq .

Let Γ be a locally compact group. The size of TIMpeΓ q for the Fourier algebra ApΓq has been studied; the following theorem of Hu is given in [175, Theorem 5.9].

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For a locally compact group Γ, we denote by bpΓq the minimum cardinality of a neighbourhood base of eΓ . Theorem 4.3.39 Let Γ be group that is not discrete, and consider a locally compact bpΓq the algebra ApΓq. Then TIMpeΓ q “ 22 . By Theorem 3.4.19, in the case where Γ is discrete, TIMpeΓ q “ 1. Set in the above theorem. It also follows from Proposition 3.4.14(iii) A “ ApΓq bpΓq that AA ě 22 . In the case where Γ is amenable, A has a bounded approximate bpΓq identity, and so AA “ A · A by Theorem 2.1.46, whence |A · A | ě 22 . For a result related to the above, see Theorem 4.4.8. The following theorem of Lau and Ülger is taken from [219], where several other equivalent properties are given. Theorem 4.3.40 Let Γ be a locally compact group. Then the following conditions on Γ are equivalent: (a) Γ is compact; (b) BpΓq has the Schur property; (c) Bρ pΓq has the Schur property; (d) ApΓq has the Schur property; (e) BpΓq has the Dunford–Pettis property and the Radon–Nikodým property. Corollary 4.3.41 Let Γ be a compact group, and let A be a closed subalgebra of BpΓq. Suppose that A is faithful. Then  · op and  ·  are equivalent on A. Proof By Theorem 4.3.40, (a) ⇒ (b), BpΓq has the Schur property, and so A has the Schur property. The result now follows from Theorem 3.1.23. Let Γ be a locally compact group. We can consider (weak) Segal algebras with respect to ApΓq in the sense of Definitions 3.1.27 and 3.1.31. We give an example. Example 4.3.42 Let Γ be a locally compact group. We define LApΓq “ ApΓq ∩ L1 pΓq ,

 f  “  f ApΓq +  f 1

p f P LApΓqq ,

as in [130]. Set A “ ApΓq and I “ LApΓq, and denote the norms on A and I by  · A and  · I , respectively. Then it is clear that pI,  · I q is a Banach space and that I is an ideal in A. Since I contains J8 pAq, the ideal I is dense in A. Take f P A and g P I. Then  f gI ď  f A gA + | f |Γ g1 ď  f A gA +  f A g1 ď  f A gI , and so I is a Banach A-module. Hence I is a weak Segal algebra with respect to A. In particular, I is a natural, normal Banach function algebra on Γ. The algebra I is called the Lebesgue–Fourier algebra on Γ.

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Suppose that Γ is discrete. Then it follows immediately from equation (4.3.4) that LApΓq “  1 pΓq. We claim that LApΓq “ ApΓq if and only if Γ is compact. Clearly LApΓq “ ApΓq if and only if ApΓq Ă L1 pΓq. Suppose that Γ is compact. Then ApΓq Ă CpΓq Ă L1 pΓq. Suppose that ApΓq Ă L1 pΓq. Then there exists a constant C > 0 such that  f 1 ď C  f A p f P Aq. In the case where Γ is not compact, there exists a sequence pKn q of compact, symmetric subspaces of Γ such that mΓ pKn q Ñ 8 as n Ñ 8. Set fn “

1 χ K  χ Kn mΓ pKn q n

pn P Nq .

Take n P N. By Theorem 4.3.12(ii), fn P ApΓqr1s . However,  fn 1 “ mΓ pKn q following equation (4.1.5), so it is not true that ApΓq Ă L1 pΓq. This gives the claim. Now suppose that the group Γ is amenable, so that ApΓq has a contractive approximate identity consisting of functions of the form upK,εq P J8 pApΓqq, as described after Theorem 4.3.29. Clearly this contractive approximate identity forms an approximate identity in I, and so I is a Segal algebra with respect to ApΓq. Thus, by Proposition 3.1.29, I has a multiplier-bounded approximate identity and, by Corollary 3.1.33, I is strongly regular. Further, let F be a non-empty, finite subset  of Γ, and take ε > 0. Then there exists u P ApΓqr1s with |u ´ 1F |F < ε and |u| < ε. In the case where Γ is not discrete, F |u| “ 0, and hence uI < 1 + ε. Γ\F This shows that I has a contractive pointwise approximate identity. We shall discuss the BSE properties of LApΓq in Example 5.3.32 and whether it is Arens regular in Example 6.4.5.

We shall conclude this section by considering the Rajchman algebra; it is related to ApΓq and BpΓq. Definition 4.3.43 Let Γ be a locally compact group. The Rajchman algebra, B 0 pΓq, is BpΓq ∩ C 0 pΓq. Thus B 0 pΓq is a closed ideal of BpΓq and ApΓq Ă B 0 pΓq, so that B 0 pΓq is a Banach function algebra on Γ. This example was defined in the case where Γ is abelian in Example 4.2.7. For studies of this algebra in the general case, see [133] and [199]. It follows from Proposition 4.3.15 that both ApΓq and B 0 pΓq are complemented as Banach subspaces of BpΓq. Moreover, since the dual of B 0 pΓq is a von Neumann algebra, the same theorem shows that ApΓq is a complemented subspace of B 0 pΓq. This implies that ApΓq is a complemented subspace of B 0 pΓq . The character space of B 0 pΓq has the form ΦB 0 pΓq “ ΦApΓq ∪ H “ Γ ∪ H ,

(4.3.9)

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285

where H is the hull of ApΓq when ApΓq is considered as a closed ideal in B 0 pΓq. As noted in §3.1, both Γ ∩ H “ ∅ and H is closed in ΦB 0 pΓq . There are some examples of infinite locally compact groups Γ such that B 0 pΓq coincides with ApΓq, but, in general, B 0 pΓq is considerably larger than ApΓq. For example, the quotient B 0 pΓq/ApΓq can be far from being a radical Banach algebra, and, in particular, B 0 pΓq  ApΓq, whenever Γ is abelian and not compact, as noted in Example 4.2.7. The quotient BpΓq/B 0 pΓq is always a semisimple Banach algebra [199, Proposition 1.2]. It is shown in [199, Corollary 2.2] that B 0 pΓq is regular as a Banach function algebra on Γ if and only if B 0 pΓq is natural. For examples where B 0 pΓq is natural, see [199, Example 2.6].

4.4 Figà-Talamanca–Herz algebras We shall now introduce some Banach function algebras on a locally compact group Γ that generalize the Fourier and Fourier–Stieltjes algebras of the previous section. These are the Figà-Talamanca–Herz algebras, A p pΓq and B p pΓq, that are defined for 1 < p < 8; the case where p “ 2 returns us to the algebras ApΓq and BpΓq, respectively. After defining the algebras A p pΓq and B p pΓq, and some related algebras, we shall show that these algebras often have properties that allow us to apply our general, earlier results, at least when Γ is amenable. There are several C ∗ -algebras associated with ApΓq and BpΓq, and the well-known theory of C ∗ -algebras can be used to establish properties of these algebras; it is harder to establish analogous properties of the algebras A p pΓq and B p pΓq because C ∗ -algebra techniques are rarely available. For a study by Daws of the algebras A p pΓq in the setting of ‘p-operator spaces’, see [69]. Let Γ be a locally compact group with left Haar measure mΓ , take p such that 1 < p < 8, set q “ p , and then define p Lq pΓq,  · π q , X p “ pL p pΓq ⊗ so that X p is the Banach space that is the projective tensor product of L p pΓq and Lq pΓq. For f P L p pΓq and g P Lq pΓq, we have  f pxyqgpyq dmΓ pyq px P Γq , pf  q gqpxq “ Γ

so that, by Proposition 4.1.4, f  q g P C 0 pΓq and f  q g Γ ď  f  p gq . Hence there is a continuous linear map π p : X p Ñ C 0 pΓq such that π p p f ⊗ gq “ f  q g p f P L p pΓq, g P Lq pΓqq . The range of this map is defined to be A p pΓq; the norm on A p pΓq is the quotient norm and the product is the pointwise product. It is a theorem of Herz that, in this

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way, A p pΓq is a Banach function algebra on Γ; see [50, Theorem 4.5.30]. In the case where Γ is compact, the function 1Γ is the identity of A p pΓq, and then A p pΓq is an unital Banach function algebra. Definition 4.4.1 Let Γ be a locally compact group, and take p with 1 < p < 8. Then pA p pΓq, · q is a Figà-Talamanca–Herz algebra on Γ. The algebra A p pΓq is described in [50, pp. 493–494], in the books of Derighetti [79] and Pier [262, §10pBq]; in each source, it is shown that A p pΓq is indeed a natural Banach function algebra on Γ. A different approach to these algebras is given by Gilbert in [138]; a seminal paper is that of Herz [165]. By Theorem 4.3.12(ii), the Fourier algebra ApΓq is the algebra A2 pΓq. The following theorem is contained in [50, Theorem 4.5.31]; see also [79, Chapter 3], [98], [165, Propositions 2 and 3], and [219, §8]. Theorem 4.4.2 Let Γ be a locally compact group, and take p with 1 < p < 8. Then A p pΓq is a natural, self-adjoint, translation-invariant, strongly regular Tauberian Banach function algebra on Γ, and A p pΓq is dense in pC 0 pΓq, | · |Γ q. For each u P A p pΓq, we have ⎫ ⎧8 ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ f ,   g  uA p pΓq “ inf ⎪ k k p q⎪ ⎪ ⎪ ⎭ ⎩ k“1

where the infimum is taken over all sequences p fk q in L p pΓq and pgk q in Lq pΓq,  gk . Further, uA p pΓq ě |u|Γ . respectively, such that u “ 8 k“1 fk  q Corollary 4.4.3 Let Γ be a locally compact group, and take p with 1 < p < 8. Then A p pΓqr2s is dense in A p pΓq. Proof Since A p pΓq is strongly regular, this follows from Proposition 3.1.10(ii). Proposition 4.4.4 Let Γ be a locally compact group, and take p with 1 < p < 8. Then A p pΓq has the strong separating ball property, and so TIMpγqr1s  ∅ for each γ P Γ. Proof Take x P Γ and U P N x . There is a symmetric, open, relatively compact neighbourhood V of eΓ such that xV 2 Ă U, say α “ 1/mΓ pVq. Set f “ α1/p χ xV and g, so that g“q g “ α1/q χV , where q “ p . Then  f  p “ gq “ 1. Now set u “ f  q u P A p pΓqr1s . Clearly  1/p+1/q upxq “ α χ xV pxyq dmΓ pyq “ α · p1/αq “ 1 V

and supp u Ă xV 2 Ă U. Hence A p pΓq has the SSBP. By Theorem 4.4.2, A p pΓq is strongly regular, and so TIMpγqr1s  ∅ for each γ P Γ by Theorem 3.4.16.

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The next proposition was first proved by Forrest in [120, Lemma 3.3]. Proposition 4.4.5 Let Γ be a locally compact group, and take p with 1 < p < 8. Then the Banach function algebra A p pΓq is an ideal in its bidual if and only if Γ is discrete. Proof Suppose that the group Γ is discrete. Then A p pΓq is a Tauberian Banach sequence algebra, and so it is an ideal in its bidual. By Proposition 4.4.4, A p pΓq has the SBP, and so, by Theorem 3.4.5(ii), Γ is weakly discrete. By Proposition 2.3.17, the weak and weak-∗ topologies on Γ coincide whenever A p pΓq is an ideal in its bidual, and so, in this case, Γ is discrete.  The following result was proved by Herz as [165, Theorem 1 and Proposition 6]; see [79, §7.8]. Theorem 4.4.6 Let Γ be a locally compact group, and take p with 1 < p < 8. (i) Suppose that Γ0 is a closed subgroup of Γ. Then the map u Þ→ u | Γ0 , defined for u P A p pΓq, is a continuous epimorphism from A p pΓq onto A p pΓ0 q, and A p pΓ0 q is identified with the quotient algebra A p pΓq/IpΓ0 q. (ii) Suppose that N is a compact, normal subgroup of Γ. Then A p pΓ/Nq is identified with a closed subalgebra of A p pΓq. It seems not to be known whether the Figà-Talamanca–Herz algebras A p pΓq are always weakly sequentially complete for each locally compact group Γ. However we do have the following theorem of Granirer [149, Lemma 18], which gives a partial result. We recall that A p pΓqH denotes the space of elements of A p pΓq whose support is contained in a closed subspace H of Γ. Theorem 4.4.7 Let Γ be a locally compact group, take p with 1 < p < 8, and take a compact subspace H of Γ. Then the Banach space A p pΓqH is weakly sequentially complete. Theorem 4.4.8 Let Γ be a metrizable locally compact group, take p such that 1 < p < 8, and set A “ A p pΓq. Then the following conditions are equivalent: (a) AA Ă WAPpAq ; (b) there exists γ P Γ such that TIMpγq “ 1 ; (c) Γ is discrete; (d) A is an ideal in its bidual. Proof Recall from Theorem 4.4.2, Proposition 4.4.4, and Theorem 4.4.7 that A is strongly regular, has the SSBP, and is such that AH is weakly sequentially complete for each compact subset H of ΦA . (a) ⇒ (b) This follows from Corollary 3.4.17.

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(b) ⇒ (c) Since Γ is metrizable, {γ} is a Gδ -subset of Γ, and so it follows from Theorem 3.4.19 that γ is an isolated point in Γ. Hence Γ is discrete. (c) ⇔ (d) This is given within Proposition 4.4.5. (d) ⇒ (a) By Theorem 2.3.7, L f is weakly compact for each f P A, and so this follows from Proposition 2.3.37. The above proof used the metrizability of Γ only for the implication (b) ⇒ (c); in the case where p “ 2, metrizability can be avoided [214, Corollary 4.11]. The following theorem of Herz [164, Theorem C] is also proved in [79, §8.3, Theorems 8 and 9]. Theorem 4.4.9 Let Γ be an amenable locally compact group, and take p and q with 1 < p ď q ď 2 or 2 ď q ď p < 8. Then ApΓq Ă Aq pΓq Ă A p pΓq and uA p pΓq ď uAq pΓq ď uApΓq pu P ApΓqq. In particular, ApΓq Ă A p pΓq whenever 1 < p < 8, and A p pΓq is a Banach ApΓq-module. The above inclusions also hold for some non-amenable groups [167], but are false for certain non-amenable groups [229]. As is the case for the algebra ApΓq, there are many properties of the algebras A p pΓq that depend on whether Γ is amenable. We first give a result of Chou and Xu [36]. Proposition 4.4.10 Let Γ be a locally compact group, and take p with 1 < p < 8. Then Γ “ ΦA p pΓq is weakly closed in A p pΓq if and only if Γ is amenable. In the following, the implication (b) ⇒ (a) follows from Propositions 3.5.8 and 4.4.10, and the implication (e) ⇒ (a) is due to Losert [231]; see also [262, Theorem 10.4]. Theorem 4.4.11 Let Γ be a locally compact group, and take p with 1 < p < 8. Then the following are equivalent: (a) Γ is amenable; (b) A p pΓq has a bounded pointwise approximate identity; (c) A p pΓq has a bounded approximate identity; (d) A p pΓq has a contractive approximate identity; (e) A p pΓq factors weakly. Let Γ be an amenable locally compact group, and take p with 1 < p < 8. It is shown in [262, Theorem 10.4] that the functions upK,εq that were described after Theorem 4.3.29 also form a contractive approximate identity for the algebra A p pΓq.

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Corollary 4.4.12 Let Γ be an amenable locally compact group with |Γ| ě 2, and take p such that 1 < p < 8. Then each maximal modular ideal in A p pΓq has a bounded approximate identity of bound 2. Proof Since Γ is amenable, Theorem 4.4.11 shows that A p pΓq has a CAI. By Proposition 4.4.4, TIMpγqr1s  ∅ pγ P Γq, and so Proposition 3.4.15(i) applies to give the result. In [63, Example 3.15], it is shown that the minimum bound of a bounded pointwise approximate identity in a maximal modular ideal of ApΓq is 2 whenever Γd is amenable. We do not know the minimum bound in general for A p pΓq when Γ is amenable. See also [77]. Consider the special case where the locally compact group Γ has the approximation property (see [240]). For such groups, it is shown by Miao in [240, Proposition 3.7] that the given norm on A p pΓq is equivalent to the operator norm if and only if Γ is amenable. We do not know whether this is true for groups Γ that do not have the approximation property; by Theorem 4.3.29, this is the case when p “ 2. Let Γ be a locally compact group. We can also consider weak Segal and Segal algebras with respect to A p pΓq. We give an example that generalizes the Lebesgue– Fourier algebra of Example 4.3.42. Example 4.4.13 Let Γ be a locally compact group, and take p with 1 < p < 8 and r with 1 ď r < 8. We define Arp pΓq “ A p pΓq ∩ Lr pΓq ,

 f  “  f A p pΓq +  f r

p f P Arp pΓqq ,

following Granirer in [151]. Essentially as in Example 4.3.42, we see that Arp pΓq is a weak Segal algebra with respect to A p pΓq, and so Arp pΓq is a natural, normal Banach function algebra on Γ. Suppose that Γ is discrete. Then clearly  p pΓq ∪  q pΓq Ă A p pΓq, where q “ p , and hence Arp pΓq “  r pΓq whenever 1 ď r ď p ∨ q. Now suppose that the locally compact group Γ is amenable, so that A p pΓq again has a contractive approximate identity consisting of functions of the form upK,εq P J8 pA p pΓqq. Clearly this contractive approximate identity forms an approximate identity in Arp pΓq, and so Arp pΓq is a Segal algebra with respect to A p pΓq, Arp pΓq has a multiplier-bounded approximate identity, and Arp pΓq is strongly regular. Further, Arp pΓq has a contractive pointwise approximate identity when Γ is also non-discrete. We shall discuss the BSE properties of Arp pΓq in Example 5.5.14 and whether it is Arens regular in Example 6.4.5.

Each element f P L1 pΓq defines a bounded linear functional ϕ f on A p pΓq by  xu, ϕ f y “ f u dm pu P A p pΓqq , Γ

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and the norms of ϕ f in A p pΓq and the operator norm of λ pf are equal, where λ pf : g Þ→ f  g ,

L p pΓq Ñ L p pΓq ,

and so L1 pΓq can be regarded as a linear subspace of A p pΓq . As in [79], we denote by PF p pΓq and PM p pΓq the spaces of p-pseudo-functions and p-pseudo-measures, respectively; they are the closures of L1 pΓq in A p pΓq in the norm and weak-∗ topologies, respectively, so that we have PF2 pΓq “ Cρ∗ pΓq and PM2 pΓq “ V NpΓq. The space PM p pΓq is identified in [79, Chapter 4] with the dual space to A p pΓq, so that (4.4.1) A p pΓq “ PM p pΓq . The spaces PF p pΓq and PM p pΓq are Banach A p pΓq-modules: the module action is defined by xu · T, vy “ xT, uvy

pu, v P A p pΓq, T P PM p pΓqq .

The following definition and remarks are taken from the book of Derighetti [79]. Definition 4.4.14 Let Γ be a locally compact group, and take p with 1 < p < 8. An operator T P BpL p pΓqq is a p-convolution operator if T pt f q “ t pT f q pt P Γ, f P L p pΓqq . The collection of these p-convolution operators is denoted by CV p pΓq. Clearly CV p pΓq is a closed subalgebra of BpL p pΓqq. Take T P BpL p pΓqq. Then, by [79, Theorem 5, §1.2], T P CV p pΓq if and only if T p f  gq “ f  T g p f P L1 pΓq, g P L p pΓqq . As in [79, p. 48], there is a natural embedding of PM p pΓq into CV p pΓq for each locally compact group Γ. By [79, Corollary 3, §5.4], we have the identification CV p pΓq “ PM p pΓq whenever the locally compact group Γ is amenable; this result is generalized by Daws and Spronk to groups with the approximation property in [74, Theorem 1.1]. There are varying definitions of a Banach function algebra B p pΓq on a locally compact group Γ for 1 < p < 8 in the literature; these definitions all agree whenever the group Γ is amenable. First Herz gave a definition of B p pΓq for each locally compact group Γ in [166, p. 146], and showed in [166, Théoreme 1(iii)] that A p pΓq is a closed ideal in B p pΓq and that, in the case where Γ is amenable, B p pΓq is identified with the multiplier algebra MpA p pΓqq of A p pΓq; see also [262, Proposition 19.9] and [78, 79]. In [46], Cowling defined B p pΓq to be MpA p pΓqq for each locally compact group Γ; see also [262, §10pBq]. Further, Runde [281] defined B p pΓq in terms of representations; the advantage of Runde’s definition is that B2 pΓq “ BpΓq for every locally compact group Γ, and so there is consistency in the notation. We shall not refer to B p pΓq in the case that Γ is not amenable. In the case where Γ is compact, we have A p pΓq “ B p pΓq “ MpA p pΓqq.

4.4 Figà-Talamanca–Herz algebras

291

Theorem 4.4.15 Let Γ be an amenable locally compact group, and take p with 1 < p < 8. Then: (i) the embedding of A p pΓq in B p pΓq is isometric, and B p pΓq “ MpA p pΓqq isometrically; (ii) B p pΓq is a dual Banach function algebra, with Banach-algebra predual PF p pΓq; (iii) B p pΓq “ B p pΓd q ∩ C b pΓq, and uB p pΓq “ uB p pΓd q pu P B p pΓqq. Proof These were essentially first proved by Herz in [166]; clause (i) is [166, Théoreme 1(iii)] and [166, Théorem 2(v)] shows that B p pΓq is the dual Banach space of PF p pΓq, and again (ii) follows from the fact that L1 pΓq is dense in PF p pΓq. Further, clause (iii) is [78, Theorem 7]. Corollary 4.4.16 Let Γ be a locally compact group such that Γd is amenable, and take p with 1 < p < 8. Then B p pΓqr1s is closed in pC b pΓq, τ p q. Proof Since Γd is amenable, B p pΓd q “ PF p pΓd q by Theorem 4.4.15(ii). From the definition of the space PF p pΓd q , it follows that a bounded net puα q in B p pΓd q converges weak-∗ to u P B p pΓd q if and only if puα q converges to u in pC b pΓq, τ p q. By using Theorem 4.4.15(iii), we see that B p pΓqr1s is closed in pC b pΓq, τ p q.

Chapter 5

BSE norms and BSE algebras

Let G be a locally compact abelian group, with dual group Γ. The classical Bochner– μ Schoenberg–Eberlein theorem states the following. Take f P C b pΓq. Then f “ p for some μ P MpGq if and only if there is a constant β ě 0 with the following property: for each n P N, each γ1 , . . . , γn P Γ, and each α1 , . . . αn P C, necessarily     n n      αi f pγi q ď β  αi γi  .   8  i“1  i“1 L pGq

Further, in this case, the infimum of the constants β that satisfy the above inequality is μ. This theorem is proved in the text of Rudin [276, Theorem 1.9.1], for example. This basic theorem for abelian groups was proved by Bochner [22] in the case where Γ “ R; an integral analogue was given by Schoenberg [291]; the case for general abelian groups was given by Eberlein in [92]. Takahasi and Hatori [297], taking this result as a starting point, introduced the notions of BSE functions, BSE norms, and BSE algebras associated with a Banach function algebra. These topics now form a significant strand within commutative Banach algebra theory. Our aim in this chapter is to present an account of these topics; we shall include important known results, sometimes with shorter proofs, a number of new theorems, and many examples. Let A be a Banach function algebra. We shall start in §5.1 by defining the key space LpAq “ lin ΦA as a linear subspace of the dual space A ; for λ P LpAq,  we have λ “ ni“1 αi ϕi in LpAq, where n P N, α1 , . . . , αn P C, and ϕ1 , . . . , ϕn are distinct points in ΦA . The annihilator LpAqK of LpAq in the bidual space A is in fact a closed ideal in pA , l q, and so we can define the quotient Banach algebra QpAq as A /LpAqK ; we shall see that QpAq is also a Banach function algebra (on ΦQpAq ) and its restriction to ΦA is contained in 8 pΦA q as a subalgebra and containing A as a subalgebra. The basic properties of QpAq will be set out in §5.1, and we shall calculate a number of examples. From §5.2 onwards, we shall consider the more familiar algebra C BS E pAq, which is the closed subalgebra of QpAq equal to QpAq ∩ C b pΦA q, so that C BS E pAq is a © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 H. G. Dales and A. Ülger, Banach Function Algebras, Arens Regularity, and BSE Norms, CMS/CAIMS Books in Mathematics 12, https://doi.org/10.1007/978-3-031-44532-3_5

293

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5 BSE norms and BSE algebras

Banach function algebra on ΦA . The norm on C BS E pAq is denoted by  · BSE , and the algebra A has a BSE norm if this norm is equivalent on A to the given norm. The multiplier algebra MpAq of a Banach algebra A was defined in §2.1. In the case where A is a Banach function algebra, we have MpAq “ { f P C b pΦA q : f A Ă A} , as in §3.1; the norm on MpAq is the operator norm,  · op . The algebra A is defined to be a BSE algebra if MpAq “ C BS E pAq as algebras. The properties of these algebras will be explored in §5.2. In §5.3, we shall give a considerable number of examples, including the seminal example, generalizing the classical Bochner–Schoenberg–Eberlein theorem, of the Fourier algebra A “ ApΓq on a locally compact group Γ. Indeed, we shall show in Theorem 5.3.18 that ApΓq has a BSE norm for every locally compact group Γ and in Theorem 5.3.23 that ApΓq is a BSE algebra if and only if Γ is amenable. We shall seek to extend these results to the Figà-Talamanca–Herz algebras A p pΓq, where 1 < p < 8, but are only partially successful in this; see Theorem 5.3.38. Also in §5.3, we shall discuss when various Banach function algebras, including uniform algebras, have the mentioned properties. We shall conclude the section by considering the Varopoulos algebra VpK, Lq, where K and L are non-empty, compact spaces; we shall show in Theorem 5.3.40 that FBSE “ Fπ pF P VpK, Lqq, but it will be shown in Theorem 5.3.42 that VpK, Lq is usually not a BSE algebra. In §5.4, we shall consider when the 1 -norm on the space LpAq is equivalent to the given norm from A for a Banach function algebra A; we shall show that this is equivalent to the fact that C BS E pAq “ C b pΦA q, and we shall exhibit examples of Banach function algebras that do not have a BSE norm. In §5.5, we shall consider Banach function algebras that are ideals in their biduals, especially those that have a multiplier-bounded approximate identity. This enables us to determine when various Banach sequence algebras that were introduced in §3.2 are BSE algebras and when they have a BSE norm.

5.1 The space LpAq and the algebra QpAq Let A be a Banach function algebra. In this section, we shall consider not just the space ΦA as a subset of A , but also its linear span, to be called LpAq, and then we shall consider the quotient algebra QpAq “ A /LpAqK , showing that QpAq is also a Banach function algebra (on ΦQpAq ); we shall give some examples and some properties of this algebra. Here A is again the bidual of A with the first Arens product, l . A key theorem, Theorem 5.1.35, will identify LpIq and QpIq when I is a weak Segal algebra with respect to A and has a contractive pointwise approximate identity. The space LpAq and the algebra QpAq are discussed in [64], where many of the results of this section are given; there is an implicit definition of our algebra QpAq

5.1 The space LpAq and the algebra QpAq

295

in [254, Theorem 3.1.14], where some of the properties of QpAq that we develop are recorded. Definition 5.1.1 Let A be a Banach function algebra, and take a non-empty subset Ω of ΦA . Then LpA, Ωq is the linear span of Ω as a subset of A , with LpAq for LpA, ΦA q.  When we write λ “ ni“1 αi ϕi for a non-zero element in LpA, Ωq, we shall always suppose that n P N, that α1 , . . . , αn P C, and that ϕ1 , . . . , ϕn are distinct points in Ω. Of course, the annihilator LpAqK is a closed linear subspace of A . As Banach spaces, we have the canonical identification LpAq – A /LpAqK .

(5.1.1)

Since f · ϕ “ ϕ · f “ f pϕq ϕ p f P A, ϕ P ΦA q, the space LpAq is an A-submodule of A and (5.1.2) LpAq Ă AA “ A A . Further, LpAq is a closed A-submodule of A ; in the case where A has a bounded approximate identity, LpAq Ă A · A. Indeed, it follows from equation (2.3.4) that LpAq is an introverted submodule of A ; clearly LpAq Ă APpAq Ă WAPpAq . By Corollary 1.2.21(ii), LpAqJK is the weak-∗ closure of LpAq in A . However it is clear that LpAqJ “ {0}, and so LpAq is always weak-∗ dense in A . Also LpAqK “ ΦKA “ {M P A : x M, ϕy “ 0 pϕ P ΦA q} . Clearly A ∩ LpAqK “ {0} in A , and hence A + LpAqK is an algebraic direct sum of Banach spaces. Example 5.1.2 Let Γ be a locally compact group, with Fourier algebra ApΓq. Then, in the notation of Definition 4.3.17, we have LpApΓqq “ Cδ∗ pΓq, so that the space LpApΓqq is a C ∗ -subalgebra of V NpΓq “ ApΓq . Further, by equation (4.3.7), we have   u “ sup {xu, λy : λ P LpApΓqqr1s } pu P Bρ pΓqq .

Theorem 5.1.3 Let A be a Banach function algebra. Then LpAqK is a closed ideal in A , the space A ‘ LpAqK is a subalgebra of A , and the quotient space QpAq :“ A /LpAqK – LpAq is a commutative, semisimple Banach algebra.

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5 BSE norms and BSE algebras

Proof Since LpAq is a closed, introverted submodule of A , Theorem 2.3.60 shows that LpAqK is a closed ideal in A , and so A ‘ LpAqK is a subalgebra of A and QpAq (with the quotient norm) is a Banach algebra. Take M, N P A . It follows from equation (2.3.5) that M l N ´ N l M P LpAqK , and so QpAq is a commutative algebra. For each ϕ P ΦA , the map r : M + LpAqK Þ→ xM, ϕy , ϕ

QpAq Ñ C ,

rpM + LpAqK q “ 0 for all is a character on QpAq, and M + LpAqK “ 0 whenever ϕ ϕ P ΦA , and so QpAq is semisimple. Thus QpAq is a Banach function algebra, with character space ΦQpAq (where ΦQpAq has the relative weak-∗ topology, σpQpAq , QpAqq ). For M P A , the corresponding element in QpAq “ A /LpAqK is denoted by rMs, so that rM l Ns “ rMs rNs We define

pM, N P A q .

    rMsQpAq “ M + LpAqK 

pM P A q ,

the quotient norm for rMs. Note that, as in Proposition 1.3.11, given f P QpAq, there exists M P A with rMs “ f and M “  f QpAq . Each element f P A determines r f s in QpAq, and r f s | ΦA “ f , and so we can regard A as a subalgebra of QpAq by identifying each f P A with r f s P QpAq. In particular,   (5.1.3) | f |ΦA ď  f QpAq “  f + LpAqK  ď  f  p f P Aq . We define a map R : M Þ→ M | ΦA ,

A Ñ QpAq ,

by setting RpMqpϕq “ xrMs, ϕy pϕ P ΦA q. This map R is a contraction and an algebra epimorphism with kernel LpAqK , and so we can make the identification QpAq “ { f P 8 pΦA q : f “ M | ΦA for some M P A } . It follows from equation (1.3.3) that    f QpAq “ sup{x f, λy : λ P LpAqr1s }

p f P QpAqq .

(5.1.4)

Thus a function f P 8 pΦA q belongs to QpAq if and only if there is a constant β f > 0  such that, for each λ “ ni“1 αi ϕi in LpAq, the inequality   n    αi f pϕi q ď β f λ  i“1  holds, and then  f QpAq is the infimum of these constants β f .

5.1 The space LpAq and the algebra QpAq

297

Proposition 5.1.4 Let A be a Banach function algebra. Then  f  “  f QpAq

p f P Aq

if and only if LpAqr1s is weak-∗ dense in Ar1s . Proof By Corollary 1.2.14, the linear space LpAq is norming if and only if LpAqr1s is weak-∗ dense in Ar1s , so this follows from equation (5.1.4). Example 5.1.5 Note that it may be that LpAq “ A , so that LpAqK “ {0}. For example, take A “ c 0 , so that LpAq “ c 00 Ă A “ 1 . Then LpAq “ A and QpAq “ A “ Cpβ Nq.

We shall determine the corresponding algebra QpAq for some other Banach function algebras A later in this section. The following is immediate from the definitions. Proposition 5.1.6 Let pA,  · q be a reflexive Banach function algebra. Then pQpAq,  · QpAq q “ pA,  · q . Example 5.1.7 For example, let G be a compact abelian group. For 1 < p < 8, the algebra A :“ pL p pGq,  q is a Segal algebra on G, and A is identified with pF p pΓq, · q, as in Example 4.2.16. Since A is reflexive as a Banach space, it follows from Proposition 5.1.6 that QpAq “ A.

r, Let A be a Banach function algebra, and take ϕ P ΦA . By identifying ϕ with ϕ defined in Theorem 5.1.3, we can, and shall, regard ΦA as a determining subset of ΦQpAq . We shall write ΦA for the closure of ΦA in ΦQpAq with respect to the weak∗ topology of ΦQpAq ; we shall see in Example 5.1.19 that the embedding of ΦA in ΦQpAq need not be continuous and in Example 5.1.24 that the image of ΦA in ΦQpAq need not be dense. Further, given ϕ P ΦQpAq , define ιpϕq on A by ιpϕqpMq “ x rMs, ϕy pM P A q . Then ιpϕq P ΦA , and the map ι : ΦQpAq Ñ ΦA is a homeomorphism that identifies ΦQpAq as closed subspace of ΦA . Take a function f on ΦA . We associate to f a linear functional τ f on LpAq by setting ⎛ ⎞ n n   ⎜⎜⎜ ⎟⎟ αi f pϕi q ⎜⎜⎝λ “ αi ϕi P LpAq⎟⎟⎟⎠ , xτ f , λy “ i“1

and we then define

i“1

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5 BSE norms and BSE algebras

    τ f  “ sup{xτ f , λy : λ P LpAqr1s } P r0, 8s . Proposition 5.1.8 Let A be a Banach function algebra. Then   QpAq “ { f P 8 pΦA q : τ f  < 8} . 8 Proof  It follows from the Hahn–Banach theorem that, for each f P pΦA q such  that τ f  < 8, there exists M P A with f “ M | ΦA . This implies the result.

Since QpAq “ LpAq and LpAq is identified with the annihilator of LpAqK in A , each Λ P LpAq acts on QpAq through the formula 

x rMs, ΛyQpAq,QpAq “ xM, ΛyA ,A

pM P A q .

(5.1.5)

Clearly F :“ LpAq is a closed QpAq-submodule of QpAq , and the map that identifies QpAq with F  is exactly the canonical map T F : QpAq Ñ F  of equation (2.4.1), and so the space LpAq is an isometric Banach-algebra predual of QpAq. This gives the following theorem. Theorem 5.1.9 Let A be a Banach function algebra. Then QpAq is an isometric dual Banach function algebra, with Banach-algebra predual LpAq. Further, Ar1s is weak∗ dense in QpAqr1s . In particular, equation (5.1.5) holds when Λ “ ψ P ΦQpAq . Since we have ΦA Ă ΦQpAq Ă LpAq Ă A , each ψ P ΦQpAq is of the form ψ “ ϕ + ξ, where ϕ “ ψ | A P ΦA ∪ {0} and ξ P AK , and so xr f s, ψy “ x f, ϕy p f P Aq, essentially as on page 108. It follows that   f pΦA q Ă r f spΦQpAq q Ă f pΦA q ∪ {0} and  r f s Φ “ | f |ΦA p f P Aq . QpAq (5.1.6) Since ΦQpAq Ă LpAq , the Gel’fand topology on ΦQpAq is the relative weak-∗ topology σpLpAq , LpAq q, and the topology on ΦA as a subset of ΦQpAq is the weak topology σpA , A q. Recall that τ p is the topology of pointwise convergence on the space ΦA . Theorem 5.1.10 Let A be a Banach function algebra. Then QpAq is the set of functions f P 8 pΦA q for which there is a bounded net p fν q in A with limν fν “ f in p 8 pΦA q, τ p q; for f P QpAq, the infimum of the bounds of such nets is equal to  f QpAq . Proof Take f P 8 pΦA q such that there is a net p fν q in Arms for some m > 0 with limν fν “ f in p 8 pΦA q, τ p q. Then we may suppose that p fν q converges in pQpAq, σpQpAq, LpAqqq, say to g. Since ΦA Ă LpAq, necessarily g | ΦA “ f , and so f P QpAq. For each λ P LpAqr1s , we have     x f, λy “ lim x fν , λy ď m , ν

5.1 The space LpAq and the algebra QpAq

299

and hence  f QpAq ď m. Conversely, suppose that f P QpAq with  f QpAq “ m. Then there exists M P A with rMs “ f and M “ m. There is a net p fν q in Arms such that M “ limν fν in pA , σpA , A qq, and in particular, for each ϕ P ΦA , we have lim fν pϕq “ xM, ϕy “ f pϕq . ν

This shows that f “ limν fν in p 8 pΦA q, τ p q. Further, there cannot be a net in Arcs , where c < m, that converges pointwise to f . The result follows. Corollary 5.1.11 Let A be a Banach function algebra. For each f P QpAq, there is a net p fν q in A with limν fν “ f in p 8 pΦA q, τ p q such that lim  fν  “ lim  fν QpAq “  f QpAq . ν

ν

(5.1.7)

Proof Take f P QpAq, and let p fν q be a net in A as specified in Theorem 5.1.10, so that  fν  ď  f QpAq for each ν. Then p fν q is a net in QpAq – LpAq that converges weak-∗ to f , and so, by using equation (1.2.6), we see that  f QpAq ď lim inf  fν QpAq ď lim inf  fν  ď lim sup  fν  ď  f QpAq , ν

ν

ν

and this implies equation (5.1.7). Corollary 5.1.12 Let A be a Banach function algebra, and let B be a closed subalgebra of A that is a Banach function algebra on ΦA . Then QpBq Ă QpAq and  f QpAq ď  f QpBq p f P QpBqq. We have seen in Theorem 3.6.26 that there may be many non-zero idempotents in Ar1s when A is a uniform algebra. We now give a similar result for a general Banach function algebra. Theorem 5.1.13 Let A be a Banach function algebra. Take ϕ P ΦA , and suppose that A has the weak separating ball property at ϕ. Then there is an idempotent element eϕ P QpAqr1s ∩ TIMpϕ, QpAqq such that eϕ pψq “ δϕ,ψ pψ P ΦQpAq q, and ϕ is an isolated point of ΦQpAq . Proof We shall again apply Theorem 3.4.10, with pA, σpA, Fqq of that theorem taken to be the present dual Banach function algebra pQpAq, σpQpAq, LpAqqq. Since A has the WSBP at ϕ, we have ϕ “ 1 as an element of ΦQpAq . Since ϕ P ΦA , Theorem 3.4.10 shows that there is an idempotent eϕ P QpAqr1s such that f eϕ “ eϕ whenever f P QpAqr1s with f pϕq “ 1. In particular, again because A has the WSBP at ϕ, for ψ P ΦA \{ϕ}, there is net in p fα q in pMψ qr1s such that limα fα pϕq “ 1. Let fψ be a weak-∗ accumulation point in QpAq of the net p fα q. Then fψ pϕq “ 1.

300

5 BSE norms and BSE algebras

This implies that eϕ pψq “ δϕ,ψ pψ P ΦA q, and hence pgeϕ ´ gpϕqeϕ q | ΦA “ 0 pg P QpAqq. Since ΦA is determining for ΦQpAq , it follows that geϕ “ gpϕqeϕ pg P QpAqq, and so eϕ P TIMpϕ, QpAqq. Since QpAq separates strongly the points of ΦQpAq , this implies that eϕ pψq “ δϕ,ψ pψ P ΦQpAq q. Clearly ϕ is an isolated point of ΦQpAq . We shall now determine some circumstances when ΦA is open and discrete in ΦQpAq ; this is equivalent to showing that each point of ΦA is isolated in ΦQpAq . Corollary 5.1.14 Let A be a Banach function algebra with the weak separating ball property. Then ΦA is the set of isolated points of ΦQpAq . Proof By Theorem 5.1.13, each point of ΦA is isolated in ΦQpAq . Let ϕ be an isolated point of ΦQpAq . By Šilov’s idempotent theorem, the characteristic function of {ϕ} is in QpAq. Since ΦA is a determining set for ΦQpAq , necessarily ϕ P ΦA . Corollary 5.1.15 Let A be a Banach function algebra that is an ideal in its bidual and has the weak separating ball property. Then ΦA is discrete, and so A is a Banach sequence algebra. Proof By Proposition 2.3.17, the weak and weak-∗ topologies on ΦA coincide, and so this follows from Corollary 5.1.14. Corollary 5.1.16 Let A be a Tauberian Banach function algebra with the weak separating ball property. Then the following conditions on A are equivalent: (a) A is a compact algebra; (b) L f is weakly compact for each f P A; (c) A is an ideal in its bidual; (d) ΦA is discrete. Proof (a) ⇒ (b) is trivial; (b) ⇔ (c) is Theorem 2.3.7; (c) ⇒ (d) is Corollary 5.1.15; (d) ⇒ (a) is Proposition 3.2.3. We remark that, in Example 4.2.22, we exhibited some natural, reflexive Banach function algebras on D, and hence they are ideals in their biduals. This shows that we cannot remove the hypothesis that A have the weak separating ball property in the above corollary. Proposition 5.1.17 Let A be a natural Banach sequence algebra on N, and suppose that A0 has an identity, E. Then A0 is Arens regular. Further, A “ A0  LpAqK ,

E l A “ A0 ,

and

QpAq “ A0 .

5.1 The space LpAq and the algebra QpAq

301

Proof Certainly A0 has a BAI, say peα q. Since A0 is an ideal in its bidual, it follows from Corollary 2.3.73, (b) ⇒ (e), that A0 is Arens regular. Since A0 is a closed ideal in A, clearly A0 is a closed ideal in A , and LpAqK is a weak-∗-closed ideal in A . Take M P A0 ∩ LpAqK . Then eα · M P A0 because A0 is an ideal in A0 , and so eα · M “ 0, whence M “ E l M “ 0. This shows that A0 + LpAqK “ A0  LpAqK . Define P : M Þ→ E l M, A Ñ A . Then P P BpA q and P2 “ P, so that P is a projection, and (5.1.8) A “ pE l A q ‘ ppI ´ Eq l A q . We claim that E l A “ A0 and that pI ´ Eq l A “ LpAqK . Clearly E l A Ă because A0 is an ideal in A and E P A0 , and A0 “ E l A0 Ă E l A . Thus E l A “ A0 . Since ΦA “ N and εn · pI ´ Eq “ 0 pn P Nq, clearly we have pI ´ Eq l A Ă LpAqK . Now take M P LpAqK . By (5.1.8), there exist N0 P A0 and N1 P A such that M “ N0 + pI ´ Eq l N1 , and then N0 P A0 ∩ LpAqK “ {0}. Thus LpAqK Ă pI ´ Eq l A , and the claim follows. The proposition follows from the claim. A0

Example 5.1.18 Now suppose that A2 “ A0 , as in Example 3.2.11. We stated that A0 is Arens regular. It follows from Corollary 2.3.73, (e) ⇒ (b), that A0 has an identity, so we can apply the above the proposition. Clearly pA q2 “ A0 , and the product in A is given by pM1 + N1 q l pM2 + N2 q “ M1 l M2

pM1 , M2 P A0 , N1 , N2 P LpAqK q .

Similarly, we have pM1 + N1 q  pM2 + N2 q “ M1  M2

pM1 , M2 P A0 , N1 , N2 P LpAqK q .

Hence A is also Arens regular.

We now give some constructions of the algebra QpAq for a Banach function algebra A. Example 5.1.19 Set A “ C 0 pKq, where K is a non-empty, locally compact space. Then A “ MpKq and LpAq – Md pKq “ 1 pKq, so that QpAq “ 8 pKq “ Cpβ Kd q , where, as usual, Kd denotes the space K with the discrete topology. Hence ΦA and ΦQpAq are identified with K and β Kd , respectively, so that ΦA “ ΦQpAq . Indeed, K “ ΦA is exactly the set of isolated points in β Kd “ ΦQpAq . Thus r : F | β Kd “ 0} “ Ipβ Kd q LpAqK “ {F P CpKq

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5 BSE norms and BSE algebras

and A “ QpAqLpAqK as a Banach algebra. In fact, since β Kd is a clopen subspace r (see Example 3.6.5), we can identify QpAq with the closed ideal of K r :F|K rc “ 0} “ IpK rc q , {F P CpKq rc “ K r \ β Kd . where K Here, the embedding of ΦA in ΦQpAq is continuous only in the special case that K is discrete. Again set A “ CpKq for a compact space K, so that QpAq “ Cpβ Kd q. Then QpQpAqq is equal to Cpβ ppβ Kd qd qq, usually a far bigger space than QpAq.

Example 5.1.20 Let A be a natural uniform algebra on a non-empty, compact space K such that A is pointwise contractive, equivalently, by Theorem 3.6.32, such that each singleton in K is a one-point Gleason part. Then it follows immediately from Theorem 3.6.31 that pQpAq,  · QpAq q “ p 8 pKq, | · |K q . We can have QpAq “ 8 pKq more generally. Indeed, let A be a natural uniform algebra on an infinite, compact space K such that there is a two-point part P “ {x, y} in K and such that the set K \ P consists of one-point parts, as in Example 3.6.41. In that example, we effectively showed that χ x , χy P QpAqrms for some m > 0. Now take f P 8 pKqr1s , say f “ g + αχ x + βχy , where g P 8 pKqr1s is such that gpxq “ gpyq “ 0 and where α, β P D. By Corollary 3.6.28, g P QpAqr1s , and so f P QpAqr2m+1s , which shows that QpAq “ 8 pKq.

Example 5.1.21 Let A “ ApDq be the disc algebra. Then it is shown in [64, Example 7.5] that the corresponding algebra QpAq is identified with the uniform algebra H 8 pDq ‘8 8 pTq.

Example 5.1.22 Let A “ p 1 , · q, as in Example 3.2.7(ii), so that A is a natural, Tauberian Banach sequence algebra on N, and hence A is an ideal in A . Here LpAq “ c 00 and A “ Mpβ Nq, so that LpAqK “ MpN∗ q. Also, QpAq “ A, ΦA “ ΦQpAq “ N, and A “ QpAq  LpAqK as a Banach algebra. Since pΔn q is a multiplier-bounded approximate identity for A, this example is a special case of Theorem 5.5.17.

5.1 The space LpAq and the algebra QpAq

303

Example 5.1.23 Let A “ lipα I and A “ Lipα I be the Lipschitz algebras of Example 3.1.36. Here LpAqK “ {0} and QpAq “ A . Take f P QpA qrms , say f “ limν fν in pCpIq, τ p q, where p fν q is a net in Arms . For each x, y, z P I with y  z, we have 

        f pxq +  f pyq ´ f pzq / |y ´ z|α “ lim  fν pxq +  fν pyq ´ fν pzq / |y ´ z|α ď m , ν

and so f P Arms . Thus QpA q “ A . We shall now calculate QpAq for A “ L1 pGq, where G is a compact abelian group. Generalizations of this calculation will be given in Examples 5.1.25 and 5.1.26. Example 5.1.24 Let G be a compact abelian group, with discrete dual group Γ, and set A “ pL1 pGq,  q. Then A “ ApΓq is a Tauberian Banach sequence algebra on Γ, and hence an ideal in A . Here LpAq – CpGq and A “ MpGq ‘1 CpGqK , so that QpAq “ QpL1 pGqq “ MpGq .

(5.1.9)

Since pMpGq,  q is a closed subalgebra of pA , l q, again A “ QpAq  LpAqK as a Banach algebra. We see that ΦQpAq “ Γ ∪ H, where H is the hull of A when A is considered as an ideal in MpGq; the embedding of ΦA “ Γ into ΦQpAq is continuous. As in Corollary 5.1.14, Γ is the set of isolated points of ΦQpAq . Since MpGq is an unital Banach algebra, ΦQpAq is compact and ΦA is also compact. As we remarked in Example 4.2.7, Γ is not dense in ΦQpAq whenever G is infinite.

Example 5.1.25 Let Γ be a locally compact group, and set A “ ApΓq. As in Example 5.1.2, we can identify the space LpAq with the C ∗ -algebra Cδ∗ pΓq, a closed C ∗ subalgebra of V NpΓq. Thus the algebra QpAq is identified as a Banach space with Cδ∗ pΓq . It was shown by Bédos in [15] that there is a surjection R : Cδ∗ pΓq Ñ Cρ∗ pΓd q, and so we can say that the algebra QpAq contains the reduced Fourier–Stieltjes algebra Bρ pΓd q “ Cρ∗ pΓd q , and hence that Bρ pΓd q Ă QpAq . In the two cases where Γ is discrete and where Γd is amenable, it is shown in [15, Theorem 3] that Cδ∗ pΓq “ Cρ∗ pΓd q, and so LpAq “ Cρ∗ pΓd q. It follows that, in the case where Γ is discrete, QpAq “ Cδ∗ pΓq “ Cρ∗ pΓq “ Bρ pΓq ,

(5.1.10)

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5 BSE norms and BSE algebras

and that, when Γd is amenable, QpAq “ LpAq “ Cδ∗ pΓq “ Cρ∗ pΓd q “ C ∗ pΓd q “ BpΓd q .

(5.1.11)

This recovers equation (5.1.9) in the special case that Γ is abelian and discrete. We prove that QpAq “ BpΓd q in the latter case, following [63, Example 7.10]. Indeed, for u P BpΓd q, define Lu : λ Þ→ xu, λy ,

Cδ∗ pΓq Ñ C .

Since Cδ∗ pΓq “ C ∗ pΓd q and C ∗ pΓd q “ BpΓd q, we have Lu  “ u p f P BpΓd qq. ∗ Since Cδ∗ pΓq is a closed C ∗ -subalgebra of V  NpΓ  d q, the functional Lu on C pΓd q has an extension r u P V NpΓd q “ ApΓd q with r u “ u. Suppose that v P A is another such extension of u. Then clearly r u ´ v P LpApΓd qqK , and so the map θ : u Þ→ r u,

BpΓd q Ñ QpAq ,

is a well-defined linear isometry, easily seen to be an algebra homomorphism. To show that θ is a surjection, take M P A . Then there is a bounded net, say p fα q, in A that converges weak-∗ to M. Since A Ă BpΓq Ă BpΓd q isometrically, the net p fα q is bounded in BpΓd q “ C ∗ pΓd q , and so has a weak-∗ accumulation point, say u, in BpΓd q, and we see that θpuq “ rMs P QpAq. Thus θ : BpΓd q Ñ QpAq is an isometric algebra isomorphism, giving equation (5.1.11). We also see that ΦQpAq “ ΦBpΓd q , and so Γ “ ΦA Ă ΦBpΓd q . Since ApΓd q is a closed ideal in BpΓd q, it follows that ΦQpAq “ Γd ∪ H, where H is the hull of ApΓd q considered as a closed ideal in BpΓd q, and again Γ is identified with the set of isolated points in ΦQpAq , as in Corollary 5.1.14.

Example 5.1.26 Let G be a locally compact abelian group. We have noted that APpGq is C ∗ -isomorphic to CpbGq, where bG is the Bohr compactification of G, that bG is a compact group, and that the map from G into bG is a continuous injection. Again set A “ pL1 pGq,  q, identified with the Banach function algebra ApΓq, p Then LpAq is a closed, self-adjoint, unital subalgebra of APpGq, and where Γ “ G. so LpAq “ APpGq by Stone–Weierstrass. This confirms that bG is the dual group of Γd . As Banach spaces, we have QpAq “ LpAq “ CpbGq “ MpbGq, and pQpAq, · q is Banach-algebra isometrically isomorphic to pMpbGq,  q and to pBpΓd q, · q. In particular, take G “ pZ, + q. Then Qp 1 pZqq “ MpbZq. We recall that bZ is the space of all group homomorphisms from T to T.

5.1 The space LpAq and the algebra QpAq

305

We now return to some more general results about QpAq, where A is a Banach function algebra. Let A be a Banach function algebra with a non-zero ideal I that is a Banach A-module. We recall that we identify ΦI with ΦA \ hpIq, and so we can regard LpIq as a linear subspace of LpAq Ă A . Proposition 5.1.27 Let A be a Banach function algebra with a non-zero ideal I that is a Banach A-module. Suppose that I has a contractive pointwise approximate identity. Then λI  “ λA pλ P LpIqq. Proof The CPAI in I is peα q. Take λ P LpIq, so that  λI ď λA . For each ε > 0, there exists f P Ar1s such that x f, λy > λA ´ ε. The net peα f q is in I, and also eα f I ď  f A eα I ď 1, and hence     x f, λy “ lim xeα f, λy ď lim sup eα f I λI  ď λI  , α

α

which shows that λA ´ ε < λI  . It follows that λA ď λI  , and hence that λI  “ λA . Let A be a Banach function algebra. Then we set ⎫ ⎧ n  n ⎪ ⎪   ⎪ ⎪ ⎬ ⎨    αi  : αi ϕi P LpAqr1s ⎪ β “ sup ⎪ “ sup{ x1K , λy  : λ P LpAqr1s } , (5.1.12) ⎪ ⎪ ⎭ ⎩  i“1

i“1

so that β P r0, 8s. Proposition 5.1.28 Let A be a Banach function algebra. Then the following conditions are equivalent: (a) QpAq has an identity 1ΦA ; (b) β < 8; (c) A has a bounded pointwise approximate identity peν q with   lim eν QpAq “ lim eν  “ 1ΦA QpAq “ β . α

α

(5.1.13)

Proof (a) ⇔ (b) This follows from Proposition 5.1.8. (a) ⇔ (c) This follows from Theorem 5.1.10 and Corollary 5.1.11. The next result is similar to Proposition 3.4.15(i). Corollary 5.1.29 Let A be a Banach function algebra with |ΦA | ě 2 such that A has a bounded pointwise approximate identity of bound m. Take ϕ P ΦA , and suppose that A has the weak separating ball property at ϕ. Then Mϕ has a bounded pointwise approximate identity of bound m + 1.

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5 BSE norms and BSE algebras

  Proof By Proposition 5.1.28, QpAq has an identity 1ΦA with 1ΦA QpAq “ m. By Theorem 5.1.13, there is an idempotent eϕ P QpAqr1s with eϕ pψq “ δϕ,ψ pψ P ΦA q. Since QpAq “ A /LpAqK , there exists an element M P Arm+1s such that xM, ϕy “ 0 and xM, ψy “ 1 pψ P ΦA \ {ϕ}q. Clearly M P pMϕ qrm+1s , and any net in pMϕ qrm+1s that converges weak-∗ to M is a BPAI of bound m + 1 for Mϕ . We set H “ { f P A :  f QpAq ď 1}; in general, H is much larger than Ar1s , as will be seen in Example 5.1.36, but we do have the following result. Proposition 5.1.30 Let pA,  · q be a Banach function algebra. Then   λ “ sup{x f, λy : f P H} pλ P LpAqq .

(5.1.14)

The set H is closed in p 8 pΦA q, τ p q if and only if pA,  · q “ pQpAq,  · QpAq q. Proof Let the supremum on the right of equation (5.1.14) be k. Take λ P LpAq. Since  f QpAq ď  f  p f P Aq, certainly λ ď k. On the other hand, since LpAq – A /LpAqK , we have     λ “ sup{xM + LpAqK , λy : M P A , M + LpAqK  ď 1}   ě sup{x f + LpAqK , λy : f P H} ě k . Equation (5.1.14) follows. Certainly H is closed in p 8 pΦA q, τ p q when A “ QpAq. Now suppose that the set H is closed in p 8 pΦA q, τ p q, and take f P QpAqr1s . By Corollary 5.1.11, there is a net p fν q in H with limν fν “ f in p 8 pΦA q, τ p q. Thus f P H Ă A, and so A “ QpAq. Further, the inclusion of Ar1s in H implies that  f QpAq “  f  p f P Aq, and so pA,  · q “ pQpAq,  · QpAq q. The next theorem is one of the main results of this section. Theorem 5.1.31 Let A and B be two Banach function algebras, and suppose that θ : B Ñ A is a continuous homomorphism with dense range such that θ pΦA q “ ΦB . Then the map θ : LpAq Ñ LpBq is a surjective isometry if and only if   θp f qQpAq “  f QpBq p f P Bq . (5.1.15) In this case, pQpBq,  · QpBq q “ pQpAq,  · QpAq q , and B has a contractive pointwise approximate identity if and only if A has a contractive pointwise approximate identity. Proof Since θpBq  “ A, it follows that θpBq is also dense in pA,  · QpAq q, and so {θp f q : f P B, θp f qQpAq ď 1} is dense in the set {g P A : gQpAq ď 1}.

5.1 The space LpAq and the algebra QpAq

307

Now suppose that equation (5.1.15) holds, so that {θp f q : f P B,  f QpBq ď 1} is dense in the set {g P A : gQpAq ď 1}. Next take λ P LpAq, so that θ pλq P LpBq. By Proposition 5.1.30,      θ pλqB “ sup{x f, θ pλqy : f P B,  f QpBq ď 1} , and so

     θ pλqB “ sup{xg, λy : g P A, gQpAq ď 1} .   Thus, again by Proposition 5.1.30, θ pλqB “ λA , and this shows that the map

θ | LpAq : LpAq Ñ LpBq is an isometry. Since θ pΦA q “ ΦB , this map is a surjection. The converse is immediate from the definitions. Set μ “ θ | LpAq : LpAq Ñ LpBq. In the case where μ is a surjective isometry, the dual map μ : LpBq Ñ LpAq is also a surjective isometry, and hence the map μ : QpBq Ñ QpAq is a surjective isometry. Moreover, μ is a homomorphism because θ is a homomorphism. Hence the final sentence follows, using Theorem 3.5.7. Corollary 5.1.32 Let B be a Banach function algebra that is dense in C 0 pΦB q, and suppose that  f QpBq “ | f |ΦB p f P Bq. Then B is pointwise contractive. Proof Each maximal modular ideal M of B is dense in the corresponding maximal modular ideal N of C 0 pΦB q. Since N has a CPAI, it follows from Theorem 5.1.31 that M has a CPAI, and so B is pointwise contractive. Proposition 5.1.33 Let pA,  · q be a Banach function algebra, and take F to be a closed linear subspace of A that is a concrete predual of A as a Banach space. (i) Suppose that Ω Ă ΦA is determining for A with Ω Ă F. Then LpA, Ωq “ F, the space F is a submodule of A , and A is a dual Banach algebra, with Banachalgebra predual F. (ii) Suppose that ΦA Ă F. Then F “ LpAq, A “ A  LpAqK , and pA,  · q “ pQpAq,  · QpAq q . Proof (i) Certainly LpA, Ωq Ă F. Take μ P F  with μ | LpA, Ωq “ 0. Then μ corresponds to an element f P A with f pϕq “ 0 pϕ P Ωq, and so f “ 0 because Ω is determining for A. Thus μ “ 0. It follows from the Hahn–Banach theorem that LpA, Ωq “ F. Since LpA, Ωq is a submodule of A , so is F, and hence A is a dual Banach algebra, with Banach-algebra predual F. (ii) Since ΦA Ă F, necessarily we have LpAq “ lin ΦA “ F by (i), and so A “ A ‘ LpAqK by (1.3.7). Take f P QpAq, with  f QpAq “ m, say. By Theorem 5.1.10, there is a bounded net p fν q in Arms with limν fν “ f in p 8 pΦA q, τ p q. Let g be an accumulation point

308

5 BSE norms and BSE algebras

of this net in pA, σpA, Fqq. Then gpϕq “ f pϕq pϕ P ΦA q, and so g “ f because ΦA Ă F. Thus f P Arms . It follows that pA,  · q “ pQpAq,  · QpAq q. Example 5.1.34 Set Bω “ p 1 pZ, ωq,  · ω ,  q, as in Example 4.2.20. We identify QpBω q in certain cases. Indeed, suppose that ρ2 “ ρ1 “ 1 and that lim|n|Ñ8 ωn “ 8 (for example, we can take ω “ pωn : n P Zq, where ωn “ p1 + |n|qα

pn P Zq

for some α > 0). Then Xω “ T is a subset of c 0 pZ, 1/ωq, a concrete predual of Bω , and so, by Proposition 5.1.33(ii), pBω ,  · ω q “ pQpBω q,  · QpBω q q. We do not have an identification of the algebra QpBω q when ρ2 “ ρ1 “ 1 and ω is unbounded, but it is not the case that lim|n|Ñ8 ωn “ 8; such weights exist.

We conclude this section with some applications of Theorem 3.5.7. Let A be a Banach function algebra, and suppose that I is a weak Segal algebra with respect to A, so that ΦI “ ΦA , and hence we can regard LpAq as a linear subspace of both A and I  . Let j : LpAq Ñ LpIq to be the identity mapping, so that j is a contraction. The dual map is j : LpIq Ñ LpAq , so that j pM + LpIqK q “ M + LpAqK

pM P I  q .

The contraction j : QpIq Ñ QpAq is an algebra monomorphism. In general, the image j pQpIqq is not dense in QpAq. It is clear that j pΦQpAq q Ă ΦQpIq ∪ {0}. If ΦQpAq is compact, then j p ΦA q is a compact subset of ΦQpIq , but it can be that 0 P j p ΦA q. For example, take A “ c 0 and I “ 1 ; here we have ΦQpAq “ β N and ΦQpIq “ N, as in Example 5.1.22. The following theorem implies that, in the case where I has a contractive pointwise approximate identity, the map j : LpAq Ñ LpIq is an isometry, and so the map j : QpIq Ñ QpAq is an isometric Banach-algebra isomorphism. Theorem 5.1.35 Let pA,  · A q be a Banach function algebra, and let pI,  · I q be a weak Segal algebra with respect to A. Then the following are equivalent: (a) I has a contractive pointwise approximate identity; (b) A has a contractive pointwise approximate identity and λI  “ λA

pλ P LpAqq ;

(5.1.16)

(c) A has a contractive pointwise approximate identity and  f QpIq “  f QpAq

p f P Iq .

Suppose that the equivalent conditions hold. Then

(5.1.17)

5.2 Basic definitions and results about BSE norms

309

pLpIq,  · I  q – pLpAq,  · A q and pQpIq,  · QpIq q “ pQpAq,  · QpAq q. Proof (a) ⇒ (b) This follows from Proposition 5.1.27. (b) ⇒ (a) By Theorem 3.5.7, λA “ 1 pλ P co ΦA q, and so, immediately from equation (5.1.16), λI  “ 1 pλ P co ΦI q. Hence, by Theorem 3.5.7 again, I has a CPAI. (b) ⇔ (c) This equivalence follows from equation (5.1.15). Now suppose that the equivalent conditions hold. Then the identification of LpIq and LpAq is an isometry, and so LpIq – LpAq, whence LpIq – LpAq . Hence pQpIq,  · QpIq q “ pQpAq,  · QpAq q. Example 5.1.36 Let M “ { f P A : f p0q “ 0} be the Banach function algebra that is the maximal ideal in the algebra A of Example 3.5.13. Set K “ p0, 1s. Then we saw in Example 3.5.13 that M is a Segal algebra with respect to C 0 pKq and that M is pointwise contractive; in particular, M has a contractive pointwise approximate identity. By Example 5.1.19, QpC 0 pKqq “ Cpβ Kd q. Thus, by Theorem 5.1.35, QpMq “ Cpβ Kd q and  f QpMq “ | f |K p f P Mq. Note in particular that QpMq is a uniform algebra, even though M is not a uniform algebra.

5.2 Basic definitions and results about BSE norms Let A be a Banach function algebra. In this section, we shall describe a central Banach function algebra, C BS E pAq, that is a closed subalgebra of QpAq; the algebra C BS E pAq is the BSE-algebra of A; the elements of C BS E pAq are the BSE-functions on ΦA , and A is a BSE-algebra if MpAq “ C BS E pAq. We shall characterize C BS E pAq in Theorem 5.2.9, and determine when C BS E pAq has an identity in Theorem 5.2.16. We shall compare the BSE properties of related Banach function algebras, and we shall consider when these properties pass to ideals in a Banach function algebra. Examples exhibiting various of these properties will be given in §5.3. As before, we write A for the Banach algebra pA , l q. We start with the definition of C BS E pAq and of a subalgebra of A , namely CpA q; this latter definition was first givenby Kanuith and Ülger  in [200, §2]. Recall that τ f  “ sup{xτ f , λy : λ P LpAqr1s } for a function f on ΦA Definition 5.2.1 Let A be a Banach function algebra.   Then C BS E pAq is the set of bounded, continuous functions f on ΦA such that τ f  < 8, and then

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5 BSE norms and BSE algebras

   f BSE “ τ f  Further,

p f P C BS E pAqq .

CpA q “ {M P A : M | ΦA P C b pΦA q} .

Clearly, CpA q is a closed A-submodule and a subalgebra of A , A + LpAqK is a subalgebra of CpA q, so that CpA q/LpAqK is a closed subalgebra of QpAq, and C BS E pAq “ CpA q/LpAqK . Thus pC BS E pAq,  · BSE q is a Banach function algebra on ΦA that contains A as a subalgebra, and C BS E pAq “ QpAq ∩ C b pΦA q . Further, for each f P C BS E pAq, we have      f BSE “  f QpAq “  f + LpAqK  “ sup{x f, λy : λ P LpAqr1s } .

(5.2.1)

To be explicit, a function f P C b pΦA q belongs to C BS E pAq if and only if there is  a constant β f > 0 such that, for each λ “ ni“1 αi ϕi in LpAq, the inequality   n    αi f pϕi q ď β f λ   i“1 holds, and then  f BSE is the infimum of these constants β f . A function f on ΦA belongs to C BS E pAq if and only if f is continuous on ΦA and there exists M P A such that xM, ϕy “ f pϕq pϕ P ΦA q, so that C BS E pAq “ { f P C b pΦA q : f “ M | ΦA for some M P A } . (5.2.2)   Further,  f BSE “ rMsQpAq when f “ M | ΦA . In particular, the restriction of  · BSE to A is an algebra norm on A, and, of course, | f |ΦA ď  f BSE ď  f 

p f P Aq .

(5.2.3)

The Banach function algebra C BS E pAq on ΦA is usually not natural. We shall write  · BSE,A for the norm on C BS E pAq if it be necessary to specify the algebra A in the notation. Example 5.2.2 It is not always the case that CpA q “ A , and so C BS E pAq may be a proper subalgebra of QpAq. For example, take K to be an infinite compact space. Then C BS E pCpKqq “ CpKq, but QpCpKqq “ 8 pKq “ Cpβ Kd q, as in Example 5.1.19, and so C BS E pCpKqq  QpCpKqq.

Definition 5.2.3 Let A be a Banach function algebra. The function  · BSE , defined on C BS E pAq, is the BSE norm of A and the elements of C BS E pAq are the BSE functions for A; C BS E pAq is the BSE algebra of A. The algebra A has a BSE norm if the

5.2 Basic definitions and results about BSE norms

311

norms  ·  and  · BSE are equivalent on A, so that there is a constant C > 0 with  f  ď C  f BSE

p f P Aq .

Clearly A has a BSE norm if and only if it is closed as a subalgebra of the Banach function algebra pC BS E pΦA q,  · BSE q on ΦA . The class of Banach function algebras A such that  f  “  f BSE p f P Aq was previously considered in [299]; as in Proposition 5.1.4, we see that this occurs if and only if LpAqr1s is weak-∗ dense in Ar1s . Trivially  · BSE “ | · |ΦA for every uniform algebra A, and so every uniform algebra has a BSE norm. We summarize the situation concerning C BS E pAq as follows. Let pA,  · A q be a Banach function algebra, and set pB,  · B q “ pC BS E pAq,  · BSE q . Then B is a Banach function algebra on ΦA with | f |ΦA ď  f B ď  f A p f P Aq; as usual, we regard ΦA as a subspace of ΦB , and clearly ΦA is determining for B. We can have either A “ B or A  B. We note that B always has a BSE norm and that B “ C BS E pBq whenever ΦA is dense in ΦB ; see Theorem 5.2.15. We can have ΦA “ ΦB , a compact space, but with A a proper closed subalgebra of B, as in Example 5.3.3. In the case where A “ C 0 pKq for a non-compact, locally compact space K, we have B “ C b pKq and K  ΦB “ β K, so that ΦA  ΦB and ΦA is dense in ΦB , as in Example 5.3.1(i). In the case where A “ ApΓq for a non-compact, locally compact abelian group Γ, we have B “ BpΓq (by Example 5.3.20); here, the space ΦA “ Γ is open, but not dense, in ΦB , as in Example 4.2.7; further, B  C BS E pBq, as in Theorem 5.3.28. Let A be a non-unital Banach function algebra, and take K “ ΦA ∪ {8} “ ΦA and define ϕ8 and z1K + f  for z P C and f P A as in equation (3.1.2). Proposition 5.2.4 Let A be a non-unital Banach function algebra such that A has a BSE norm. Then A has a BSE norm. Proof Since the Banach function algebra A has a BSE norm, there exists C > 0 such that  f  ď C  f BSE p f P Aq. We start with an element z1K + f P A , where z P C and f P A, and we consider two cases. Case 1 Suppose that |z| ě  f  /3C, so that z1K + f  ď p3C + 1q |z|. Since ϕ8 P LpA qr1s , we have |z| “ xz1K + f, ϕ8 y ď z1K + f BSE,A , and so z1K + f  ď p3C + 1q z1K + f BSE,A .

(5.2.4)

 Case 2 Suppose that |z| ď  f  /3C. There exists an element λ P LpAqr1s with 2 x f, λy ě  f BSE . Define μ “ λ + p1 ´ x1K , λyqϕ8 ,

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 so that μ P LpA q and x1  K , μy “ 1. For each w1K + g P A , where w P C and g P A,  we have xw1K + g, μy ď |w| + xg, λy ď |w| + g “ w1K + g, and so μ “ 1 and μ P LpA qr1s . We also have

    1 1 1 z1K + f BSE,A ě xz1K + f, μy ě x f, λy ´ |z| ě f ´ f “ f , 2C 3C 6C and so  f + z1k  ď

3C + 1  f  ď p6C + 2q  f + z1K BSE,A . 3C

(5.2.5)

It follows from equations (5.2.4) and (5.2.5) that A has a BSE norm. Proposition 5.2.5 Let A be a natural Banach sequence algebra on a non-empty set S . Suppose that there is a constant C > 0 such that, for each f P A, there is a finite subset T of S with χT op ď C and  f  ď C χT f . Then A has a BSE norm. Proof Let f P A, and take T as specified, say T “ {s1 , . . . , sn }. The map   n    · n : z “ pz1 , . . . , zn q Þ→  zi δ si  , C n Ñ R+ ,  i“1  is a norm on C n , with dual norm  · n . Take w “ pw1 , . . . , wn q P C n with wn “ 1 and   n     f  . f psi qwi  “ p f ps1 q, . . . , f psn qqn “ χT f  ě C  i“1   Set λ “ ni“1 wi ε si , so that λ P LpAq. For each g P A, we have   n      xg, λy  “  gpsi qwi  ď χF g wn ď C g ,   i“1   and so λ ď C. Thus C  f BSE ě  x f, λy  ě  f  /C, and hence A has a BSE norm. Proposition 5.2.6 Let pA,  · q be a Banach function algebra. Suppose that A has a multiplier-bounded approximate identity and that AA “ LpAq. Then A and A have BSE norms. Further,  f BSE “  f  p f P Aq whenever A has such a multiplierbounded approximate identity of multiplier-bound 1. Proof Take peν q to be a MBAI for A with bound  m.  Let f P A. Then there exists λ P Ar1s with x f, λy “  f . For each index ν, we have eν · λ P A · A Ă LpAq, and also eν · λ ď m λ ď m by Proposition 2.1.42, and hence     x f eν , λy “ x f, eν · λy ď m  f BSE .     Thus  f  “  x f, λy  “ limν  x f eν , λy  ď m  f BSE , and so A has a BSE norm. By Proposition 5.2.4, A also has a BSE norm.

5.2 Basic definitions and results about BSE norms

313

Clearly  · BSE “  ·  whenever A has such a MBAI of bound 1 with respect to the multiplier norm. Proposition 5.2.7 Let pA,  · q be a Banach function algebra. Suppose that Ω Ă ΦA is a determining set for A, that A is von Neumann algebra, and that lin LpA, Ωq is a C ∗ -subalgebra of A . Then  f BSE,A “  f 

p f P Aq .

(5.2.6)

Proof Let V “ lin LpA, Ωq be the C ∗ -subalgebra of A . Since Ω is a determining set for A, the C ∗ -algebra V is weak-∗ dense in A . By Kaplanky’s density theorem, Theorem 2.2.14, the set Vr1s is weak-∗ dense in Ar1s . Since LpAqr1s ⊃ Vr1s , it follows that LpAqr1s is also weak-∗ dense in A , and this implies equation (5.2.6). Definition 5.2.8 Let A be a Banach function algebra. The algebra A is a BSE algebra when MpAq “ C BS E pAq. Note that, for a BSE algebra A, the norms  · op and  · BSE are equivalent on MpAq “ C BS E pAq, but they are not necessarily equal. In the case where ΦA is compact, so that A has an identity, the Banach function algebra A is a BSE algebra if and only if A “ C BS E pAq “ { f P CpΦA q :  f BSE < 8} . We shall see that a BSE algebra need not have a BSE norm (e.g., Examples 5.3.32, 5.3.33, and 5.4.13) and that a Banach function algebra with a BSE norm need not be a BSE algebra (e.g., Example 5.3.1(ii)). Of course, the abbreviation ‘BSE’ stands for ‘Bochner–Schoenberg–Eberlein’, and the terminology is used because of their famous theorem. This theorem shows that, for each locally compact abelian group Γ, we have  f BSE “  f  p f P BpΓqq and that ApΓq is a BSE algebra; the theorem will be generalized in Theorem 5.3.23 (when these results will be given whenever Γ is an amenable locally compact group), and will be partially extended to Figà-Talamanca–Herz algebras in Theorem 5.3.38. As we mentioned, the BSE norm and BSE algebras for arbitrary Banach function algebras were introduced by Takahasi and Hatori in [297]; for further study of BSE algebras, see [63, 178, 179, 297–299] and, especially, [200]. The following result is immediate from the characterization of QpAq in Theorem 5.1.10 and Corollary 5.1.11. Theorem 5.2.9 Let A be a Banach function algebra. Then C BS E pAq is the set of functions f P C b pΦA q for which there is a bounded net p fν q in A with limν fν “ f in pC b pΦA q, τ p q; for f P C BS E pAq, the infimum of the bounds of such nets is equal to  f BSE . Further, for each f P C BS E pAq, there is a net p fν q in A with limν fν “ f in pC b pΦA q, τ p q such that lim  fν  “ lim  fν BSE “  f BSE . ν

ν

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5 BSE norms and BSE algebras

We see from Theorem 5.2.9 that, given f P C b pΦA q and m > 0, we can conclude that f P C BS E pAqrms if and only if, for each  finite subset F of ΦA and each ε > 0,   there exists g P Arm+εs with f pϕq ´ gpϕq < ε pϕ P Fq. Corollary 5.2.10 Let A be a Banach function algebra, and suppose that peα q is a pointwise approximate identity for A with eα BSE ď 1 for each α. Then lim  f eα BSE “  f BSE α

p f P C BS E pAqq .

Proof Take f P C BS E pAq, so that limα f eα “ f pointwise. Clearly we have lim supα  f eα BSE ď  f BSE . Now suppose that lim inf α  f eα BSE “ m. Then it follows from Theorem 5.2.9 that  f BSE ď m. This gives the result. p

As before, we shall denote by B the closure in pC b pΦA q, τ p q of a subset B of b C pΦA q. The following corollary is immediate from the above theorem. As in §5.1, we set H “ { f P A :  f BSE ď 1}, so that Ar1s Ă H Ă C BS E pAqr1s . Note also that Ar1s Ă MpAqr1s . Corollary 5.2.11 Let A be a Banach function algebra. Then p

p

p

Ar1s “ H “ C BS E pAqr1s Ă MpAqr1s . Further, C BS E pAqr1s is closed in pC b pΦA q, τ p q, and C BS E pAq “ A if and only if Ar1s is closed in pC b pΦA q, τ p q. Corollary 5.2.12 Let pA,  · A q and pB,  · B q be two Banach function algebras on ΦA such that A is a subalgebra of B with  f A ě  f B p f P Aq. Suppose that Br1s is closed in pC b pΦA q, τ p q. Then C BS E pAqr1s Ă Br1s . Proof Take f P C BS E pAqr1s . By Theorem 5.2.9, there is a net p fν q in Ar1s with limν fν “ f in pC b pΦA q, τ p q. The net p fν q is contained in Br1s and Br1s is closed in pC b pΦA q, τ p q, and so f P Br1s . Thus C BS E pAqr1s Ă Br1s . The algebra C BS E pAq satisfies the conditions imposed on B in the above corollary, and so C BS E pAq is the smallest Banach function algebra on ΦA that satisfies these conditions. Corollary 5.2.13 Let A be a Banach function algebra. Suppose that MpAqr1s is closed in pC b pΦA q, τ p q. Then C BS E pAqr1s Ă MpAqr1s . Proposition 5.2.14 Let pA,  · A q and pB,  · B q be two Banach function algebras on ΦA such that A is a subalgebra of B with  f A ě  f B p f P Aq. Then pC BS E pAq,  · BSE q is the largest such algebra B with λB “ λA pλ P LpAqq.

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315

Proof Set pB,  · B q “ pC BS E pAq,  · BSE q, and take λ P LpAq. Take f P Br1s and ε > 0. By Theorem 5.2.9, there exists g P Ar1s with x f ´ g, λy < ε, and so x f, λy < λA + ε. It follows that λB “ λA . Now let B be a Banach function algebra satisfying the   conditions and such that λB “ λA pλ P LpAqq. Take f P Br1s . Then x f, λy ď λA pλ P LpAqq, and so f P C BS E pAqr1s . Hence Br1s Ă C BS E pAqr1s . The conclusion follows. Theorem 5.2.15 Let pA,  · A q be a Banach function algebra, and set pB,  · B q “ pC BS E pAq,  · BSE q . Then  f BSE,B “  f B p f P Bq, and so C BS E pAq has a BSE norm. Suppose, further, that ΦA is dense in ΦB . Then C BS E pBq “ B Proof Take λ P LpAq. By Proposition 5.2.14, λA “ λB . We claim that LpAqr1s is σpB , Bq-dense in Br1s . For assume that this is false, σ

and take Λ0 P Br1s \ LpAqr1s Then, by Theorem 1.2.15(ii), there exists an element   b P pB , σpB , Bqq “ B with  xλ, by  ď 1 pλ P LpAq  r1s q and xΛ0 , by > 1. The first inequality shows that bB ď 1, and hence we have  xΛ0 , by  ď Λ0 B bB ď 1, a contradiction. The claim follows. Since LpAqr1s Ă LpBqr1s Ă Br1s , the set LpBqr1s is σpB , Bq-dense in Br1s . As before, it follows that  f BSE,B “  f B p f P Bq, as required. Now suppose, further, that ΦA is dense in ΦB . Take f P C BS E pBq, and set f1 “ f | ΦA P C b pΦA q. For each finite subset F of ΦA and each ε > 0, there exist g P B  f pϕq ´ gpϕq < ε pϕ P Fq and h P A with gpϕq ´ hpϕq < ε pϕ P Fq. Thus with 1    f1 pϕq ´ hpϕq < 2ε pϕ P Fq. This shows that f1 P B, and so f P B because ΦA is dense in ΦB . Thus C BS E pBq “ B. It is not true that the above B is always equal to C BS E pBq: see Theorem 5.3.28. We now consider when the algebra C BS E pAq has an identity. Clearly this is the case if and only if QpAq has an identity, and so the following theorem is related to Proposition 5.1.28; most of the theorem was first proved by Takahasi and Hatori in [297, Corollary 5]. Theorem 5.2.16 Let A be a Banach function algebra, and take m ě 0. Then the following conditions are equivalent:   (a) C BS E pAq has an identity 1ΦA with 1ΦA BSE ď m; (b) A has a bounded pointwise approximate identity, of bound m; (c) MpAqr1s Ă C BS E pAqrms ; (d)  f BSE ď m  f op p f P MpAqq .

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5 BSE norms and BSE algebras

Proof (a) ⇔ (b) This follows from Proposition 5.1.28. (b) ⇒ (c) Take peα q to be a BPAI in Arms . For f P MpAqr1s , we have f eα P Arms for each α and limα f eα “ f in pC b pΦA q, τ p q, and so f P C BS E pAqrms by Theorem 5.2.9. (c) ⇒ (d) This is immediate.   (d) ⇒ (a) By (d), 1ΦA BSE ď m, and so 1ΦA P C BS E pAqrms . Corollary 5.2.17 Let A be a Banach function algebra with a contractive pointwise p approximate identity. Then MpAqr1s “ C BS E pAqr1s . p

Proof As above, MpAqr1s Ă C BS E pAqr1s , and so MpAqr1s Ă C BS E pAqr1s . By Corolp

lary 5.2.11, C BS E pAqr1s Ă MpAqr1s , giving the result. Corollary 5.2.18 Let A be a Banach function algebra that is a BSE algebra. Then A has a bounded pointwise approximate identity. Corollary 5.2.19 Let pA,  · q be a Banach function algebra, and suppose that A has a BSE norm and a bounded pointwise approximate identity. Then  · op ∼  ·  on A. Proof By Theorem 5.2.16, (b) ⇒ (d),  · BSE  · op , and also  ·  ∼  · BSE . Hence  · op ď  ·  ∼  · BSE  · op . Thus  ·  and  · op are equivalent on A. In Example 5.4.13, we shall exhibit an unital Banach function algebra A, so that MpAq “ A, but such that A is not closed in C BS E pAq and A does not have a BSE norm. We shall also see in Example 5.4.13 that the Banach function algebra pM,  · q on K “ p0, 1s of that example, which has a contractive pointwise approximate identity, is such that | · |K “  · BSE ď  · op “  ·  on M, and  · BSE is not equivalent to  · op . Corollary 5.2.20 Let A be a Banach function algebra such that Ar1s is closed in pC b pΦA q, τ p q and A has a bounded pointwise approximate identity. Then A is a BSE algebra with an identity. Proof By Corollary 5.2.11, A “ C BS E pΦA q. Also MpAq Ă C BS E pΦA q by Theorem 5.2.16, (b) ⇒ (c), and so A “ MpAq is a BSE algebra with an identity.

5.2 Basic definitions and results about BSE norms

317

Corollary 5.2.21 Let A be a Banach function algebra. Then pC BS E pAq,  · BSE q “ pMpAq,  · op q

(5.2.7)

if and only if A has a contractive pointwise approximate identity and MpAqr1s is closed in pC b pΦA q, τ p q. Proof Suppose that equation (5.2.7) holds. Then A has a CPAI by Theorem 5.2.16, (c) ⇒ (b), and MpAqr1s is closed in pC b pΦA q, τ p q. Conversely, by Theorem 5.2.16, (b) ⇒ (c), MpAqr1s Ă C BS E pAqr1s when A has a CPAI, and, by Corollary 5.2.13, C BS E pAqr1s Ă MpAqr1s when MpAqr1s is closed in pC b pΦA q, τ p q. Thus (5.2.7) holds when both these conditions are satisfied. It is not true that every Banach function algebra with a bounded pointwise approximate identity and a BSE norm is a BSE algebra. For example, an unital uniform algebra is not necessarily a BSE algebra; examples to show this will be given in §5.3. Proposition 5.2.22 Let A be a Banach function algebra with a bounded pointwise approximate identity, and suppose that A has a BSE norm. Then MpAq is closed as a subalgebra of pC BS E pAq,  · BSE q. Proof By Theorem 5.2.16, (b) ⇒ (c), MpAq is a subalgebra of C BS E pAq. Consider a sequence p fn q in MpAq such that fn Ñ f in pC BS E pAq,  · BSE q as n Ñ 8. For each g P A, the sequence p fn gq is contained in A, and we have limnÑ8  fn g ´ f gBSE “ 0, whence limnÑ8  fn g ´ f g “ 0 because A has a BSE norm. Thus f g P A, and so f P MpAq. This shows that MpAq is closed as a subalgebra of C BS E pAq. Definition 5.2.23 Let A be a Banach function algebra. Then Ar1s is τ p -closed in A if Ar1s is closed in pA, τ p q. We note that the above definition is not the same as saying that ‘Ar1s is closed in pC b pΦA q, τ p q’. Theorem 5.2.24 Let pA,  · q be a Banach function algebra. Then Ar1s is τ p -closed in A if and only if pA,  · q “ pA,  · BSE q. Proof Suppose that Ar1s is τ p -closed in A, and take f P A with  f BSE ď 1. By Theorem 5.2.9, there is a net p fα q in Ar1s such that limα fα “ f in pC b pΦA q, τ p q. Since Ar1s is τ p -closed in A, we have f P Ar1s . It follows that  f BSE “  f . Conversely, suppose that  f BSE “  f  p f P Aq. Let p fα q be a net in Ar1s that converges to f P A in the topology τ p . For each λ P LpAqr1s , it follows that x f, λy “  limα x fα , λy, and so x f, λy ď 1. Hence  f BSE ď 1, and so f P Ar1s . This shows that Ar1s is τ p -closed in A.

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5 BSE norms and BSE algebras

We shall now compare C BS E pAq and C BS E pBq for two related Banach function algebras, A and B. Let A and B be two natural Banach function algebras on the same space such that B is a dense subalgebra of A. It is tempting to seek to show that one of A and B has a BSE norm whenever the other does. However Examples 5.4.13 and 5.4.14 will show that neither of these implications holds. Theorem 5.2.25 Let A and B be Banach function algebras, and suppose that there is a Banach-algebra homomorphism θ : A Ñ B such that θ ď 1. Then   θp f qBSE,B ď  f BSE,A p f P Aq . (5.2.8) Proof We have noted that θ pΦB q Ă ΦA ∪ {0}. Take λ P LpBqr1s . Then clearly θ pλq P LpAq1s , and so, for each f P A, we have     θp f qBSE,B “ sup{xθp f q, λy : λ P LpBqr1s }   “ sup{x f, θ pλqy : λ P LpBqr1s }   ď sup{x f, μy : μ P LpAqr1s } “  f BSE,A , giving equation (5.2.8). Corollary 5.2.26 Let pA,  · A q and pB,  · B q be Banach function algebras on ΦB such that A is a subalgebra of B with  f A ě  f B p f P Aq. Suppose that ΦB is a subspace of ΦA . Then C BS E pAqr1s | ΦB Ă C BS E pBqr1s .

(5.2.9)

f P C BS E pAq Ă C b pΦA q. Then f | ΦB P C b pΦB q. By equation (5.2.8), Proof Take   f | ΦB BSE,B ď  f BSE,A . Thus equation (5.2.9) follows. Corollary 5.2.27 Let B be a Banach function algebra with a BSE norm, and suppose that A is a closed subalgebra of B. Then A has a BSE norm. In the case where  f BSE,B “  f  p f P Bq, also  f BSE,A “  f  p f P Aq. Proof There exists C > 0 such that  f B ď C  f BSE,B p f P Bq. Now take f P A. Then  f A “  f B ď C  f BSE,B ď C  f BSE,A by equation (5.2.8), and so A has a BSE norm. The final sentence follows by taking C “ 1. Proposition 5.2.28 Let A be a Banach function algebra with a contractive pointwise approximate identity. Then  f BSE,MpAq “  f BSE,A

p f P Aq .

(5.2.10)

5.2 Basic definitions and results about BSE norms

319

Proof Take f P A such that  f BSE,MpAq “ 1. By Theorem 5.2.25,  f BSE,A ě 1. Now fix a finite  subset F of ΦA Ă ΦMpAq and ε > 0. Then there exists g P MpAqr1s with  f pϕq ´ gpϕq < ε  pϕ P Fq. ByCorollary 5.2.17, g P C BS E pAqr1s , and so there  exists h P Ar1s with gpϕq ´ hpϕq < ε pϕ P Fq. It follows that   f pϕq ´ hpϕq < 2ε pϕ P Fq, and hence  f BSE,A ď 1. Equation (5.2.10) follows. The requirement that A have a CPAI to obtain equation (5.2.10) cannot be dispensed with. For example, set A “ p p , · q, so that MpAq “ 8 . For each f P A, we have  f BSE,A “  f  p , but  f BSE,MpAq “  f 8 . Proposition 5.2.29 Let A be a Banach function algebra. Then the following conditions on A are equivalent: (a) A has a BSE norm; (b) A is closed as a subalgebra of C BS E pAq; (c) the subalgebra A + LpAqK of CpA q is norm-closed in CpA q. Proof The equivalence of (a) and (b) is immediate, and the equivalence of (b) and (c) follows from Corollary 1.3.17. Let A be a Banach function algebra. We consider when C BS E pAq Ă MpAq, equivalently when A is an ideal in C BS E pAq. We shall exhibit Banach function algebras A in Examples 5.3.14 and 5.4.13 that show that A need not be an ideal in C BS E pAq; in these two examples, A will be, respectively, closed and dense in C BS E pAq. We recall that A2 is dense in A when A has a bounded approximate identity or is Tauberian. Proposition 5.2.30 Let pA,  · q be a Banach function algebra such that A2 is dense in A and  f BSE “  f  p f P Aq. Then C BS E pAq Ă MpAq if and only if MpAqr1s is closed in pC b pΦA q, τ p q. Proof Suppose that MpAqr1s is closed in pC b pΦA q, τ p q. Then C BS E pAq Ă MpAq by Corollary 5.2.13. Now suppose that C BS E pAq Ă MpAq, and let p fα q be a net in MpAqr1s that converges to f in pC b pΦA q, τ p q. Take g P Ar1s . Then p fα gq is a net in Ar1s that converges to f g in pC b pΦA q, τ p q, and so f g P C BS E pAqr1s Ă MpAq. It follows that f A2 Ă A. There is a sequence pgn q in A2 such that limnÑ8 gn ´ g “ 0. We have f gn P A pn P Nq and limnÑ8  f gn ´ f gBSE “ 0. It follows that lim  f gn BSE ď 1, and so lim sup  f gn  ď 1. This implies that f g P Ar1s , and hence f P MpAqr1s . Thus MpAqr1s is closed in pC b pΦA q, τ p q. Corollary 5.2.31 Let A be a Banach function algebra with a contractive approximate identity that is a BSE algebra. Then the following conditions are equivalent: (a)  f BSE “  f  p f P Aq ; (b) MpAqr1s is closed in pC b pΦA q, τ p q ; (c) Ar1s is τ p -closed in A.

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5 BSE norms and BSE algebras

Proof (a) ⇒ (b) Since A has a CAI, A2 “ A, and so, by Proposition 5.2.30, MpAqr1s is closed in pC b pΦA q, τ p q. (b) ⇒ (a) By Corollary 5.2.17 and (b), C BS E pAqr1s “ MpAqr1s , and hence (a) follows. (a) ⇔ (c) This follows from Theorem 5.2.24. Clause (i) of the following result is [200, Theorem 4.8]. Proposition 5.2.32 Let A be a non-unital Banach function algebra. (i) The algebra A is a BSE algebra if and only if C BS E pAq ∩ C 0 pΦA q “ A. (ii) Suppose that  f BSE “  f  p f P Aq. Then C BS E pAq ∩ C 0 pΦA q “ A if and only if Ar1s is closed in pC 0 pΦA q, τ p q. Proof Set K “ ΦA , so that A Ă C BS E pAq ∩ C 0 pKq. (i) Suppose that A is a BSE algebra, and take f P C BS E pAq ∩ C 0 pKq. There are m > 0 and a net p fν q in Arms such that fν Ñ f pointwise on K; extend f and each fν to CpK8 q, each taking the value 0 at 8. Then fν Ñ f pointwise on K8 , and so f P C BS E pA q “ A . Thus C BS E pAq ∩ C 0 pKq “ A. Conversely, suppose that C BS E pAq ∩ C 0 pKq “ A, and take f P C BS E pA q. Then f ´ f p8q1K8 P A, and so f P A . Thus A is a BSE algebra. (ii) Suppose that C BS E pAq ∩ C 0 pKq “ A, and take a net in Ar1s that converges pointwise to f in C 0 pKq. Then f P C BS E pAq ∩ C 0 pKq “ A, and so  f  “  f BSE ď 1. Hence Ar1s is closed in pC 0 pKq, τ p q. Conversely, suppose that Ar1s is closed in pC 0 pKq, τ p q, and take a function f P C BS E pAq ∩ C 0 pKq, say  f BSE ď 1. Then there is a net in Ar1s that converges pointwise to f in C 0 pKq. Thus f P Ar1s , and so C BS E pAq ∩ C 0 pKq “ A. Example 5.3.1(ii) will show that p is not a BSE algebra, but that p p q is a BSE algebra. Theorem 5.3.28 will show that the BSE algebra L1 pGq is such that L1 pGq is not a BSE algebra for each non-discrete LCA group, and Example 5.4.13 will also exhibit a non-unital Banach function algebra M that is a BSE algebra, but such that M  is not a BSE algebra. Proposition 5.2.33 Let A be a Banach function algebra such that Ar1s is closed in pC 0 pΦA q, τ p q. Then MpAq is a BSE algebra. Proof Since MpAq is an unital Banach function algebra, it follows from Corollary 5.2.21 that pC BS E pMpAqq,  · BSE,MpAq q “ pMpAq,  · op q if and only if MpAqr1s is closed in pCpΦMpAq q, τ p q, in which case MpAq is a BSE algebra. Let p fν q be a net in MpAqr1s that converges pointwise on ΦMpAq to f P CpΦMpAq q, regarded as a function in C b pΦA q. Take g P Ar1s . Then f g P C 0 pΦA q, and p fν gq is a net in Ar1s that converges pointwise on ΦA to f g. By hypothesis, f g P Ar1s , and hence f P MpAqr1s , as required.

5.2 Basic definitions and results about BSE norms

321

Corollary 5.2.34 Let A be a non-unital Banach function algebra such that A is a BSE algebra and  f BSE “  f  p f P Aq. Then MpAq is a BSE algebra. Proof By Proposition 5.2.32(ii), Ar1s is closed in pC 0 pΦA q, τ p q, and so MpAq is a BSE algebra by Proposition 5.2.33. Let A be a Banach function algebra. The algebras M 00 pAq and M 0 pAq were defined in Definition 3.1.24 as closed ideals in MpAq, and we have A Ă M 00 pAq Ă M 0 pAq “ MpAq ∩ C 0 pΦA q Ă MpAq . Proposition 5.2.35 Let pA,  · q be a Banach function algebra with a contractive pointwise approximate identity. (i) Suppose that Ar1s is closed in pC 0 pΦA q, τ p q. Then M 0 pAqr1s “ Ar1s . (ii) Suppose that pMpAq,  · op q “ pC BS E pAq,  · BSE q. Then Ar1s is closed in pC 0 pΦA q, τ p q if and only if M 0 pAqr1s “ Ar1s . Proof The CPAI in A is peα q. (i) Certainly Ar1s Ă M 0 pAqr1s . Now take f P M 0 pAqr1s Ă C 0 pΦA q. Then p f eα q is a net in Ar1s which converges pointwise to f on ΦA , and so f P Ar1s . Hence Ar1s “ M 0 pAqr1s . (ii) By (i), M 0 pAqr1s “ Ar1s whenever Ar1s is closed in pC 0 pΦA q, τ p q. Conversely, suppose that M 0 pAqr1s “ Ar1s . Take a net p fα q in Ar1s that converges pointwise on ΦA to f P C 0 pΦA q. Then f P C BS E pAqr1s “ MpAqr1s , and so f P M 0 pAqr1s “ Ar1s . Thus Ar1s is closed in pC 0 pΦA q, τ p q. Theorem 5.2.36 Let A be a Banach function algebra that is a BSE algebra, and suppose that M 00 pAq  M 0 pAq. Then neither A nor MpAq is a BSE algebra. Proof Since by hypothesis M 00 pAq  M 0 pAq, necessarily A  MpAq, and so A is non-unital. Since A is a BSE algebra, A  M 0 pAq “ C BS E pAq ∩ C 0 pΦA q, and so, by Proposition 5.2.32(i), A is not a BSE algebra. It follows from Corollary 5.2.18 that A has a BPAI, say peα q; we can regard peα q as a bounded net in MpAq such that each eα | H “ 0, where H “ ΦMpAq \ ΦA . Assume that MpAq is a BSE algebra, so that C BS E pMpAqq “ MpAq. Now take f P M 0 pAq. Set g “ f on ΦA and g “ 0 on H, so that g P CpΦMpAq q because f P C0 pΦA q. Clearly the net p f eα q converges pointwise to g on ΦMpAq , and so g P C BS E pMpAqq “ MpAq. Since ΦA is a determining set for MpAq, it follows that f “ g P M 00 pAq. Hence M 0 pAq “ M 00 pAq, a contradiction. So MpAq is not a BSE algebra.

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5 BSE norms and BSE algebras

Theorem 5.2.37 Let pA,  · q be a dual Banach function algebra with Banachalgebra predual F, and suppose that ΦA ∩ F is dense in ΦA . Then pA,  · q “ pC BS E pAq,  · BSE q .

(5.2.11)

Further, the following are equivalent: (a) A is a BSE algebra; (b) A has an identity; (c) A has a bounded pointwise approximate identity. Proof Let p fα q be a net in Ar1s with limα fα “ f in pA, τ p q. Let g be an accumulation point of the net p fα q in pA, σpA, Fqq. Then gpϕq “ f pϕq pϕ P ΦA ∩ Fq, and so g “ f because ΦA ∩ F is dense in ΦA . By equation (1.2.6), g ď 1, and so f P Ar1s . This shows that Ar1s is τ p -closed in A, and so equation (5.2.11) follows from Theorem 5.2.24. Since A “ C BS E pAq, the equivalence of (a), (b), and (c) follows from Theorem 5.2.16. Proposition 5.2.38 Let A be a Banach function algebra such that CpA q “ A . Then C BS E pAq is an isometric dual Banach function algebra, with Banach-algebra predual LpAq. In particular, this holds whenever A is an ideal in its bidual or A is a natural Banach sequence algebra. Proof Since CpA q “ A , it follows that pC BS E pAq,  · BSE q “ pQpAq,  · QpAq q as Banach function algebras, and so the main result is immediate from Theorem 5.1.9. In the case where A is an ideal in its bidual, it follows from Proposition 2.3.17 that CpA q “ A , and this is immediate when A is a natural Banach sequence algebra, giving the result. We next characterize the BSE algebras that have a BSE norm. Proposition 5.2.39 Let A be a BSE algebra. Then A has a BSE norm if and only if A is closed in pMpAq,  · op q. Proof Since A is a BSE algebra, MpAq “ C BS E pAq, and so the norms  · op and  · BSE are equivalent on MpAq. Since A has a BSE norm if and only if A is closed in pC BS E pAq,  · BSE q, this holds if and only if A is closed in pMpAq,  · op q. The second clause of the following corollary was given in [297, Theorem 3]. Corollary 5.2.40 Let A be a BSE algebra with a bounded approximate identity. Then A has a BSE norm. In the case where also C BS E pAq “ C b pΦA q, then necessarily A “ C 0 pΦA q. Proof Since A has a BAI, A is closed in pMpAq,  · op q by Proposition 2.1.30, and so, by Proposition 5.2.39, A has a BSE norm. Suppose now that C BS E pAq “ C b pΦA q. Then A is a closed ideal in the space MpAq “ C BS E pAq “ C b pΦA q, and so A is a C ∗ -algebra. Thus A “ C 0 pΦA q.

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323

Let A be a Banach function algebra with a contractive approximate identity, so that MpAqr1s Ă C BS E pAqr1s by Theorem 5.2.16, (b) ⇒ (c). We next consider the question whether  f BSE “  f op p f P MpAqq. Theorem 5.2.41 Let A be a Banach function algebra with a contractive approximate identity. Then the following conditions are equivalent: (a)  f BSE,MpAq “  f op p f P MpAqq; (b)  f BSE “  f  p f P Aq; (c) LpAqr1s is weak-∗ dense in MpAqr1s . Further, when these conditions are satisfied,  f BSE “  f BSE, MpAq p f P MpAqq. Proof (a) ⇒ (b) By Proposition 2.1.30,  f op “  f  p f P Aq because A has a CAI, and so this is immediate. (b) ⇒ (a) Take f P MpAq. By Theorem 5.2.16, (b) ⇒ (d),  f BSE ď  f op . For each ε > 0, there is g P Ar1s with  f g >  f op ´ ε. Since  f g “   f gBSE , there exists λ P LpAqr1s with x f g, λy >  f g ´ ε. Now  f BSE ě x f, g · λy because g · λ P LpAqr1s , and so    f BSE ě x f g, λy >  f g ´ ε >  f op ´ 2ε . Hence  f BSE “  f op , as required. (a) ⇔ (c) This follows from Corollary 1.2.14 (where we take E “ MpAq and F “ LpAq). Now suppose that the equivalent conditions are satisfied, and take f P MpAq. By Proposition 5.1.27, LpAqr1s Ă LpMpAqqr1s , and so  f BSE ď  f BSE, MpAq . Further,    f BSE, MpAq “ sup{x f, λy : λ P LpMpAqqr1s }    σ ď sup x f, λy : λ P LpAqr1s by pcq   “ sup{x f, λy : λ P LpAqr1s } “  f BSE , and so  f BSE “  f BSE,MpAq p f P MpAqq. Let A be a Banach function algebra with a mixed identity E P A . Then, as in Corollary 2.3.79, the map f Þ→ f · E “ Lf pEq, MpAq Ñ A , is a Banach-algebra embedding, and so, in the next theorem, we regard MpAq as a closed subalgebra of A . The result is [200, Theorem 2.1]. Theorem 5.2.42 Let A be a Banach function algebra with a mixed identity. Then A is a BSE algebra if and only if CpA q “ MpAq ‘ LpAqK . In the case where A has an identity, A is a BSE algebra if and only if CpA q “ A ‘ LpAqK .

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5 BSE norms and BSE algebras

Proof Since there exists a mixed identity for A, it follows that A has a BAI, and so it follows from Theorem 5.2.16, (b) ⇒ (c), that the isomorphism R : M Þ→ M + LpAqK ,

CpA q Ñ C BS E pAq ,

always maps MpAq into C BS E pAq, and A is a BSE algebra if and only if this map is a surjection. In this case, CpA q “ MpAq + LpAqK . Since MpAq and LpAqK are linear subspaces of CpA q and since xE, ϕy “ 1 pϕ P ΦA q, it follows that MpAq ∩ LpAqK “ {0}, and hence CpA q “ MpAq ‘ LpAqK . Conversely, suppose that CpA q “ MpAq ‘ LpAqK . Then the map R identifies MpAq with C BS E pAq, and so A is a BSE algebra. We finally consider when some BSE properties pass to ideals in Banach function algebras. Proposition 5.2.43 Let A be a Banach function algebra, and let I be a weak Segal algebra with respect to A. Then C BS E pIq is an ideal in C BS E pAq, and  f BSE, A ď  f BSE, I

p f P C BS E pIqq .

(5.2.12)

Suppose that A has a bounded pointwise approximate identity. Then I also has a bounded pointwise approximate identity if and only if C BS E pIq “ C BS E pAq. Proof Since  f A ď  f I p f P Iq and ΦI “ ΦA , it follows from Corollary 5.2.26 that C BS E pIq Ă C BS E pAq and that equation (5.2.12) holds. Take f P C BS E pAqr1s and g P C BS E pIqr1s . For each finite  subset F of ΦA and each    ε > 0, there are f1 P Ar1s and g1 P Ir1s with f pϕq ´ f1 pϕq 0. Then C BS E pAq | ΦI “ C BS E pIq and  f BSE,A ď  f BSE,I ď k  f BSE,A

p f P Iq .

(5.2.13)

Proof Certainly C BS E pIq Ă C BS E pAq | ΦI and  f BSE,A ď  f BSE,I p f P Iq. Now take f P C BS E pAqrms , where m > 0. Then, foreach finite subset F of ΦI and each ε > 0, there exists g P Arms with  f pϕq ´ gpϕq < ε pϕ P Fq. Since I has a

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325

  BPAI of bound k > 0, there exists h P Irks with 1 ´ hpϕq < ε pϕ P Fq. It follows that gh P Irkms and        f pϕq ´ pghqpϕq ď  f pϕq ´ gpϕq + m 1 ´ hpϕq < pm + 1qε pϕ P Fq . Thus, by Theorem 5.2.9, f | ΦI P C BS E pIqrkms , and hence C BS E pAq | ΦI Ă C BS E pIq. It follows that C BS E pIq “ C BS E pAq | ΦI . n  Next,  take f P I. Then, for each ε > 0, there exists λ “ i“1 αi ϕi P LpIqr1s with x f, λy >  f BSE,I ´ ε ; we regard λ as an element of LpAq. Take h P Ar1s . Then there exists g P Irks such that n 

   |αi | hpϕi q 1 ´ gpϕi q < ε .

i“1

       , and so xgh, λy ď ghI ď k. Thus xh, λy < xgh, λy + ε. But gh P I and λ P Ir1s     This shows that xh, λy < k + ε, and so λ P LpAqrks , whence x f, λy ď k  f BSE,A . It follows that  f BSE,I ď k  f BSE,A , giving equation (5.2.13). The following result is close to [180, Theorem 9.10] and [198, Proposition 2.8]. Theorem 5.2.45 Let A be a Banach function algebra that is a BSE algebra with a bounded approximate identity, and let I be a non-zero ideal in A that is a Banach A-module. Then I is a BSE algebra if and only if I has a bounded pointwise approximate identity. In this case, C BS E pIq “ MpIq “ MpAq | ΦI . Suppose, further, that I is dense in A and that C BS E pIq “ MpAq. Then I is a BSE algebra. Proof Suppose that I is a BSE algebra. Then I has a BPAI by Corollary 5.2.21. Conversely, suppose that the ideal I has a BPAI. Then MpIq Ă C BS E pIq by Theorem 5.2.16, (b) ⇒ (c). Also C BS E pIq Ă C BS E pAq | ΦI , and C BS E pAq “ MpAq because A is a BSE algebra. Further, Proposition 3.1.26 applies because A has a BAI, and so MpAq | ΦI Ă MpIq. Hence C BS E pIq “ MpIq, I is a BSE algebra, and MpIq “ MpAq | ΦI . Now suppose that I is dense in A, so that it is a weak Segal algebra with respect to A. By Proposition 5.2.43, I has a BPAI, and so I is a BSE algebra. Corollary 5.2.46 Let A be an unital Banach function algebra, and let I be a closed ideal in A. Suppose that C BS E pAq “ A. Then I is a BSE algebra if and only if I has a bounded pointwise approximate identity. Clause (i) of the following corollary is an extension of [200, Theorem 4.1]. Corollary 5.2.47 Let A be a Banach function algebra with a bounded approximate identity. (i) Suppose that MpAq is a BSE algebra. Then A is a BSE algebra. (ii) Suppose that ΦA is dense in ΦMpAq and that A is a BSE algebra. Then MpAq is a BSE algebra.

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5 BSE norms and BSE algebras

Proof (i) By Proposition 2.1.30, A is a closed ideal in the unital algebra MpAq, and so this is immediate from Corollary 5.2.46. (ii) Take f P C BS E pMpAqq. Then f | ΦA P C BS E pAq by Theorem 5.2.44, and so there exists g P MpAq such that f | ΦA “ g. Since ΦA is dense in ΦMpAq , we have f “ g on ΦMpAq , and so f P MpAq. Thus MpAq is a BSE algebra. Proposition 5.2.48 Let A be a Banach function algebra that is a BSE algebra, and let I be a closed ideal in A with ΦI “ ΦA and such that I has a bounded approximate identity. Then I is a BSE algebra and C BS E pIq “ C BS E pAq. Proof Certainly we have C BS E pIq Ă C BS E pAq “ MpAq. By Proposition 3.1.26, MpAq Ă MpIq. By Theorem 5.2.16, (b) ⇒ (c), MpIq Ă C BS E pIq. It follows that C BS E pIq “ C BS E pAq “ MpIq, and so I is a BSE algebra.

5.3 Examples In this section, we shall give various examples of Banach function algebras for which we can calculate the BSE norm and determine whether they are BSE algebras. After some easy examples and some examples on closed intervals of R, we shall consider when uniform algebras are BSE algebras; we shall give some examples that are and are not BSE algebras, but we have no general characterization. A major class of examples includes the Fourier and Fourier–Stieljes algebras associated to a locally compact group, and various related Segal algebras, as described in §4.3, and then the Figà-Talamanca–Herz algebras that were discussed in §4.4. A key tool will be the Kaplansky density theorem that is available for Fourier algebras, but not for the more general Figà-Talamanca–Herz algebras. The final part of this section will be devoted to showing that a Varopoulos algebra VpK, Lq is always such that  f BSE “  f π for all f P VpK, Lq for every non-empty, compact spaces K and L, but that VpK, Lq is not a BSE algebra, at least when both K and L are infinite, compact, metrizable spaces. Some further examples, including some Banach sequence algebras, will be given in §5.5. We begin with some easy examples, in particular showing that some basic Banach sequence algebras are not BSE algebras. Example 5.3.1 (i) Consider the Banach sequence algebra c 0 . Then it is clear that Mpc 0 q “ C BS E pc 0 q “ 8 , and so c 0 is a BSE algebra with a BSE norm. More generally, let K be a non-empty, locally compact space. Then the natural uniform algebra C 0 pKq is a BSE algebra such that C BS E pC 0 pKqq “ MpC 0 pKqq “ C b pKq “ Cpβ Kq

5.3 Examples

327

with  f BSE “ | f |K p f P C b pKqq. Set A “ C 0 pKq and B “ C BS E pAq. Then ΦA “ K and ΦB “ β K, so that ΦA is dense in ΦB . (ii) Take p with 1 ď p < 8, and consider the Tauberian Banach sequence algebra p p ,  ·  p q, so that p is a Segal algebra with respect to c 0 . p Take α P C BS E p p qr1s . There is a net in r1s that converges coordinatewise to n   p α  ď 1 pn P Nq, and hence α P p . This shows that α; clearly we have j“1

j

r1s

C BS E p p q “ p and that αBSE “ α p pα P p q, so that p has a BSE norm. Clearly Mp p q “ 8 , and so p is not a BSE algebra. Since p does not have a bounded pointwise approximate identity, it also follows from Proposition 5.2.43 that C BS E p p q  C BS E pc 0 q. Since C BS E p p q ∩ c 0 “ p , it follows from Proposition 5.2.32(i) that p p q is a BSE algebra. Example 5.3.2 (i) Consider the Beurling algebra Bω “ p 1 pZ, ωq,  · ω ,  q for a weight ω on Z, as in Example 4.2.20, so that Bω is an isometric dual Banach function algebra, with Banach-algebra predual c 0 pZ, 1/ωq, and Bω is regarded as an unital Banach function algebra on the set Xω . Using the notation of that example, we know from Example 5.1.34 that pBω ,  · ω q “ pQpBω q,  · QpBω q q “ pC BS E pBω q,  · BSE q whenever ρ1 “ ρ2 “ 1 and lim|n|Ñ8 ωn “ 8. Now suppose that ρ2 < ρ1 . Then int Xω is a dense subset of Xω that is contained in c 0 pZ, 1/ωq, and so it follows from Theorem 5.2.37 that pBω ,  · ω q “ pC BS E pBω q,  · BSE q and that Bω is a BSE algebra in this further case. : n P N} “ 1, so that Now let ω be a weight on pZ+ , + q such that inf{ω1/n n 1 pZ+ , ωq is a natural Banach function algebra on D. Then, similarly, 1 pZ+ , ωq is a BSE algebra with a BSE norm. In particular, this covers the Banach function algebra A+ pDq of absolutely convergent Taylor series, as in Example 3.6.10. p 8 “ p ⊗ p Cpβ Nq, as in (ii) Take p with 1 < p < 8, and set A “ p ⊗ Example 3.3.6(ii), so that A is a dual Banach algebra with Banach-algebra predual F “ Kpc 0 , q q, where q “ p . Further, A is a natural Banach function algebra on N × β N. Take K “ N × N Ă ΦA . As in Example 5.5.8, the map εpm,nq is a rank-one operator in Kpc 0 , q qr1s for pm, nq P N × N, and so ΦA ∩ F is dense in ΦA . It follows from Theorem 5.2.37 that A “ C BS E pAq and that A has a BSE norm. However A does not have a bounded pointwise approximate identity for any p, and so A is not a BSE algebra. p q pZ+ , ω s q of Exam(iii) The unital Banach function algebra A “ p pZ+ , ωr q ⊗ q G for ple 4.2.23 is a dual Banach function algebra with Banach-algebra predual F ⊗ q G. Thus, by Theorem 5.2.37, A is a certain Banach spaces F and G, and ΦA Ă F ⊗ BSE algebra with a BSE norm.

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5 BSE norms and BSE algebras

We now consider some standard function algebras on closed intervals in R. The first example was given in [200, Example 6.1]. Example 5.3.3 Take α such that 0 < α < 1, and consider the Lipschitz algebras A “ lipα I and A “ Lipα I on the interval I “ r0, 1s, as in Example 3.1.36. Here LpAqK “ {0} and CpA q “ A , and so A Ă C BS E pAq “ QpAq “ A . Hence A has a BSE norm, and A has a BSE norm by Theorem 5.2.15. Since A and A are unital, MpAq “ A and MpA q “ A , and so A is not a BSE algebra. As in Example 5.1.23, QpA q “ A , and so C BS E pA q “ A . Thus A is a BSE algebra.

Example 5.3.4 Let I “ ra, bs be a closed interval in R, where a < b, and let BVCpIq and ACpIq be, respectively, the natural, unital Banach function algebras on I of continuous functions of bounded variation and absolutely continuous functions on I, as described in Example 3.1.35, so that ACpIq is a proper, closed subalgebra of BVCpIq. Let f P C BS E pBVCpIqqr1s , say limν fν “ f in pCpIq, τ p q where p fν q is a net in BVCpIqr1s . Take x P I and x1 , . . . , xn in I with a “ x0 < x1 < · · · < xn “ b. Then ⎛ ⎞ n  n  ⎜⎜⎜    ⎟⎟⎟        f pxq +  f px j q ´ f px j´1 q “ lim ⎜⎜⎝⎜ fν pxq +  fν px j q ´ fν px j´1 q⎟⎟⎠⎟ ď 1 , ν j“1

j“1

and so f P BVCpIqr1s . It follows that BVCpIq is a BSE algebra with a BSE norm. Now take f P BVCpIqr1s . Then it is easy to see that there is a net p fν q in ACpIqr1s such that limν fν “ f in pCpIq, τ p q, and so C BS E pACpIqq “ BVCpIq. This shows that ACpIq is not a BSE algebra.

For n P N, we write pC pnq pIq,  · n q for the algebra of n-times continuously differentiable, as in Example 3.1.37, where I “ r´1, 1s, an unital Banach function algebra on I. A more general form of the following remark is given in [179, Theorem 6.6]. Example 5.3.5 Consider the sequence p fk q in C p1q pIq, where fk is defined for k P N by  2 1/2 kt + 1 pt P Iq . fk ptq “ k+1   Then it is easily checked that | f | “  f   “ 1 pk P Nq, so that p f q is a bounded kI

k I

k

sequence in C p1q pIq, and that limkÑ8 fk ptq “ |t| pt P Iq. Set f ptq “ |t| pt P Iq. Then f P C BS E pC p1q pIqq, but f  C p1q pIq, and so C p1q pIq is not a BSE algebra.

5.3 Examples

329

Before giving the next example, we require a lemma. In the lemma, I “ ra, bs. Lemma 5.3.6 Let f P CpIq, and suppose that there is a net p fα q in C p2q pIqr1s that converges pointwise to f . Then f P C p1q pIqr1s , and we may suppose that f “ limα fα in pC p1q pIq,  · 1 q. Proof The net p fα q is a pointwise bounded and equicontinuous set in CpIq, and so, by the Ascoli–Arzelà theorem, Theorem 1.1.11, it forms a relatively compact set in CpIq. By passing to a subnet, we may suppose that limα fα “ f in pCpIq, | · |I q. Now consider the net p fα q. This net is also pointwise bounded and equicontinuous set in CpIq (because it is bounded in C p1q pIq), and so, by passing to a subnet, we    may suppose that it converges to a function h P CpIq with |h|I ď sup  α fα I .  Take a point t0 P I and ε > 0. Then there exists δ > 0 such that hptq ´ hpt0 q < ε α with  such a t (with t  t0). Then there exists  when t  P I with |t ´ t0 | < δ. Take  fα ´ hI < ε, with  fα ptq ´ f ptq < ε |t ´ t0 |, and with  fα pt0 q ´ f pt0 q < ε |t ´ t0 |. We see that    f ptq ´ f pt0 q  ´ hpt0 q < 4ε.   t ´ t0 This shows that f is differentiable at t0 with f  pt0 q “ hpt0 q. Hence f  “ h, and so f P C p1q pIqr1s with f “ limα fα in pC p1q pIq,  · 1 q. Example 5.3.7 Let I “ ra, bs be a closed interval in R, where a < b, and let D :“ DpI; pMk qq be a natural Dales–Davie algebra on I, as in Example 3.1.38. Take f P C BS E pDqr1s . Then there is a net p fα q in Dr1s such that limα fα “ f in pCpIq, τ p q. Take n P N. Then it follows from successive applications of Lemma 5.3.6 that f P C pnq pIq. Now take t0 , t1 , . . . , tn P I. Then, for each α, we have n  1  pkq   fα ptk q ď 1 . Mk k“0

      It follows that nk“0  f pkq ptk q /Mk ď 1, and so nk“0  f pkq I /Mk ď 1. This holds for each n P N, and so f P Dr1s . Thus D is a BSE algebra with a BSE norm.

We now consider when uniform algebras are BSE algebras; we recall that certainly the uniform norm and the BSE norm of a uniform algebra are equal. We caution that it is claimed in [200, Theorem 2.6] that each uniform algebra is a BSE algebra; Examples 5.3.11, 5.3.14, and 5.3.15 will show that this is not correct. However, to give an intrinsic characterization of uniform algebras that are BSE algebras is a challenging problem.

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The following example, which shows that some uniform algebras are BSE algebras, slightly generalizes [297, Theorem 7] of Takahasi and Hatori. Example 5.3.8 Let K be a non-empty, compact subset of C, and consider the algebra A “ ApKq of Example 3.6.9; we recall that A is a natural, unital uniform algebra on K. Set U “ int K, and suppose that U  ∅; this class of examples includes the disc algebra, ApDq. Alternatively, let U be a non-empty, open set in C, and consider the unital uniform algebra A “ H 8 pUq. For each of these algebras A, let S be a countable, dense subset of U, say  S “ {S n : n P N}, where each S n is a finite set and S n Ă S n+1 pn P Nq.  f P C BS E pAqr1s . For each n P N, there exists fn P Ar1s such that Take a function  f pzq ´ fn pzq < 1/n pz P S n q. By Montel’s theorem, Theorem 1.1.12, the sequence p fn q has a subsequence that converges in OpUq with respect to the compact–open topology, say with limit g. Clearly gpzq “ f pzq pz P S q, and so f | U “ g P OpUq. Thus f P A in each case. This shows that ApKq and H 8 pUq are BSE algebras. However, when K Ă C, we do not know whether RpKq is always a BSE algebra. In particular, this is open whenever K has an empty interior (and RpKq  CpKq). Assume that C BS E pRpKqq “ CpKq. By Corollary 5.3.13, each point of K is isolated with respect to the Gleason metric. By [295, Theorem 26.12], each point of K is then a peak point for RpKq, and so RpKq “ CpKq by [295, Corollary 26.15]. Thus RpKq Ă C BS E pRpKqq  CpKq for each such K whenever RpKq  CpKq. Let M “ { f P ApDq : f p0q “ 0}, a maximal ideal of the disc algebra. Clearly C BS E pMq “ M, but MpMq “ ApDq, and so M is not a BSE algebra.

Definition 5.3.9 Let A be a natural uniform algebra on a non-empty, compact space K. Then Ac is the algebra of all continuous functions f on K such that f | P is constant on P for each Gleason part P, and alg{A, Ac } consists of all finite sums of finite products of element of A ∪ Ac . Thus alg{A, Ac } is the algebra generated by A and Ac . Of course, it may be that Ac Ă A, and so alg{A, Ac } “ A. For example, this is the case when A is the disc algebra, but A  Ac “ CpKq for each non-trivial Cole algebra. Theorem 5.3.10 Let A be a natural uniform algebra on a non-empty, compact space K. Then alg{A, Ac } Ă C BS E pAq. Proof Let f P pAc qr1s . By Corollary 3.6.28, there exists an element F P Ar1s such that F | K “ f , and so f P C BS E pAqr1s . Since C BS E pAq is an algebra and since certainly A Ă C BS E pAq, it follows that alg{A, Ac } Ă C BS E pAq, as required. Example 5.3.11 The unital uniform algebra A “ H 8 +C was described in Example 3.6.39. It is shown in [105] that Ac is not contained in A, and so, by Theorem 5.3.10, A is not a BSE algebra.

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Proposition 5.3.12 Let A be a natural uniform algebra on a non-empty, compact space K. Then the following conditions are equivalent: (a) every singleton in K is a one-point Gleason part; (b) pQpAq,  · QpAq q “ p 8 pKq, | · |K q ; (c) pC BS E pAq,  · BSE q “ pCpKq, | · |K q. Proof (a) ⇒ (b) This is given in Example 5.1.20. (b) ⇒ (c) This is immediate from the definition of C BS E pAq. (c) ⇒ (a) We may suppose that |K| ě 2. Take x, y P K with x  y. Then there exists f P CpKqr1s with f pxq “ 1 and f pyq “ ´1. By (c), f P C BS E pAqr1s , and so there exists F P CpA qr1s with F | K “ f . This shows that dA px, yq “ 2, where dA is the Gleason metric on K, and so x  y. Thus (a) follows. Corollary 5.3.13 Let A be a natural uniform algebra on a non-empty, compact space K with |K| ě 2, and suppose that C BS E pAq “ CpKq. Then there exists d > 0 such that dA px, yq > d whenever x, y P K with x  y, and so each point of K is isolated with respect to the Gleason metric. Proof There is a constant C > 0 such that  f BSE,A ď C | f |K p f P CpKqq. As in the proof of Proposition 5.3.12, (c) ⇒ (a), dA px, yq ě 2/C whenever x, y P K with x  y, and so each point of K is isolated in pK, dA q. Example 5.3.14 Let A be a natural uniform algebra on a compact space K such that A  CpKq and such that each singleton in K is a one-point part. Then, by Proposition 5.3.12, A is not a BSE algebra. Here A is an unital, closed subalgebra of CpKq “ C BS E pAq, and so A is not an ideal in C BS E pAq. As in §3.6, there are well-known examples of separable Cole algebras A on certain compact spaces K, so that every point of K is a strong boundary point for A, and hence a one-point part, but such that A  CpKq. These Cole algebras are uniform algebras that are not BSE algebras. These examples include the example of Basener mentioned on page 218: there is a compact subspace K of C2 such that RpKq  CpKq, but Γ0 pRpKqq “ K, and so C BS E pRpKqq “ CpKq, which shows that RpKq is not a BSE algebra. By Theorem 3.6.20, all Cole algebras are contractive. For Cole algebras with additional properties, see Example 3.6.40. In Example 3.6.37, it was noted that there are natural, separable uniform algebras A on K such that each point of K is a one-point part, but ΓpAq  K; these uniform algebras also fail to be BSE algebras.

Example 5.3.15 Let A be the separable, regular, natural uniform algebra on a compact space K such that there is a two-point part in K, and such that all other points of K are one-point Gleason parts, as mentioned in Examples 3.6.41 and 5.1.20, so that

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A  CpKq. As in Example 5.1.20, QpAq “ 8 pKq, and hence C BS E pAq “ CpKq. However, by Proposition 5.3.12, pC BS E pAq,  · BSE q  pCpKq, | · |K q. In [105], it is shown by using such algebras that there is a separable uniform algebra A on a compact space K such that C BS E pAq is a dense subalgebra of CpKq, but such that C BS E pAq  CpKq, and so C BS E pAq is not equivalent to a uniform algebra. It follows that QpAq is not equivalent to a uniform algebra.

We now give some examples of unital uniform algebras that are dual Banach algebras, but are not BSE algebras. Example 5.3.16 Let A be an unital uniform algebra on a non-empty, compact space K. It follows from Proposition 2.4.9 that A (regarded as a uniform algebra on ΦA ) is a dual Banach algebra. Let π : ΦA Ñ K be the continuous projection of Proposition 3.6.13(i). As in Proposition 3.6.13(i), Φ{x} is a fibre in ΦA for each x P K; we are identifying A with the functions in A that are constant on each fibre in ΦA . Suppose that A is a BSE algebra on ΦA , and take a function f P C BS E pAqr1s , say f “ limα fα in pCpKq, τ p q, where p fα q is a net in Ar1s , so that p fα q is also a net in Ar1s . Define F on ΦA by setting Fpϕq “ f pπpϕqq pϕ P ΦA q, so that F P CpΦA qr1s . For each ϕ P ΦA , we have Fpϕq “ f pπpϕqq “ lim fα pπpϕqq “ lim fα pϕq , α



α



and so F P C BS E pA qr1s . By hypothesis, F P A . But F is constant on all fibres in ΦA , and so F P A, i.e., f P A. Thus C BS E pAq “ A, and A is a BSE algebra on K. Now suppose that A is not a BSE algebra, say A is a non-trivial Cole algebra. Then A is an unital uniform algebra that is a dual Banach algebra, but it is not a BSE algebra.

Example 5.3.17 Let A be a natural uniform algebra on a compact space K, and suppose that there exists x P K such that {x} is a singleton Gleason part, but x  Γ0 pAq; see Examples 3.6.37, 3.6.38, and 3.6.42. As we remarked, by Theorem 3.6.29, the maximal ideal M x has a contractive pointwise approximate identity, but, by Theorem 3.6.16, M x does not have a bounded approximate identity. The Banach function algebra M x has a BSE norm, and, by Corollary 5.2.19, the operator norm is equivalent to the uniform norm on M x , despite the fact that M x does not have a bounded approximate identity; cf. Proposition 2.1.30.

We now turn to the BSE properties of some algebras that arise in harmonic analysis. Let Γ be a locally compact group. The Fourier algebra ApΓq, the Fourier–Stieljes algebra BpΓq, and related algebras were introduced in §4.3; the Figà-Talamanca– Herz algebras A p pΓq were introduced in §4.4.

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The first result gives a version of the Bochner–Schoenberg–Eberlein theorem that applies to certain Fourier algebras and generalizes the theorem for locally compact abelian groups mentioned in the introduction to this chapter; it is based on results contained in the seminal thesis of Eymard [97]. Theorem 5.3.18 Let Γ be a locally compact group. (i) The Banach function algebra C :“ C BS E pApΓqq is a closed subalgebra of BpΓq, and so  f BSE “  f BpΓq p f P Cq . Further, C is the smallest closed subalgebra of BpΓq containing ApΓq such that Cr1s is closed in pC b pΓq, τ p q. (ii) The Banach function algebra Bρ pΓq is a closed subalgebra of C, with equality when Γ is discrete. Proof (i) By Theorem 4.3.28(ii), BpΓqr1s is closed in pC b pΓq, τ p q, and so, by Corollary 5.2.12, C is a subalgebra of BpΓq and  f BpΓq ď  f BSE p f P Cq. Now take f P Cr1s and ε > 0, and choose a net p fα q in ApΓq  r1s with  limα fα “ f in pC b pΓq, τ p q. There exists λ P LpApΓqqr1s with  f BSE < x f, λy + ε. Since λ P Cρ∗ pΓd qr1s and Cρ∗ pΓd q “ Bρ pΓd q, we have     x f, λy “ lim x fα , λy ď  f BpΓd q “  f BpΓq α

using Theorem 4.3.28(i), and so  f BSE ď  f BpΓq . It follows that  f BSE “  f BpΓq p f P Cq, and so C is a closed subalgebra of BpΓq. Certainly ApΓq is a closed subalgebra of C. The fact that C is the smallest closed subalgebra of BpΓq containing ApΓq such that Cr1s is closed in pC b pΓq, τ p q follows from Corollary 5.2.12. (ii) The fact that  f BSE “  f BpΓq p f P Bρ pΓqq is exactly equation (4.3.7) of Theorem 4.3.23, noting that LpApΓqq “ Cδ∗ pΓq as in Example 5.1.2, and so Bρ pΓq is a closed subalgebra of C. That C “ Bρ pΓq when Γ is discrete follows from Corollary 4.3.26 because the compact-open topology is the same as τ p when Γ is discrete. Corollary 5.3.19 Let Γ be a locally compact group. Then  f BSE “  f 

p f P ApΓqq ,

and so ApΓq has a BSE norm. Example 5.3.20 It follows that ApΓq Ă Bρ pΓq Ă C BS E pApΓqq Ă BpΓq Ă MpApΓqq for every locally compact group Γ. In the case where Γ is amenable, we have

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ApΓq Ă Bρ pΓq “ C BS E pApΓqq “ BpΓq “ MpApΓqq , so ApΓq is a BSE-algebra.

By Theorem 4.3.28(ii), BpΓqr1s is closed in C b pΓ, τ p q; as a related result, we have the following. Proposition 5.3.21 Let Γ be a locally compact group. Then MpApΓqqr1s is closed in pC b pΓq, τ p q. Proof Set A “ ApΓq. Since A is Tauberian and regular, it follows from Proposition 3.1.10(ii) that Ar2s is dense in A. By Theorem 5.3.18(i),  f BSE “  f  p f P Aq and C BS E pAq Ă BpΓq. Thus C BS E pAq Ă MpAq, and so the result follows from Proposition 5.2.30. The following proposition gives a proof of the equivalence (g) ⇔ (a) in Theorem 4.3.29. Proposition 5.3.22 Let Γ be a locally compact group. Then ApΓq has a contractive pointwise approximate identity if and only if Γ is amenable. Proof Suppose that ApΓq has a CPAI, and take f P ApΓq. Then it follows from Corollary 5.2.17 and Proposition 5.3.21 that  f BSE “  f op . However, by Theorem 5.3.18(i),  f BSE “  f , and so  f op “  f . By Theorem 4.3.29, (j) ⇒ (a), Γ is amenable. Conversely, suppose that Γ is amenable. Then ApΓq has a CAI by Theorem 4.3.29, (a) ⇒ (f), and so certainly ApΓq has a CPAI. The following theorem extends [200, Theorem 5.1]. Theorem 5.3.23 Let Γ be a locally compact group. Then the following are equivalent: (a) Γ is amenable; (b) pBpΓq,  · BpΓq q “ pC BS E pApΓqq,  · BSE,ApΓq q; (c) BpΓqr1s Ă C BS E pApΓqqr1s ; (d) pC BS E pApΓq,  · BSE,MpApΓqq q “ pMpApΓqq,  · op q. In this case, Bρ pΓq “ C BS E pApΓqq “ BpΓq “ MpApΓqq, and so ApΓq is a BSE algebra. Further, when Γ is amenable,  f op “  f BSE,ApΓq “  f BSE,BpΓq “  f BpΓq

p f P BpΓqq .

(5.3.1)

Proof Set A “ ApΓq and B “ BpΓq. (a) ⇒ (b) Since Γ is amenable, B “ Bρ pΓq by Theorem 4.3.29, (a) ⇔ (c), and so (b) follows from Theorem 5.3.18(i).

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(b) ⇒ (c) This is immediate. (c) ⇒ (a) The algebra B is unital, and so 1Γ BSE,ApΓq “ 1 by (c). By Theorem 5.2.16, (a) ⇒ (b), A has a CPAI, and so Γ is amenable by Proposition 5.3.22. (d) ⇒ (a) By Corollary 5.2.21, A has a CPAI, and so again Γ is amenable. (a) ⇒ (d) Since Γ is amenable, A has a CAI by Theorem 4.3.29, (a) ⇒ (f), and so MpAqr1s Ă C BS E pAqr1s by Theorem 5.2.16, (b) ⇒ (c). By Theorem 5.3.18, C BS E pAq is a closed subalgebra of B. But Br1s “ MpAqr1s by Theorem 4.3.29, (a) ⇒ (h), and so (d) follows. Now suppose that the conditions are satisfied. Then  f op “  f  p f P ApΓqq by Theorem 4.3.29, (a) ⇒ (j), and so, by Theorem 5.2.41, equation (5.3.1) holds.  Corollary 5.3.24 Let Γ be an amenable locally group, and let I be a non-zero ideal in ApΓq that is a Banach A-module. Then I is a BSE algebra if and only if I has a bounded pointwise approximate identity. Proof By Theorem 5.3.23, ApΓq is a BSE algebra, and ApΓq has a BAI by Theorem 4.3.29. Thus this follows from Theorem 5.2.45. Corollary 5.3.25 Let Γ be a locally group. Then pBpΓq,  · BpΓq q “ pC BS E pApΓqq,  · BSE,ApΓq q

(5.3.2)

if and only if Γ is amenable. Proof Suppose that Γ is amenable. Then equation (5.3.2) follows by Theorem 5.3.23, (a) ⇒ (b). Suppose that equation (5.3.2) holds. Then 1Γ BSE,ApΓq ď 1, and so ApΓq has a CPAI by Theorem 5.2.16. This implies that Γ is amenable by Proposition 5.3.22.  Let Γ be a locally compact group. Then we have shown in Theorem 5.3.18(ii) that Bρ pΓq is a closed subalgebra of C BS E pApΓqq, with equality when Γ is discrete. We also obtain equality when Γ is amenable because in this case Bρ pΓq “ BpΓq by Theorem 4.3.29, (a) ⇒ (c). We do not have an example of a locally compact group Γ such that Bρ pΓq  C BS E pApΓqq. The following corollary is one formulation of the Bochner–Schoenberg–Eberlein theorem. Corollary 5.3.26 Let G be a locally compact abelian group, and take A to be the group algebra pL1 pGq,  q. Then C BS E pAq “ MpAq “ MpGq, so that L1 pGq is a BSE algebra, and, further, μ “ μBSE,A pμ P MpGqq. In the case where Γ is an amenable locally compact group, equation (5.3.1) shows that  f BSE,BpΓq “  f BpΓq p f P BpΓqq. In fact, this is true for each locally compact group Γ. The following result is close to [97, Lemme (2.13)] of Eymard.

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Proposition 5.3.27 Let Γ be a locally compact group. Suppose that B is a closed subalgebra of BpΓq. Then  f BSE,B “  f B p f P Bq. Proof By Corollary 5.2.27, it suffices to show this for B “ BpΓq. Since Γ is a determining set for B, since B “ C ∗ pΓq is a von Neumann algebra, and since lin {ε x : x P Γ} is a C ∗ -subalgebra of B , this follows from Proposition 5.2.7. The following result is given by Takahasi and Hatori in [297, p. 157]. Theorem 5.3.28 Let Γ be a locally compact abelian group. Then the algebra BpΓq is a BSE algebra if and only if Γ is compact. In the case where Γ is not compact, ApΓq  C BS E pApΓq q and BpΓq  C BS E pBpΓqq. Proof Set A “ ApΓq. Suppose that Γ is compact. Then BpΓq “ A by Theorem 4.3.12(iii), and ApΓq is a BSE algebra by Theorem 5.3.23, (a) ⇒ (d). Now suppose that Γ is not compact. Since BpΓq is an unital algebra, BpΓq “ C BS E pAq “ MpAq “ MpBpΓqq Ă C BS E pBpΓqq . It follows that M 00 pAq  M 0 pAq, as in Example 4.2.7, and so, by Theorem 5.2.36, A and MpAq are not a BSE algebra. Let Γ be a locally compact group that is not compact. Then, as shown in [200, §5], there is surprising diversity: there are amenable groups for which BpΓq (regarded as a Banach function algebra on ΦBpΓq q is and is not a BSE algebra, and there are non-amenable groups for which BpΓq is and is not a BSE algebra. In the cases where BpΓq is not a BSE algebra, it follows that BpΓq  C BS E pBpΓqq. We shall now exhibit another BSE-algebra; a more general form of the result was given by Kamali and Bami in [194]. Proposition 5.3.29 The Banach function algebra pL1 pR+ q,  q is a BSE algebra, and  f 1 “  f BSE p f P L1 pR+ qq. Proof Set A “ L1 pRq and B “ L1 pR+ q, a closed subalgebra of A. Since B has a CAI, it follows from Theorem 5.2.16, (b) ⇒ (c), that MpBqr1s Ă C BS E pBqr1s . Recall from page 256 that B is a natural Banach function algebra on Π. Take G P C BS E pBqr1s , so that G P C b pΠqr1s . Then G P C BS E pAqr1s , and so, by Theorem 5.3.23, there exists μ P MpRq with p μpiyq “ Gpiyq py P Rq. As in Example 5.3.8, it follows from Montel’s theorem that G | Π is analytic, and so G P A b pΠq. By Proposition 4.2.2, (b) ⇒ (a), αpμq ě 0, and so μ P MpR+ q. This implies that C BS E pBqr1s Ă MpR+ qr1s . Finally MpR+ qr1s Ă MpBqr1s , and so the result follows.

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Theorem 5.3.30 Let Γ be an amenable locally compact group, and let I be a weak Segal algebra with respect to ApΓq. Then the following are equivalent: (a) I is a BSE algebra; (b) I has a bounded pointwise approximate identity; (c) C BS E pIq “ BpΓq. Proof By Theorem 4.3.29, ApΓq has a CAI and BpΓq “ MpApΓqq. Further, ApΓq is a BSE algebra with a BSE norm by Theorem 5.3.23. Thus the equivalence of (a), (b), and (c) follows from Theorem 5.2.45. Theorem 5.3.31 Let Γ be an amenable locally compact group, and let S be a Segal algebra with respect to ApΓq. Suppose that S has a contractive pointwise approximate identity. Then | f |Γ ď  f BSE,S “  f op,S “  f ApΓq ď  f S

pf P Sq,

(5.3.3)

and so S has a BSE norm if and only if S “ ApΓq. Proof Let f P S . Since ApΓq has a CAI, it follows from Proposition 3.1.29 and equation (3.1.12) that  f op,S ď  f ApΓq and from Theorem 5.2.16, (b) ⇒ (d), that  f BSE,S ď  f op,S . By Theorem 5.3.23,  f BSE,ApΓq “  f ApΓq . Further, S satisfies the conditions on I in Theorem 5.2.44 (with k “ 1), and so  f BSE,S “  f BSE,ApΓq . Thus equation (5.3.3) follows. We now present two examples of Banach function algebras that are BSE algebras but do not have a BSE norm. Example 5.3.32 Let Γ be an amenable locally compact group that is not discrete, and set S “ LApΓq “ ApΓq ∩ L1 pΓq , the Lebesgue–Fourier algebra on Γ, as in Example 4.3.42, so that S is a Segal algebra with respect to ApΓq and S has a contractive pointwise approximate identity. By Theorem 5.3.30, S is a BSE algebra and C BS E pS q “ BpΓq. In the case where Γ is also not compact, S  ApΓq, as in Example 4.3.42, and so S does not have a BSE norm by Theorem 5.3.31. The case where Γ is discrete will be considered in Example 5.5.14.

Example 5.3.33 Take p with 1 ď p < 8. The Segal algebra S p “ S p pRq was defined in Example 4.2.12. By Theorem 4.2.14, S p has a CPAI, and so it follows from Theorem 5.3.30 that S p is a BSE algebra such that C BS E pS p q is isometrically Banach-algebra isomorphic to pMpRq,  q. The norm on the algebra S p is not equivalent to the norm  · 1 on L1 pRq, but, by equation (5.3.3),  · BSE, S p “  · 1 on S p , and so S p does not have a BSE norm.

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5 BSE norms and BSE algebras

The following theorem is proved by Kaniuth and Ülger in [200, Theorem 5.3]. Theorem 5.3.34 Let Γ be an amenable locally compact group, and let I be a nonzero, closed ideal in ApΓq. Then the following conditions are equivalent: (a) I is a BSE algebra; (b) I has a bounded pointwise approximate identity; (c) I has a bounded approximate identity; (d) I has the form IpEq for some E P Rc pΓq. Proof By Theorem 4.3.29, ApΓq has a CAI, and, by Theorem 5.3.23, ApΓq is a BSE algebra. Thus the equivalence of (a) and (b) follows from Theorem 5.2.45, and the equivalence of (c) and (d) follows from Theorem 4.3.32(iii). Certainly (c) implies (b). (b) ⇒ (d) Let peα q be a BPAI in the ideal I. Since I is a closed linear subspace of BpΓq and of BpΓd q, we may suppose that peα q converges in the weak-∗ topology σpBpΓd q, C ∗ pΓd qq, say to f P BpΓd q. It follows that limα eα “ f pointwise on Γ, and hence f pγq “ 0 pγ P Eq and f pγq “ 1 pγ P Γ \ Eq, where E “ hpIq. Thus the idempotent χE “ 1Γ ´ f belongs to BpΓd q. By Theorem 4.3.32(i), E P RpΓq. Since E is closed in Γ, it follows that E P Rc pΓq. By Theorem 4.3.32(ii), I “ IpEq, and so (d) follows. Recall that B 0 pΓq “ BpΓq ∩ C 0 pΓq is the Rajchman algebra of a locally compact group Γ; see Definition 4.3.43. By Theorem 5.3.27, B 0 pΓq has a BSE norm. We now consider when B 0 pΓq is a BSE algebra, following Kaniuth, Lau, and Ülger [199, Theorem 3.7]. Lemma 5.3.35 Let Γ be a locally compact group such that B 0 pΓq has a bounded pointwise approximate identity. Then MpB 0 pΓqq “ C BS E pB 0 pΓqq “ BpΓq . Proof Since B 0 pΓq is a closed ideal in BpΓq, certainly BpΓq Ă MpB 0 pΓqq. By Theorem 5.2.16, (b) ⇒ (c), MpB 0 pΓqq Ă C BS E pB 0 pΓqq. Take a function f P C BS E pB 0 pΓqq, so that there is a bounded net p fα q in B 0 pΓq that converges pointwise on ΦB 0 pΓq to f . Thus p fα | Γq is a bounded net in BpΓq that converges pointwise to f | Γ in C b pΓq. By Theorem 4.3.28(ii), f | Γ P BpΓq. Thus C BS E pB 0 pΓqq Ă BpΓq. The result follows. Theorem 5.3.36 Let Γ be a locally compact group. Then the following are equivalent: (a) B 0 pΓq is a BSE algebra;

5.3 Examples

339

(b) B 0 pΓq has a bounded pointwise approximate identity; (c) Γ is amenable and B 0 pΓq is natural on Γ. Proof Set Φ “ ΦB 0 pΓq “ Γ ∪ H, as in equation (4.3.9). (a) ⇒ (b) By Theorem 5.2.16, (c) ⇒ (b), B 0 pΓq has a BPAI, giving (b). (b) ⇒ (c) By Proposition 3.5.8, Φ is weakly closed in B 0 pΓq . Assume that pxα q is a net in Γ such that pε xα q is a weakly null net in ApΓq . As explained after Definition 4.3.43, ApΓq is a complemented subspace of B 0 pΓq , and so pε xα q is a weakly null net in B 0 pΓq , and hence 0 P Φ, a contradiction. So Γ is weakly closed in ApΓq . By Proposition 4.4.10, Γ is amenable We now show that Φ “ Γ. Assume to the contrary that H  ∅. Since Γ is amenable, there is a BAI peα q in ApΓq, and we know that B 0 pΓq has a BPAI, say puα q. Thus peα ´ uα q is a bounded net in B 0 pΓq, and it converges pointwise on Φ to χH P C b pΦq. So χH P C BS E pB 0 pΓqq. By Lemma 5.3.35, χH P BpΓq. Since Γ is a determining set for BpΓq, we have χH “ 0, a contradiction. Thus (c) follows. (b) ⇒ (a) This follows from Lemma 5.3.35. (c) ⇒ (b) Since Γ is amenable, the algebra ApΓq has a BAI, and this is a BPAI for B 0 pΓq because Φ “ Γ. We now consider the Figà-Talamanca–Herz algebras A p pΓq and the related algebras B p pΓq that were described in §4.4. Recall that C ∗ -algebra techniques are not available for general Figà-Talamanca–Herz algebras; in particular, we do not have a version of Kaplansky’s density theorem. Thus our results are not as complete as they were for the Fourier algebra ApΓq. For example, it seems that no analogue to Theorem 5.3.34 is known for these algebras. Let Γ be an amenable locally compact group, and take p with 1 < p < 8. We recall from Theorem 4.4.15 that A p pΓq is a closed ideal in B p pΓq “ MpA p pΓqq and that B p pΓq is a dual Banach function algebra with Banach-algebra predual PF p pΓq, and from Theorem 4.4.11, (a) ⇒ (d), that A p pΓq has a contractive approximate identity. Proposition 5.3.37 Let Γ be a locally compact group such that Γd is amenable, and take p with 1 < p < 8. Then  f BSE,A “  f 

p f P Aq ,

where A “ B p pΓq or A “ A p pΓq, and so both B p pΓq and A p pΓq have BSE norms. Proof This follows from Theorem 5.2.24 and Corollaries 5.2.27 and 4.4.16. We do not know whether A p pΓq has a BSE norm for all locally compact groups Γ; this is true for p “ 2 by Corollary 5.3.19 and by Proposition 5.3.37 when Γd is amenable. We now consider when A p pΓq is a BSE algebra.

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5 BSE norms and BSE algebras

Theorem 5.3.38 Let Γ be a locally compact group, and take p with 1 < p < 8. (i) Suppose that A p pΓq is a BSE algebra. Then Γ is amenable. (ii) Suppose that Γd is amenable. Then A p pΓq is a BSE algebra, with pC BS E pA p pΓqq,  · BSE q “ pMpA p pΓqq,  · op q “ pB p pΓq,  · q .

(5.3.4)

Proof (i) By Corollary 5.2.18, A p pΓq has a BPAI when it is a BSE algebra, and so, by Theorem 4.4.11, (b) ⇒ (a), Γ is amenable in this case. (ii) Since Γ is amenable, B p pΓq “ MpA p pΓqq isometrically by Theorem 4.4.15(i). By Corollary 4.4.16, B p pΓqr1s is closed in pC b pΓq, τ p q, and so MpA p pΓqqr1s is also closed in pC b pΓq, τ p q. By Theorem 4.4.11, A p pΓq has a CAI. It now follows from Corollary 5.2.21 that C BS E pA p pΓqq “ MpA p pΓqq isometrically, giving equation (5.3.4), and so A p pΓq is a BSE algebra. We do not know whether A p pΓq is a BSE algebra when Γ is just amenable; this is true for p “ 2 by Theorem 5.3.23, (a) ⇒ (d). Example 5.3.39 Let Γ be an amenable locally compact group that is not discrete, take p with 1 < p < 8 and r with 1 ď r < 8, and set S “ Arp pΓq “ A p pΓq ∩ Lr pΓq , as in Example 4.4.13, so that S is a Segal algebra with respect to A p pΓq and S has a contractive pointwise approximate identity. By Theorem 4.4.11, A p pΓq has a CAI, and, in the case where Γd is also amenable, A p pΓq is a BSE algebra with a BSE norm by Theorem 5.3.38(ii), and so, by Theorem 5.2.45, S is a BSE algebra. In the case where Γ is also not compact, S  A p pΓq, essentially as in Example 4.3.42, and so S does not have a BSE norm, essentially as in Theorem 5.3.31.

p B of We conclude this section by considering the projective tensor product A ⊗ two Banach function algebras A and B. We do not know an example where both A p B is a Banach function algebra, but where A ⊗ pB and B have a BSE norm, where A ⊗ does not have a BSE norm. p CpLq Our first results will show that every Varopoulos algebra VpK, Lq “ CpKq ⊗ has a BSE norm when K and L are non-empty, compact spaces, but that VpK, Lq is usually not a BSE algebra; the algebra VpK, Lq was defined in §3.3. Recall that it was shown in §3.3 that VpK, Lq is a natural, self-adjoint Banach function algebra on K × L, and that VpK, Lq  CpK × Lq whenever K and L are both infinite. Theorem 5.3.40 Let K and L be non-empty, compact spaces. Then the Varopoulos algebra VpK, Lq is such that FBSE “ Fπ

pF P VpK, Lqq .

(5.3.5)

5.3 Examples

341

Proof Set V “ VpK, Lq, so that V  “ WpCpKq, MpLqq by Corollary 2.2.17. Suppose that λ “ εpx,yq , where x P K and y P L. Then τλ is defined to be the map τλ : f Þ→ f pxqδy ,

CpKq Ñ MpLq .

The space LpVq is identified with the linear span of these operators. For the result, we shall show that LpVqr1s is weak-∗ dense in the closed unit ball BpCpKq, MpLqqr1s . Choose T P BpCpKq, MpLqqr1s , ε > 0, k P N, and functions f1 , . . . , fk P CpKqr1s and g1 , . . . , gk P CpLqr1s . As explained on page 65, it is sufficient to find S P LpVqr1s such that   xg, pT ´ S q f y < 2ε (5.3.6) whenever f P { f1 , . . . , fk } and g P {g1 , . . . , gk }. By Theorem 3.1.16, there are open covers U and V of K and L, respectively, such that UU P BpCpKqqr1s and UV P BpCpLqqr1s , such that |UU f ´ f |K < ε for each f P { f1 , . . . , fk } and |UV g ´ g|K < ε for each g P {g1 , . . . , gk }, and where UU and UV have the forms UU : f Þ→

m 

f pxi qui

and

UV : g Þ→

i“1

n 

gpy j qv j ,

j“1

respectively, where u1 , . . . , um P CpK, Iq with u1 + · · · + um “ 1K and where v1 , . . . , vn P CpL, Iq with v1 + · · · + vn “ 1L . Define  ◦ T ◦ UU , S “ UV so that S P BpCpKq, MpLqqr1s . We claim that S P LpVq. For this, take r P CpKq and s P CpLq. Then xr ⊗ s, S y “ xs, S ry “ xUV s, pT ◦ UU qprqy m  m  n n   “ ε xi prqεy j psqxv j , T ui y “ αi, j xr ⊗ s, εpxi ,y j q y , i“1 j“1

i“1 j“1

where αi, j “ xv j , T ui y pi P Nm , j P Nn q, and so S “

m  n 

αi, j εpxi ,y j q P LpVq ,

i“1 j“1

as claimed. Since S  ď 1, it follows that S P LpVqr1s , as required. We next claim that equation (5.3.6) holds. Indeed, take f P { f1 , . . . , fk } and g P {g1 , . . . , gk }. Then     xg, pT ´ S q f y “ xg, T f y ´ xUV g, pT ◦ UU qp f qy     ď xg ´ U g, T f y + xU g, T p f ´ U f qy V

V

ď |g ´ UV g|L + | f ´ UU f |K < 2ε .

U

342

5 BSE norms and BSE algebras

This establishes the claim. Equation (5.3.5) follows. Let A and B be natural uniform algebras on compact spaces K and L, respectively. p B always has a BSE norm. The first caution is that, as we It is natural to ask if A ⊗ p B is not necessarily a Banach function algebra; we shall remarked on page 181, A ⊗ p B is indeed a Banach function algebra. It would consider only the cases where A ⊗ p B has a BSE norm if A ⊗ p B were follow immediately from Corollary 5.2.27 that A ⊗ a closed subalgebra of VpK, Lq. However this is not easily seen: it is not immediate because a proper uniform algebra A on a compact space K is never complemented in CpKq [325, Corollary III.I.3]. The result is true in the special case where A and B are the disc algebra, as is shown in [325, Theorem III.I.19]. p A is a closed subTheorem 5.3.41 Let A :“ ApDq be the disc algebra. Then A ⊗ p algebra of VpTq, and so A ⊗ A has a BSE norm. p A has a BSE norm for each uniform algebra A. We do not know whether A ⊗ Next, we shall prove that the Varopoulos algebra VpK, Lq is not a BSE algebra whenever K and L are infinite, compact, metrizable spaces. This follows from results of Varopoulos in [319] and Graham in [143], where further, related results are given. Theorem 5.3.42 Let K and L be infinite, compact, metrizable spaces, and set V “ VpK, Lq. Then there exists H P C BS E pVq such that H  V, but H 2 P V. In particular, VpK, Lq is not a BSE algebra. Proof The metrics on K and L are denoted by dK and dL , respectively. Let pn j q be the sequence in N that was defined above Proposition 3.3.16. Since K is infinite and compact, it contains a non-isolated point, say x0 . Induc  tively, define a null sequence pη j q in R+• such that U j  ě n j2 , where U j “ {x P K : η j+1 < dK px, x0 q < η j } , so that the sets U j are open and pairwise-disjoint. Similarly, take a non-isolated point y0 P L, and form an analogous sequence pV j q in L. As in Lemma 3.3.10, there is a constant C > 0 such that, foreach j P N, there G  G  ě C j2 and “ 1 and is an upper-triangular function G j P V with j j   K×L  π G j π p j P Nq, so that F j π “ 1 and with supp G Ă U × V . Set F “ G / j j j j j   F j K×L ď 1/C j2 for j P N.  For each n P N, set Hn “ nj“1 F j ; by Proposition 3.3.15, Hn π “ 1, and so pHn q is a bounded sequence in V. Further, the sequence pHn q converges uniformly on K × L, say to H P CpK × Lq. Clearly H P C BS E pVq. Note that each function G2j is an upper-triangular function on an upper-triangular     set of size n j2 , and so, by Proposition 3.3.16, G2j  ď j2 , whence F 2j  ď 1/C 2 j2 . π π  2 2 It follows that the series 8 j“1 F j is convergent in pV,  · π q, and the sum is H , so 2 that H P V.

5.4 Pointwise contractive Banach function algebras and 1 -norms

343

We claim that H  V. Indeed, assume towards a contradiction that H P V. Since Hpx0 , y0 q “ 0, it follows from Theorem 3.3.13 that there is a neigbourhood W of px0 , y0 q and F P Jpx0 ,y0 q such that F | W “ 0 and H ´ HFπ < 1. There exists j P N such  that  U j × V j Ă W. Set R “ H · χU j ×V j . Then Rπ < 1. But R is equal to  F j , and F j π “ 1, the required contradiction. Since V “ MpVq, the algebra V is not a BSE algebra. The result follows. Set V “ VpK, Lq, as above. It is easy to see from the estimates in §3.3 that C BS E pVq  CpK × Lq, and so V  C BS E pVq  CpK × Lq. However, we do not have a characterization of the functions in C BS E pVq.

5.4 Pointwise contractive Banach function algebras and  1 -norms Let A be a Banach function algebra. In this section, we shall first introduce the notion of an 1 -norm on the space LpAq, and consider when this norm is equivalent to the given norm on LpAq; we shall show that this is equivalent to the fact that C BS E pAq is equal to C b pΦA q, and implies that the BSE norm on A is equivalent to the uniform norm on ΦA . This will lead to Examples 5.4.13 and 5.4.16 that exhibit further unital Banach function algebras that do not have a BSE norm. Whether A has the above property is closely related to whether A is pointwise contractive. The main result of this section is a classification in Theorem 5.4.12 of contractive and pointwise contractive Banach function algebras among those unital Banach function algebras that have a BSE norm. Most of the results in this section are taken from the memoir [64]. 1 Let A be a Banach function algebra. As  before, we denote by pΦA q the set of   functions f : ΦA Ñ C such that { f pϕq : ϕ P ΦA } < 8; we define   { f pϕq : ϕ P ΦA } p f P 1 pΦA qq ,  f 1 “ so that p 1 pΦA q,  · 1 q is a natural Banach sequence algebra on the set ΦA . There is a natural contraction  ι : f Þ→ { f pϕqϕ : ϕ P ΦA } , p 1 pΦA q,  · 1 q Ñ pLpAq,  · q Ă pA ,  · q . Clearly ιp 1 pΦA qq contains LpAq, and so ιp 1 pΦA qq is a dense linear subspace of LpAq. However, the map ι is not always an injection: it may be that there exist a  sequence pϕn q in ΦA and an element α “ pαn q P 1 such that 8 n“1 αn ϕn “ 0 (with convergence of the sum in A ), but with α  0. For example, this occurs in the case where A is the disc algebra ApDq; further, there are examples of uniform algebras with Γ0 pAq, the Choquet boundary of A, dense in ΦA having this property. For a discussion of this point, see [104]. Fortunately, the following lemma shows that this difficulty does not arise in the cases of interest to us.

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5 BSE norms and BSE algebras

Lemma 5.4.1 Let A be a Banach function algebra such that Mϕ has a bounded pointwise approximate identity for each ϕ P ΦA ∪ {8}. Then the above map ι : 1 pΦA q Ñ A is an injection.   Proof Take a subset {αϕ : ϕ P ΦA } of C such that {αϕ  : ϕ P ΦA } < 8 and  {αϕ ϕ : ϕ P ΦA } “ 0, and assume towards a contradiction that there exists ϕ0 P ΦA with αϕ0  0. Since both A and Mϕ0 have BPAIs, it follows from Proposition 3.5.2 that there exists M P A with xM, ϕy “ δϕ,ϕ0 pϕ P ΦA q. Then C G   0 “ M, αϕ ϕ “ αϕ xM, ϕy “ αϕ0 , ϕPΦA

ϕPΦA

a contradiction. Thus αϕ “ 0 pϕ P ΦA q, and so ι is an injection. It follows that, in the case where A satisfies the conditions of the above lemma, we can regard 1 pΦA q as a linear subspace of LpAq. For λ “ ιp f q, where f P 1 pΦA q,  we set λ1 “  f 1 ; in particular, for an element λ “ ni“1 αi ϕi P LpAq, we have n λ1 “ i“1 |αi |, so that λ ď λ1 . Definition 5.4.2 Let A be a Banach function algebra. Then the norm on LpAq is equivalent to the 1 -norm if there is a constant C > 0 such that λ1 ď C λ

pλ P LpAqq .

In the case where the map ι : 1 pΦA q Ñ A is an injection, the norm on LpAq is equivalent to the 1 -norm if and only if the map ι : p 1 pKq,  · 1 q Ñ pLpAq,  · q is a linear homeomorphism. Theorem 5.4.3 Let A be a Banach function algebra. Then the following are equivalent: (a) the norm on LpAq is equivalent to the 1 -norm; (b) C BS E pAq “ C b pΦA q; (c) there is a constant C > 0 such that, for each ε > 0, for each n P N, for distinct points   ϕ1 , . . . , ϕn in ΦA , and for ζ1 , . . . , ζn P D, there exists f P ArCs with  f pϕi q ´ ζi  < ε pi P Nn q. Proof (a) ⇒ (b) Since LpAq ∼ 1 pΦA q, it follows that QpAq “ 8 pΦA q, and so C BS E pAq “ C b pΦA q, giving (b). (b) ⇒ (a), (c) There is a constant C > 0 with  f BSE ď C | f |ΦA p f P C b pΦA qq. Take λ P LpAq. Then there exists f P C 0 pΦA qr1s such that λ1 “ x f, λy. By (b), f P C BS E pAqrCs . It follows that λ1 ď  f BSE λ ď C λ, giving (a). Take ϕ1 , . . . , ϕn in ΦA and ζ1 , . . . , ζn P D as specified in (c). Then there exists a function g P C b pΦA qr1s with gpϕi q “ ζi pi P Nn q, and g P C BS E pAqrCs , giving (c).  (c) ⇒ (a) Take λ  “ ni“1 αi ϕi P LpAq. By (c), for each ε > 0, there exists  f P C BS E pAqrCs with  |αi | ´ αi f pϕi q < ε pi P Nn q, and so

5.4 Pointwise contractive Banach function algebras and 1 -norms

345

  n   λ1 ď  αi f pϕi q + nε ď C λ + nε .  i“1  Thus λ1 ď C λ, giving (a). Corollary 5.4.4 Let A be a Banach function algebra for which the norm on LpAq is equivalent to the 1 -norm, and set B “ C BS E pAq. Then ΦA is dense in ΦB . Proof Here B “ C b pΦA q, and so ΦB “ β ΦA , giving the result. Theorem 5.4.5 Let A be a Banach function algebra. Then the following are equivalent: (a) λ “ λ1 pλ P LpAqq; (b) pC BS E pAq,  · BSE q “ pC b pΦA q, | · |ΦA q; (c) A is strongly pointwise contractive. Proof This is the same as the proof of Theorem 5.4.3, with C “ 1. Corollary 5.4.6 Let A be a natural Banach function algebra on a non-empty, compact space K. Then Ar1s is dense in pCpKqr1s , τ p q if and only if λ “ λ1

pλ P LpAqq .

Proof Suppose that λ “ λ1 pλ P LpAqq. By the theorem, (a) ⇒ (b), we have pC BS E pAq,  · BSE q “ pCpKq, | · |ΦA q, and so Ar1s is dense in pCpKqr1s , τ p q. Suppose that Ar1s is dense in pCpKqr1s , τ p q, and take λ P LpAq. Then     λ1 “ sup{x f, λy : f P CpKqr1s } “ sup{x f, λy : f P Ar1s } “ λ , as required. Corollary 5.4.7 Let A be a natural uniform algebra on a non-empty, compact space K. Then pC BS E pAq,  · BSE q “ pCpKq, | · |K q if and only if each singleton in K is a one-point Gleason part if and only if Ar1s is dense in pCpKqr1s , τ p q. Proof By Theorem 3.6.32, A is strongly pointwise contractive if and only if each singleton in K is a one-point Gleason part, so that the equivalences follow from the above results. Proposition 5.4.8 Let A be a natural Banach sequence algebra on a non-empty set S . Then the following are equivalent: (a) the norm on LpAq is equivalent to the 1 -norm; (b) sup{χT BSE : T Ă S , T finite, T  ∅} is finite.

346

5 BSE norms and BSE algebras

Proof (a) ⇒ (b) Take C > 0 such that λ1 ď C λ pλ P LpAqq. For a non-empty, finite subset T of S and λ P LpAqr1s , we have       xχT , λy “  λpsq ď λ1 ď C ,   sPT and so χT BSE ď C, giving (b). (b) ⇒ (a) Take m > 0 such that sup{χT BSE : T Ă S , T finite, T  ∅} ď m,  and let λ “ ni“1 αi ε si P LpAqr1s . Then, for each subset N of Nn , we have      {αi : i P N} “ xχT , λy ď m , where T “ {si : i P N}. By Proposition 1.1.1, 4m λ, giving (a).

n i“1

|αi | ď 4m, and hence λ1 ď

Let A be a Banach function algebra. We now compare the two properties that A is pointwise contractive and that the norm on LpAq is equivalent to the 1 -norm. Theorem 5.4.9 Let A be a pointwise contractive Banach function algebra. Then  f BSE ď 4 | f |ΦA

p f P C b pΦA qq ,

so that C BS E pAq “ C b pΦA q, the norm on LpAq is equivalent to the 1 -norm, and the uniform norm | · |ΦA and the BSE norm  · BSE are equivalent on A. In the case where A is strongly pointwise contractive, pC BS E pAq,  · BSE q “ pC b pΦA q, | · |ΦA q . Proof Let f P C b pΦA qr1s . Take a finite subset T  of ΦA and ε > 0. By Proposition  3.5.10(ii), there exists g P Ar4s with gpϕq ´ f pϕq < ε pϕ P T q. By Theorem 5.2.9, f P C BS E pAq with  f BSE ď 4, and so C BS E pAq “ C b pΦA q. The result now follows from Theorem 5.4.3. In the case where A is strongly pointwise contractive, we may suppose that g P Ar1s , and so  f BSE “ | f |ΦA p f P C BS E pAqq, giving the required conclusion. Corollary 5.4.10 Let A be a pointwise contractive, natural Banach function algebra on a non-empty, compact space K. Then C BS E pAq “ CpKq, and A has a BSE norm if and only if A is equivalent to a uniform algebra. Further, the following conditions on A are equivalent: (a) A is a BSE algebra; (b) A “ CpKq; (c) Ar1s is closed in pCpKq, τ p q.

5.4 Pointwise contractive Banach function algebras and 1 -norms

347

Proof By Theorem 5.4.9, C BS E pAq “ CpKq, and the BSE norm on A is equivalent to the uniform norm, so that A has a BSE norm if and only if A is equivalent to a uniform algebra. Since A is unital, MpAq “ A. (a) ⇒ (b) ⇒ (c) These are immediate. (c) ⇔ (a) This is Corollary 5.2.11. Theorem 5.4.11 Let A be a Banach function algebra with |ΦA | ě 2. Then the following conditions on A are equivalent: (a) pLpAq,  · q “ p 1 pΦA q,  · 1 q; (b) A is pointwise contractive and  f BSE “ | f |ΦA p f P Aq; (c) A has the weak separating ball property and  f BSE “ | f |ΦA p f P Aq; (d) for each ϕ P ΦA , the maximal modular ideal Mϕ has norm-one characters and  f BSE “ | f |ΦA p f P Aq. Proof Take B to be the uniform closure of A in C 0 pΦA q, so that B is a natural uniform algebra on ΦA . Then the embedding of A into B is a continuous homomorphism with dense range. (a) ⇒ (b) This is part of Theorem 5.4.5. (b) ⇒ (c) This is immediate. (c) ⇒ (a) Since A has the WSBP, B also has the WSBP, and so, by Corollary 3.6.30, B  has the WSBP. By Theorem 3.6.32, B  is strongly pointwise contractive, and this implies that pLpBq,  · q “ p 1 pΦA q,  · 1 q. For each f P A, we have  f BSE,A “ | f |ΦA by hypothesis, and  f BSE,B “ | f |ΦA , and so  f BSE,A “  f BSE,B . It follows from Theorem 5.1.31 that LpBq “ LpAq, and so pLpAq,  · q “ p 1 pΦA q,  · 1 q, giving (a) (c) ⇔ (d) This follows from Proposition 3.4.3. We now present an important theorem: it classifies contractive and pointwise contractive unital Banach function algebras among those which have a BSE norm. Theorem 5.4.12 Let A be an unital Banach function algebra with a BSE norm. (i) Suppose that A is contractive. Then A is equivalent to a Cole algebra. (ii) Suppose that A is pointwise contractive. Then A is equivalent to a uniform algebra for which each singleton in ΦA is a one-point part. Proof By the hypotheses and Theorem 5.4.9, the norm on A is equivalent to the uniform norm | · |ΦA in both cases. (i) Since A is contractive, pA, | · |ΦA q is also contractive, and so it is a Cole algebra by Theorem 3.6.20. (ii) Since A is pointwise contractive, pA, | · |ΦA q is also pointwise contractive, and so each singleton in ΦA is a one-point part by Theorem 3.6.32.

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We shall now present further examples of Banach function algebras that do not have a BSE norm. The two algebras are strongly pointwise contractive, and the second is contractive, but neither is equivalent to a uniform algebra, and so the condition that A have a BSE norm cannot be removed from either of clauses (i) and (ii) of the above theorem. Example 5.4.13 Again, let A be the unital Banach function algebra of Example 3.5.13, so that A is strongly pointwise contractive. By Theorem 5.4.9, we have C BS E pAq “ CpIq and  f BSE,A “ | f |I p f P Aq. Since the norm of A is not equivalent to the uniform norm on I, it follows that A does not have a BSE norm. Further, MpAq “ A, and so A is not a BSE algebra. Here A is a proper, dense, unital subalgebra of C BS E pAq, and so A is not an ideal in C BS E pAq. In summary, it follows that | f |I “  f BSE,A “  f QpAq ď  f op “  f 

p f P Aq .

Let M “ { f P A : f p0q “ 0} be the maximal ideal in A of Example 3.5.13, and set K “ p0, 1s, so that M is a Segal algebra with respect to C 0 pKq. By Theorem 5.4.9, C BS E pMq “ C b pKq, and  f QpMq “ | f |K , as in Example 5.1.36, and hence  f BSE,M “ | f |K p f P Mq. This implies that M does not have a BSE norm. Since MpMq “ C b pKq, the algebra M is a BSE algebra.

We can use the above example to see that there are a locally compact space K and natural Banach function algebras A and B on K such that B is a dense subalgebra of A, such that B is a BSE algebra with a BSE norm, but such that A is neither a BSE algebra nor has a BSE norm. Example 5.4.14 Let A be the natural Banach function algebra on I of the above example, so that A is not a BSE algebra and does not have a BSE norm. Now take α with 0 < α < 1, and consider the natural Banach function algebra B “ Lipα I of Lipschitz functions on I, as in Example 5.3.3. It is clear that B is a dense subalgebra of A (with  f A ď p1/αq  f B p f P Bq); as in Example 5.3.3, B is a BSE algebra with a BSE norm.

The following example of Inoue and Takahasi [181] is essentially the same as Example 5.4.13. Example 5.4.15 Take p with 1 ď p < 8, and set I p “ C 0 pRq ∩ L p pRq, taken with the norm  f I p “ | f | R +  f  p p f P I p q . We have f g P I p with  f gI p ď  f I p |g| R when f P I p and g P C 0 pRq, and so I p is a Segal algebra with respect to C 0 pRq, whence I p is a natural Banach function algebra on R.

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It is clear that I p is pointwise contractive, that C BS E pI p q “ MpI p q “ C b pRq, and that  f BSE “ | f |R p f P C BS E pI p qq, and so I p is also a BSE algebra that does not have a BSE norm.

Example 5.4.16 Let A be the natural Banach function algebra on T constructed in Example 3.5.14, so that A is contractive and strongly pointwise contractive. By Theorem 5.4.9, C BS E pAq “ CpTq and  f BSE “ | f |T p f P Aq, and so A does not have a BSE norm. Since MpAq “ A, it follows that A is not a BSE algebra.

In Theorem 5.4.12(i), we showed that a contractive Banach function algebra with a BSE norm is equivalent to a Cole algebra. We see by combining Theorems 3.3.13 and 5.3.40 that the condition that the algebra be contractive cannot be weakened to ‘all maximal ideals in the algebra have a bounded approximate identity of bound 3/2’.

5.5 Banach function algebras that are ideals in their bidual In this section, we shall discuss Banach function algebras A that are ideals in their bidual. For example, we shall show in Corollary 5.5.10 that such a Banach function algebra A is a BSE algebra if and only if it has a bounded approximate identity, and in Theorem 5.5.17 we shall describe A , at least when A has a multiplier-bounded approximate identity. In particular, we shall concentrate on the case where A is a Tauberian Banach sequence algebra, showing in Examples 5.5.15, 5.5.20, and 5.5.21 when certain specific Banach sequence algebras are BSE algebras and when they have a BSE norm. Let A be a Banach function algebra that is an ideal in its bidual. We recall from Proposition 5.2.38 that C BS E pAq “ QpAq “ A /LpAqK , and hence C BS E pAq is an isometric dual Banach function algebra, with Banach-algebra predual LpAq. Proposition 5.5.1 Let pA,  · q be a Banach function algebra that is an ideal in its bidual. Then AA “ A A “ LpAq, A + LpAqK is an ideal in A , A is an ideal in C BS E pAq, and p f + Mq l pg + Nq “ f g p f, g P A, M, N P LpAqK q ,

(5.5.1)

so that LpAqK Ă rad A . Suppose, further, that A “ A A. Then pC BS E pAq,  · BSE q “ pA ,  · q. Proof By (5.1.2), LpAq Ă AA . By Lemma 3.4.7, A · A Ă LpAq because A is an ideal in A , and hence AA Ă LpAq. Thus AA “ LpAq.

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Since A and LpAqK are ideals in A , so is A + LpAqK , and then A is an ideal in C BS E pAq. By Theorem 2.3.60 (with F “ A A “ LpAq), we have A l LpAqK “ {0}, and so equation (5.5.1) holds. Suppose, further, that A “ A A. Then LpAqK “ {0}, and so it is immediate that pC BS E pAq,  · BSE q “ pA ,  · q. Corollary 5.5.2 Let A be a dual Banach function algebra with Banach-algebra predual F, and suppose that A is an ideal in its bidual. Then A is Arens regular, F “ LpAq “ A A, and A has a unique Banach-algebra predual. Further, pA,  · q “ pC BS E pAq,  · BSE q and A “ A  LpAqK . Proof By Theorem 2.4.17, A is Arens regular and ΦA Ă F. By Proposition 5.1.33(ii), LpAq “ F and A “ A ‘ F K . By Proposition 5.5.1, A A “ F. By Theorem 5.2.37, pA,  · q “ pC BS E pAq,  · BSE q. Example 5.5.3 Consider the Banach sequence algebra A “ 1 , so that A is a dual Banach algebra with Banach-algebra predual LpAq “ c 0 and an ideal in its bidual, and so the above corollary applies to this algebra. In particular, A is Arens regular, as in Example 3.2.7(ii), A has a BSE norm, as in Example 5.3.1(ii), and the unique Banach-algebra predual is c 0 .

Corollary 5.5.4 Let A be a Banach function algebra that is an ideal in its bidual. Then A “ C BS E pAq if and only if A is a dual Banach function algebra. Proof Suppose that A “ C BS E pAq. Then A is a dual Banach function algebra by Proposition 5.2.38. Now suppose that A is a dual Banach function algebra. By Corollary 5.5.2, A “ A  LpAqK , showing that A “ A /LpAqK “ C BS E pAq. Corollary 5.5.5 Let A be a Tauberian Banach sequence algebra with a multiplierbounded approximate identity. Then A and A have BSE norms. Proof The Banach sequence algebra A is an ideal in its bidual, and so, by Proposition 5.5.1, AA “ LpAq. Thus this follows from Proposition 5.2.6. Example 5.5.6 Let pω n q be a sequence in r1, 8q such that limnÑ8 ω n “ 8, and set   A0 “ A0 pωn q “ {α “ pαn q : pω n αn q P c 0 } and αω “ pωn αn qN pα P A0 q .

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Then A0 is a Tauberian Banach sequence algebra on N and pΔn q is an MBAI for A0 with multiplier-bound 1, as in Example 3.2.8, and so αω “ αBSE, ω pα P A0 q by Corollary 5.5.5. Here MpA0 q “ 8 ,  · op “ | · |N , and   C BS E pA0 q “ A0 “ β P 8 : sup |βn | ωn < 8  8 , nPN

and so A0 is not a BSE algebra.

Corollary 5.5.7 Let A be a Banach function algebra that is an ideal in its bidual, and suppose that A has a bounded approximate identity and that A is Arens regular. Then LpAq “ A ,  f BSE “  f  p f P Aq, A is a BSE algebra, and A is a Banach function algebra with a BSE norm. Proof By Corollary 2.3.73, A “ A · A, and so LpAq “ A by Proposition 5.5.1. Hence LpAqK “ {0} and pC BS E pAq,  · BSE q “ pQpAq,  · QpAq q “ pA ,  · q . It follows that  f BSE “  f  p f P Aq and that A is a Banach function algebra on ΦA . By Proposition 2.3.74, MpAq “ A , and so A is a BSE algebra. By Theorem 5.2.15, C BS E pAq has a BSE norm, and hence A has a BSE norm. p q , as in Example 5.5.8 Take p and q such that 1 < p, q < 8, and set A “ p ⊗  q q ,  · ε q – Kp p , q q, so that A is a Tauberian Example 3.3.6(i). Set F “ p p ⊗ Banach sequence algebra that is a dual Banach algebra with Banach-algebra predual F.  Take pm, nq P N × N, so that the element T pm,nq of Bp p , q q corresponding to the character εpm,nq is the map given by T pm,nq p f qpgq “ f pmqgpnq “ εpm,nq p f ⊗ gq p f P p , g P q q . This shows that T pm,nq has one-dimensional range, and so is compact. We conclude that ΦA Ă F, and so, by Proposition 5.1.33(ii), F “ LpAq. By Corollary 5.5.2, F “ A A and A “ A ‘ LpAqK , with LpAqK “ {0} if and only if pq > p + q, as in Example 3.3.6. Here C BS E pAq “ A “ F  in each case.

Theorem 5.5.9 Let A be a Banach function algebra that is an ideal in its bidual. Then C BS E pAq Ă MpAq and | f |ΦA ď  f op ď  f BSE

p f P C BS E pAqq .

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5 BSE norms and BSE algebras

Proof Take f P C BS E pAq. Then there exists M P A with M | ΦA “ f . For each g P A, we have M · g P A because A is an ideal in A , and so f g P A. Thus f P MpAq, showing that C BS E pAq Ă MpAq. Certainly | f |ΦA ď  f op , and so it suffices to show that  f op ď  f BSE . Take g P Ar1s . Then there exists λ P Ar1s with  f g “ x f g, λy. By Proposition 5.5.1, g · λ P LpAq, and so g · λ ď g λ ď 1. Hence we have   f g “ x f, g · λy ď  f BSE . This implies that  f op ď  f BSE , as required. The following result extends [200, Theorem 3.1]. Corollary 5.5.10 Let A be a Banach function algebra that is an ideal in its bidual. Then the following are equivalent: (a) A is a BSE algebra; (b) A has a bounded pointwise approximate identity; (c) A has a bounded approximate identity. In this case, A has a BSE norm. Proof (a) ⇒ (b) This is Corollary 5.2.18. (b) ⇒ (c) This is Proposition 3.5.11. (c) ⇒ (a) By Theorem 5.5.9, C BS E pAq Ă MpAq. By Theorem 5.2.16, (b) ⇒ (c), MpAq Ă C BS E pAq. Hence C BS E pAq “ MpAq, and so A is a BSE algebra. Suppose that the equivalent conditions (a)–(c) hold. Then A has a BSE norm by Corollary 5.2.40. Corollary 5.5.11 Let A be a Banach function algebra with approximate units such that A is an ideal in its bidual, and let S be a Segal algebra with respect to A. Suppose that S is a BSE algebra. Then S “ A. Proof By Proposition 3.1.34, S is an ideal in S  , and so, by Corollary 5.5.10, (a) ⇒ (c), S has a BAI. By Proposition 2.1.41, S “ A. Corollary 5.5.12 Let A be a Banach function algebra that is an ideal in its bidual and has a bounded pointwise approximate identity contained in J8 pAq. Then A is Tauberian, A has a bounded approximate identity, and A is a BSE algebra with a BSE norm. Proof By Proposition 3.5.11, A “ J8 pAq, whence A is Tauberian, and A has a BAI. By Corollary 5.5.10, A is a BSE algebra with a BSE norm. Corollary 5.5.13 Let A be a Tauberian Banach sequence algebra on a non-empty set S such that A has a bounded pointwise approximate identity. Then: (i) A has a bounded approximate identity, and A is a BSE algebra with a BSE norm; (ii) the norm on LpAq is equivalent to the 1 -norm if and only if A “ c 0 pS q.

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Proof (i) This follows from Corollary 5.5.12. (ii) The norm on A is denoted by  · . Suppose that the norm on LpAq is equivalent to the 1 -norm. By Theorem 5.4.3, the norms | · |S and  · BSE are equivalent on A. By (i), A has a BSE norm, and so  ·  and | · |S are equivalent. Since A is dense in c 0 pS q, necessarily A “ c 0 pS q. The converse is immediate. Example 5.5.14 Let Γ be a discrete, amenable group, and take p with 1 < p < 8. By Theorem 4.4.11, the Figà-Talamanca–Herz algebra A p pΓq has a contractive approximate identity, and, by Proposition 4.4.5, A p pΓq is an ideal in A p pΓq . Let S be a Segal algebra with respect to A p pΓq, and suppose that S is a BSE algebra. Then, by Corollary 5.5.11, S “ A p pΓq; by Theorem 5.3.38(ii), A p pΓq is indeed a BSE algebra in this case. In particular, take p and r such that 1 < p < 8 and 1 ď r < 8, and consider the example Arp pΓq “ A p pΓq ∩ Lr pΓq of Example 4.4.13. When Γ is discrete and amenable, Arp pΓq is a Segal algebra with respect to A p pΓq, but Arp pΓq  A p pΓq (unless Γ is finite), and so Arp pΓq is not a BSE algebra (cf. Example 5.3.39).

Example 5.5.15 Let J be the James algebra of Example 3.2.10. We know that J is a maximal ideal in J  “ J  , that pMpJq, · q is isometrically identified with pJ  , · q, and that J has a contractive approximate identity. By Corollary 5.5.10, J is a BSE algebra with a BSE norm, and so C BS E pJq “ J  .

Example 5.5.16 Let G be an infinite, compact abelian group, with discrete dual group Γ. (i) Take p with 1 < p < 8, and set S “ pL p pGq,  q “ pF p pΓq, · q, as in Example 4.2.16, so that S is a dual Banach sequence algebra that is an ideal in its bidual. Then QpS q “ S , as in Proposition 5.1.6, and so C BS E pS q “ S ; this is also noted in [299, §5]. However, S does not have a bounded pointwise approximate identity, and so S is not a BSE algebra. (ii) Consider the algebra pL8 pGq,  q. As in Example 4.2.17, we identify this algebra with the natural Banach sequence algebra I “ F8 pΓq, which is a weak Segal algebra with respect to ApΓq. This Banach sequence algebra is not Tauberian, and does not have a bounded pointwise approximate identity. As noted, I is a dual Banach algebra that is an ideal in its bidual, and so pI,  · I q “ pC BS E pIq,  · BSE,I q and L1 pGq “ LpIq is the unique Banach-algebra predual of I by Corollary 5.5.2. (iii) Take p and q with 1 < p, q < 8; set A “ F p pΓq and B “ Fq pΓq and p B, as in Example 4.2.19. Again as noted, A is a dual Banach function A “ A⊗ algebra that is an ideal in its bidual, and A “ C BS E pAq, but A does not have a bounded approximate identity. Thus A also has a BSE norm, but it is not a BSE algebra.

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5 BSE norms and BSE algebras

The next result is based on [222, Part 2]. Theorem 5.5.17 Let A be a Banach function algebra that is an ideal in its bidual and has a multiplier-bounded approximate identity. Then: (i) A “ QpAq  LpAqK “ C BS E pAq  LpAqK ; (ii) A l LpAqK “ {0} and LpAqK “ rad A ; (iii) A is Arens regular if and only if LpAqK l QpAq “ {0}; (iv) for each f P QpAq, we have f P ZpA q if and only if f · A Ă LpAq. Proof By Theorem 5.1.3, LpAqK is a closed ideal in A , and the quotient space QpAq “ A /LpAqK is Banach function algebra. Since A is an ideal in A , it follows from Proposition 5.2.38 that QpAq “ C BS E pAq and from Proposition 5.5.1 that LpAq “ A A and that M l N “ 0 pM, N P LpAqK q. As in Theorem 2.3.84, there is a bounded projection P in M pA q such that  A “ PpA q  ker P, such that ker P “ pA AqK , and such that PpA q is identified with QpAq “ C BS E pAq. Thus (i) follows. Further, Pp f · Mq “ f · M “ f · PpMq p f P A, M P A q . It follows that PpMq · λ “ M · λ “ λ · M “ λ · PpMq pλ P A , M P A q, and hence that N l PpMq “ N l M ,

PpMq l N “ M l N

pM, N P A q .

(5.5.2)

As in Theorem 2.3.84, we have A l LpAqK “ {0}, and so LpAqK Ă rad A . Since QpAq is semisimple, it follows from Proposition 2.1.2(ii) that LpAqK “ rad A , giving (ii). Suppose that A is Arens regular. Since A l LpAqK “ {0}, it follows that LpAqK l A “ {0}, and in particular LpAqK l QpAq “ {0}. Now suppose that LpAqK l QpAq “ {0}. It follows that p f + Mq l pg + Nq “ f g p f, g P QpAq, M, N P LpAqK q . This shows that A is commutative, and so A is Arens regular. This gives (iii). For (iv), take f P QpAq. Suppose that f P ZpA q. For each N P A and λ P A , we have lim xN, eν · f · λy “ lim xeν · pN l f q, λy “ xPpN l f q, λy “ xPp f l Nq, λy ν

ν

“ x f l N, λy “ xN l f, λy “ xN , f · λy , whence limν eν · f · λ “ f · λ weakly in A . Since eν · f · λ P AA Ă LpAq for each ν, also f · λ P LpAq, and so f · A Ă LpAq. Conversely, suppose that f · A Ă LpAq. For each N P LpAqK , we have xN l f, λy “ xN, f · λy “ 0 pλ P A q, and so N l f “ 0. By (ii), f l N “ 0. Now

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355

take g P QpAq. Then f l pg + Nq “ pg + Nq l f “ f g, and so f P ZpA q. Thus (iv) follows. The next result follows from Theorem 5.5.17. In the case where Γ is amenable, certainly ApΓq has a bounded approximate identity. However, we noted in Example 4.3.37 that there are some non-amenable discrete groups Γ such that ApΓq has a multiplier-bounded approximate identity. Corollary 5.5.18 Let Γ be a discrete group, and suppose that ApΓq has a multiplierbounded approximate identity. Then ApΓq “ Bρ pΓq  Cρ∗ pΓqK . Proof Since Γ is discrete, ApΓq is a Tauberian Banach sequence algebra on Γ, and hence an ideal in its bidual. By Theorem 5.3.18(ii), C BS E pApΓqq “ Bρ pΓq, and LpApΓqq “ Cρ∗ pΓq as in Example 5.1.25. So the result follows from Theorem 5.5.17(i). p c 0 . The algebra A does not have a bounded pointExample 5.5.19 Let A “ 1 ⊗ wise approximate identity, and so, by Theorem 5.2.16, (c) ⇒ (b), it is not true that MpAq Ă C BS E pAq, and so A is not a BSE algebra. It is clear that pΔn ⊗ Δn q is a multiplier-bounded approximate identity for A, and so Theorem 5.5.17 applies to A. We have A “ 1 pc 0 q, and so p 8 , C BS E pAq “ 1 p 8 q “ 1 ⊗

whereas

p 8 , MpAq “ 8 ⊗

and so C BS E pAq  MpAq. Hence we see again that A is not a BSE algebra. p 8 q “ {0} by By Corollary 2.3.41, A is Arens regular, and so LpAqK l p 1 ⊗ Theorem 5.5.17(iii).

Example 5.5.20 Let A be the Feinstein algebra described in Example 3.2.11, so that A “ {α P c 0 : ppαq < 8}, where 1 k |αk+1 ´ αk | , n k“1 n

pn pαq “

ppαq “ sup{pn pαq : n P N} ,

and α “ |α|N + ppαq for α P A. Then A is a natural Banach sequence algebra on N. Set A0 “ {α P c 0 : limnÑ8 pn pαq “ 0}, so that A0 “ J8 pAq and A0 is a Tauberian Banach sequence algebra, and so the structure theorem, Theorem 5.5.17, applies to A0 . We noted that pΔn q is a bounded approximate identity for A0 with bound 2, and so A0 is a BSE algebra with a BSE norm by Corollary 5.5.10. In fact, A has a BSE norm. For this, we shall apply Proposition 5.2.5, with the set T taken to be Nn for suitable n P N, so that χT op ď Δn  ď 2. Now

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take α P A, and choose k P N with |Pk α|N “ |α|N and 2pk pαq ě ppαq. Then 2ppPk+1 αq ě 2pk pPk+1 αq “ 2pk pαq ě ppαq, and so α ď 2 Pk+1 α, showing that the hypotheses of Proposition 5.2.5 are satisfied with n “ k + 1. Thus A also has a BSE norm. We now set B “ {β P 8 : ppβq < 8} , and β “ |β|N + ppβq pβ P Bq. It is clear that pB,  · q is a Banach function algebra on N (but B is not natural), and that B contains A as a closed subalgebra. We claim that MpAq “ C BS E pAq “ B . By Theorem 5.2.16, (b) ⇒ (c), MpAq Ă C BS E pAq. Now take sequences α P A and β P B. Then αβ P c 0 ∩ B “ A, and so A is an ideal in B; this shows that B Ă MpAq. Finally, take β P C BS E pAq. There is a net in Ar1s that converges coordinatewise to β, whence pn pβq ď 1 pn P Nq, and so β P B. Thus C BS E pAq Ă B, giving the claim. It follows from the claim that A is a BSE algebra. Further, it follows from Proposition 5.2.48 that C BS E pA0 q “ B, and so MpA0 q “ C BS E pA0 q “ B because A0 is a BSE algebra.

Example 5.5.21 Let ω : N Ñ r1, 8q be a sequence, and take pBω ,  · ω q and Mω to be the Banach sequence algebras of Example 3.2.12, so that ⎧ ⎫ 8 ⎪ ⎪  ⎪ ⎪ ⎨ ⎬ 8 Bω “ ⎪ : ω ´ α < 8 α P , |α | ⎪ i i+1 i ⎪ ⎪ ⎩ ⎭ i“1

and Mω is the maximal ideal Bω ∩ c 0 of Bω , identified with the semigroup algebra Aω “ 1 pN∧ , ωq. For every ω, the Banach space Aω is the dual of the concrete predual Eω “ c 0 pN, 1/ωq, and Mω is a Tauberian Banach sequence algebra on N. We see that ΦAω “ N, and so LpAω q Ă Eω . The algebra Mω has a multiplier-bounded approximate identity, and so the structure theorem, Theorem 5.5.17, applies to Mω . This shows that Mω “ QpMω q  LpMω qK “ C BS E pMω q  LpMω qK . Further, by Corollary 5.5.5, Mω and Bω have BSE norms. We seek to determine C BS E pMω q “ QpMω q; take β P C BS E pMω q. We first claim that β P Bω . Indeed, let pαpνq q be a net in pMω qr1s such that pνq limν αn “ βn pn P Nq. For each m, n P N, we have

5.5 Banach function algebras that are ideals in their bidual

|βm | +

n  i“1

357

⎛ n  ⎞⎟ ⎜⎜⎜ pνq   pνq pνq ⎟    ⎜ ωi |βi+1 ´ βi | “ lim ⎜⎝αm  + ωi αi+1 ´ αi ⎟⎟⎟⎠ ď 1 , ν

i“1

and so β P pBω qr1s , giving the claim. It follows that either C BS E pMω q “ Mω or C BS E pMω q “ Bω . By Theorem 5.2.16, the constant sequence 1 belongs to C BS E pMω q if and only if Mω has a bounded pointwise approximate identity; by Proposition 3.5.12, this holds if and only if Mω has a bounded approximate identity; as noted in Example 3.2.12, this holds if and only if lim inf nÑ8 ωn < 8. Thus we have determined C BS E pMω q “ QpMω q. Consider the case where limnÑ8 ωn “ 8. Then C BS E pMω q “ Mω . As in Example 3.2.12, Bω  MpMω q, at least when ω is increasing, and hence Mω is not a BSE algebra. In effect, LpAω q and Eω are closed linear subspaces of Aω , with LpAω q Ă Eω , such that both these two spaces are Banach-space preduals of Aω . By Proposition 1.3.28, it follows that LpAω q “ Eω . We claim that Eω is a Banachalgebra predual of Aω . To see this take n0 P N and λ P Eω . We need to see that n0 · λ is also in Eω . Take n 1 P N  such that n1 ě n0 and 1/ωn < ε pn ě n1 q. Then λpn0 ∨ nq /ωpnq ď ε λpn0 q pn ě n1 q, and so n0 · λ P Eω . It follows from Corollary 5.5.2 that LpAω q “ Eω is the unique Banach-algebra predual of Aω . Consider the case where lim inf nÑ8 ωn < 8, so that, as in Example 2.4.31, Aω is not a dual Banach algebra. By Corollary 5.5.10, Mω is a BSE algebra, and so C BS E pMω q “ MpMω q “ Bω . Since Mω  Bω , it follows that, in this case, LpAω q  Eω . By Proposition 5.2.32(i), Bω is also a BSE algebra. In particular, bv0 and bv are BSE algebras with BSE p Mω is a Tauberian Banach sequence algebra on norms. Further, in this case, Mω ⊗ p Mω has a bounded approximate identity, and so it also follows from N×N and Mω ⊗ p Mω is a BSE algebra with a BSE norm. Corollary 5.5.10 that Mω ⊗ We shall discuss the Arens regularity of Mω in Example 6.1.24.

Example 5.5.22 Again, consider the semigroup algebra A “ p 1 pZ∨ q,  q and the maximal modular ideal M in A of Examples 2.1.26 and 3.2.13; we made the identification ⎧ ⎫ ⎪ ⎪   ⎪ ⎪ ⎨ ⎬  8   pZq : MpMq “ MpAq “ ⎪ βpnq ´ βpn ´ 1q < 8 β P , ⎪ ⎪ ⎪ ⎩ ⎭ nPZ

  with β “ |β|Z + nPZ βpnq ´ βpn ´ 1q pβ P MpMqq. Since M is an ideal in its bidual and M has a bounded approximate identity, it follows from Corollary 5.5.10 that M is a BSE algebra with a BSE norm. By Corollary 5.2.47(ii), MpMq is a BSE algebra, and so A is also a BSE algebra. 

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5 BSE norms and BSE algebras

p M is a Tauberian Banach sequence algebra on Z × Z with a bounded Since M ⊗ p M is a BSE approximate identity, it also follows from Corollary 5.2.47(ii) that M ⊗ p A is a BSE algebra, as proved by algebra with a BSE norm. This implies that A ⊗ Dabhi and Upadhyay in [49].

Chapter 6

Arens regularity

In this chapter, we shall discuss whether many examples of Banach algebras are Arens regular or, at the other extreme, strongly Arens irregular, a concept that will be defined in Definition 6.1.2. Some Banach algebras have neither of these properties. pq In §6.1, we shall introduce the left and right topological centres, Zt pA q and prq Zt pA q, of a Banach algebra A, and say that A is strongly Arens irregular whenever pq prq Zt pA q “ Zt pA q “ A. We shall also define in Definition 6.1.11 a subset of A to be a set that is determining for the left topological centre (a DLTC set). Often, but not always, a Banach algebra that is strongly Arens irregular has a finite DLTC set. p B of two Banach In §6.2, we shall discuss when the projective tensor product A ⊗ algebras, A and B, that are Arens regular is also Arens regular, in particular considering the Varopoulos algebra VpK, Lq for two locally compact spaces, K and L. In fact, VpK, Lq is Arens regular if and only if either of the spaces K and L is scattered; see Theorem 6.2.18. The centre of VpK, Lq is identified in a theorem of Neufang, Theorem 6.2.20. We shall also determine in Theorem 6.2.24 when the projective tensor product of two unital C ∗ -algebras is Arens regular. In § 6.3, we shall consider the above questions for semigroup algebras of the form p 1 pS q,  q, and their weighted analogues  1 pS , ωq. We shall show that the algebra  1 pS q has two-point DLTC set when S is cancellative, and discuss some examples when S is an abelian, idempotent semigroup that is usually not weakly cancellative. Finally, in §6.4, we shall discuss the Arens regularity of various Banach algebras associated with locally compact groups, in particular proving that every group algebra pL1 pGq,  q is strongly Arens irregular and has a two-point DLTC set.

6.1 Topological centres As a first step towards obtaining results on the Arens regularity of various Banach algebras, and especially group algebras, we shall define the two topological centres of an arbitrary Banach algebra; we shall then obtain some abstract theorems that © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 H. G. Dales and A. Ülger, Banach Function Algebras, Arens Regularity, and BSE Norms, CMS/CAIMS Books in Mathematics 12, https://doi.org/10.1007/978-3-031-44532-3_6

359

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will be applied later. A Banach algebra is strongly Arens irregular if these two topological centres are equal to the original algebra, and so such an algebra is strongly Arens irregular if it is ‘as far as possible’ from being Arens regular. We shall prove Theorem 6.1.20 that gives a criterion for a Banach algebra to be strongly Arens irregular, and then apply this condition to some examples. Later, we shall determine the topological centres of certain operator algebras, especially the algebra KpEq for a Banach space E, showing that the two centres may be distinct. We recall that the first and second Arens products on the bidual A of a Banach algebra A are denoted by l and , respectively, as in §2.3. The following definition is taken from [57, Definition 2.17] and [58, Definition 2.24], following [220], where the two sets are called Z1 and Z2 , respectively. Definition 6.1.1 Let A be a Banach algebra. Then the left and right topological cenpq prq tres, Zt pA q and Zt pA q, of A are   pq Zt pA q “ M P A : M l N “ M  N pN P A q ,   prq Zt pA q “ M P A : N l M “ N  M pN P A q , respectively. Note that a l M “ a  M “ a · M and M l a “ M  a “ M · a for each a P A pq prq and M P A , and so A Ă Zt pA q and A Ă Zt pA q. Take M P A . Then clearly the following are equivalent: pq

(a) M P Zt pA q; (b) L M : N Ñ M l N is weak-∗ continuous on A ; (c) wk∗ – limα M · aα “ M l N whenever paα q is a bounded net in A with wk∗ – limα aα “ N in A . pq

prq

We see that both Zt pA q and Zt pA q are closed subalgebras of both pA , l q pq and pA ,  q. For example, take M1 , M2 P Zt pA q. Then pM1 l M2 q l N “ M1 l pM2 l Nq “ M1  pM2  Nq “ pM1  M2 q  N “ pM1 l M2 q  N

pN P A q ,

pq

and so M1 l M2 P Zt pA q. Let M P A . It follows from Proposition 2.3.5 that pq ZpA , l q Ă Zt pA q, but we shall see in Example 6.1.30 that it may be that prq  ZpA , l q  Zt pA q. It is immediate from the definition that a Banach algebra pq prq A is Arens regular if and only if Zt pA q “ Zt pA q “ A . In the case where A is commutative, we have M l N “ N  M pM, N P A q and hence pq prq Zt pA q “ Zt pA q “ ZpA q , the centre of both of the algebras pA , l q and pA ,  q.

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Definition 6.1.2 Let A be a Banach algebra. Then A is left strongly Arens irregular pq prq (LSAI) if Zt pA q “ A and right strongly Arens irregular (RSAI) if Zt pA q “ A; A is strongly Arens irregular (SAI) if it is both left and right strongly Arens irregular. Note that a Banach algebra is both Arens regular and strongly Arens irregular if and only if it is reflexive. For example, suppose that A is a Banach algebra with zero pq prq product. Then A is Arens regular by Proposition 2.3.20, and Zt pA q “ Zt pA q “  A . We shall show in Theorem 6.4.18 that the group algebra L1 pGq of a locally compact group G is always strongly Arens irregular, a striking contrast to the fact that all C ∗ -algebras are Arens regular. Let A be a Banach algebra. We have defined the notions of ‘A is strongly Arens irregular’ and ‘A is extremely non-Arens regular’; see Definition 2.3.39. It is shown by Hu and Neufang in [176] that neither of these properties implies the other. We begin with an easy example of a Banach algebra that is right, but not left, strongly Arens irregular. Example 6.1.3 Let A be a Banach algebra, and define A “ A ‘1 A, with the product given by pa1 , b1 q · pa2 , b2 q “ pa1 a2 , a1 b2 q pa1 , a2 , b1 , b2 P Aq , so that A is a Banach algebra. Now suppose that A is strongly Arens irregular and non-reflexive (such as L1 pGq for an infinite locally compact group G). Then we see pq prq easily that Zt pA q “ A ‘ A , whereas Zt pA q “ A ‘ A “ A. Thus we have the required example.

Proposition 6.1.4 Let A be a Banach algebra such that A has a mixed identity E. prq pq Then Zt pA q Ă E l A . Suppose that E P Zt pA q. Then A A “ A . prq

prq

Proof Let M P Zt pA q. Then M “ E  M “ E l M, so that Zt pA q Ă E l A . pq Now suppose that E P Zt pA q, say E “ wk∗ – limα eα , where peα q is a bounded net in A, and take M P A , so that M “ E  M “ E l M. In the case where K M P A A and λ P A , we have xM, λy “ xE l M, λy “ xE, M · λy “ lim xeα , M · λy “ lim xM, λ · eα y “ 0 , α

α

and so M “ 0. Hence A A “ A . Proposition 6.1.5 Let A be a Banach algebra with a closed subalgebra B of finite codimension. Then A is strongly Arens irregular if and only if B is strongly Arens irregular.

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Proof This follows as in Proposition 2.3.23; we use equation (1.3.4). Proposition 6.1.6 Let A be a commutative Banach algebra that is strongly Arens irregular, and suppose that I is a closed ideal that has a mixed identity. Then I is strongly Arens irregular. Proof By Proposition 2.3.77, ZpI  q Ă ZpA q, and so ZpI  q Ă I  ∩ A. By equation (1.3.4), I  ∩ A “ I, and so ZpI  q “ I, showing that I is SAI. Proposition 6.1.7 Let pA,  · q be a Banach function algebra such that A is strongly Arens irregular and has the Schur property. Then  · op and  ·  are equivalent on A. Proof Suppose that M P A and M l N “ 0 pN P A q. Then M · f “ 0 p f P Aq, pq and so M  N “ 0 pN P A q. This shows that M P Zt pA q “ A. It follows that M “ 0, and so A is faithful. The result now follows from Theorem 3.1.23. Proposition 6.1.8 Let A be a Banach algebra. Then A A Ă WAPpAq if and only if pq AA Ă Zt pA q. Proof Suppose that A A Ă WAPpAq, and take a P A and M P A . For each λ P A , it follows from Theorem 2.3.29 that xM l N, λ · ay “ xM  N, λ · ay and so

pN P A q ,

xpa · Mq l N, λy “ xpa · Mq  N, λy pN P A q .

This holds for each λ P A , and so pa · Mq l N “ pa · Mq  N pN P A q, and hence pq pq pq a · M P Zt pA q by the definition of Zt pA q. Thus AA Ă Zt pA q, as required. pq Conversely, suppose that AA Ă Zt pA q. Then the reverse of the above argument shows that A A Ă WAPpAq. Corollary 6.1.9 Let A be a Banach algebra. Suppose that A is strongly Arens irregular and that AA ∪ A A Ă WAPpAq. Then A is an ideal in its bidual. pq

Proof It follows from Proposition 6.1.8 that AA Ă Zt pA q “ A. Similarly, A A Ă A, and so A is an ideal in A . Proposition 6.1.10 Let A be a faithful Banach algebra that is strongly Arens irregular. Then A A “ A .

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363

Proof Assume to the contrary that A A  A , so that there exists λ P A \ A A . Then there exists M P A with xM, λy “ 1 and xM, N · μy “ 0 pN P A , μ P A q. The latter equation implies that M l N “ 0 pN P A q, and so M · a “ 0 pa P Aq, whence M  N “ 0 pN P A q. In particular, M l N “ M  N pN P A q, and hence pq M P Zt pA q. Since A is SAI, M P A. Since A is faithful and MA “ {0}, it follows that M “ 0, a contradiction. Thus A A “ A . Definition 6.1.11 Let A be a Banach algebra. Then a subset V of A is determining pq for the left topological centre (a DLTC set) of A if M P Zt pA q whenever M P A and M l N “ M  N pN P Vq . Note that the empty set is a DLTC set when A is Arens regular. In the case where the algebra A is commutative, we shall use the phrase determining for the topological centre and refer to a ‘DTC set’. At later stages, we shall be interested in finding ‘small’ DLTC sets for various Banach algebras; see Theorem 6.3.7 and Examples 6.3.15, 6.3.34, and 6.3.35. Indeed, we shall show in Theorem 6.4.17 that the group algebra L1 pGq of locally compact, non-compact group G has a two-point DLTC set. However some commutative Banach algebras do not have ‘small’ DTC sets; see §6.3. There is a small generalization of the notion of the left topological centre. Indeed, let A be a Banach algebra, and let F be a closed A-submodule of pA ,  · q that is leftintroverted, as in Definition 2.3.57. As in Definition 2.3.61, we obtain the Banach algebra pF  , l q; in the case where F is faithful, we regard A as a subalgebra of pF  , l q. The following notion of a topological centre was introduced in [182] and discussed in [220], where it is denoted by Zr1 . Definition 6.1.12 Let A be a Banach algebra, and let F be a left-introverted A-submodule of pA ,  · q. Then the topological centre of F  is Zt pF  q “ {M P F  : L M is continuous on pF  , σpF  , Fqq} . Clearly the set Zt pF  q is a  · -closed subalgebra of the Banach algebra pF  , l q, and A Ă Zt pF  q Ă F  when F is faithful. In the case where F “ A , the set pq Zt pF  q coincides with the left topological centre, Zt pA q. When A is commutative,    Zt pF q “ ZpF q, the centre of the algebra pF , l q. Suppose that A is Arens regular. Then Zt pF  q “ F  for each left-introverted submodule F of A . In the case where F is introverted in A , so that we have two products l and  on the space F  , we have   Zt pF  q “ M P F  : M l N “ M  N pN P F  q . (6.1.1) Example 6.1.13 Let A be a Banach algebra, and suppose that F is a closed A-submodule of A such that F Ă WAPpAq. Then F is introverted by Corollary 2.3.59 and Zt pF  q “ F  by Theorem 2.3.29.

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Now suppose that the Banach algebra A has a mixed identity, say E, in A , and consider the left-introverted submodule F “ A A of A , so that F  “ E l A and A “ F   F K , as in equation (2.3.16). Then it is clear that pq

Zt pA q ∩ F  Ă Zt pF  q ;

(6.1.2)

pq

we shall see in Example 6.4.19 that we can have Zt pA q ∩ F   Zt pF  q in the above circumstances. In the case where the algebra A is also commutative, so that pq prq Zt pA q “ Zt pA q “ ZpA q, we have ZpA q Ă F  by Proposition 6.1.4, and  hence ZpA q Ă Zt pF  q. The canonical embedding θ : M r pAq Ñ pF  , l q of the next theorem was defined in Theorem 2.3.81; the definition of θ requires the algebra A to have a mixed identity. Theorem 6.1.14 Let A be a Banach algebra that has a mixed identity, and let F be a faithful, left-introverted submodule of A such that F “ FA. Then the canonical Banach-algebra embedding θ : M r pAq Ñ pF  , l q is such that the range of θ is contained in Zt pF  q. Proof Take R P M r pAq, and set M “ θpRq P F  . Let Nα Ñ N in pF  , σpF  , Fqq. Then, for each a P A and λ P F, we have xM l Nα , λ · ay “ xM, Nα · λ · ay “ xRa, Nα · λy “ xNα , λ · Ray Ñ xN, λ · Ray “ xRa, N · λy “ xM l N, λ · ay , and so L M is continuous on pF  , σpF  , Fqq, whence M P Zt pF  q. Let G be a locally compact group, and set A “ L1 pGq, the group algebra of G. As in §4.1, A has a contractive approximate identity, and so it has a mixed identity in Ar1s ; the space A is identified with L8 pGq, and M r pAq is identified with MpGq, which is regarded as a subset of X  in the following corollary. Corollary 6.1.15 Let G be a locally compact group, set A “ L1 pGq, and let X be a faithful, left-introverted A-submodule of A with X “ XA. Then MpGq Ă Zt pX  q. Corollary 6.1.16 Let A be a commutative Banach algebra that has a mixed identity E. Then the canonical Banach-algebra embedding θ : MpAq Ñ pA , l q is such that the range of θ is contained in ZpE l A q. Proof Set F “ A A, so that F  “ E l A . Since A is commutative, Zt pF  q “ ZpF  q. By Theorem 6.1.14, θpMpAqq Ă Zt pF  q, giving the result.

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Proposition 6.1.17 Let A be a Banach algebra with a closed ideal I. Suppose that A/I and I are strongly Arens irregular. Then A is strongly Arens irregular. Proof Set B “ A/I; the quotient map is q : A Ñ B. pq Let M P Zt pA q. We have q pMq l q pNq “ q pMq  q pNq pN P A q, and pq so q pMq P Zt pB q. Since B is SAI, we see that q pMq P B, say q pMq “ qpa0 q, pq where a0 P A. Thus q pM ´ a0 q “ 0, and so M ´ a0 P Zt pI  q. Since I is SAI, M ´ a0 P I, and so M P A. Hence A is LSAI. Similarly, A is RSAI, and so A is SAI. Let A be a Banach algebra, and let F be a closed submodule of pA ,  · q that is faithful and left-introverted. The next proposition provides a method by which we can determine whether an element of F  belongs to Zt pF  q. Proposition 6.1.18 Let A be a Banach algebra, and let F be a closed A-submodule of pA ,  · q that is faithful and introverted. Take M P F  . Then M P Zt pF  q if and only if, for each λ P F and each bounded net paα q in A with M “ wk∗ – limα aα in F  , the net pλ · aα q converges weakly to λ · M in F. In this case, λ · M P FA pλ P Fq. Proof Suppose that M P Zt pF  q, and take λ P F and a bounded net paα q in A such that M “ limα aα in the topology σpF  , Fq. Then, for each N P F  , we have xN , λ · aα y “ xaα · N , λy Ñ xM l N , λy “ xM  N , λy “ xN , λ · My , and so pλ · aα q converges weakly to λ · M in F. Conversely, suppose that pλ · aα q converges weakly to λ · M in F for each λ P F and each bounded net paα q in A for which M “ wk∗ – limα aα . Then again we have limα xN , λ · aα y “ xM l N , λy for each N P A . But also xM  N , λy “ xN , λ · My “ lim xN , λ · aα y “ lim xaα · N , λy “ xM l N , λy , α

α

and so M l N “ M  N, whence M P Zt pF  q. Suppose that M P Zt pF  q, and take λ P F. We have shown that λ · M belongs to the weak closure of FA; by Mazur’s theorem, Theorem 1.2.24(ii), λ · M P FA. The following corollary is [220, Theorem 5.1]; as in Definition 2.3.64, we denote by EA the set of mixed identities for A. Since A has a bounded approximate identity, the set EA is not empty. Corollary 6.1.19 Let A be a Banach algebra with a bounded approximate identity, pq and take M P A . Then M P Zt pA q if and only if the following three conditions pq are satisfied: (i) M · a P Zt pA q pa P Aq; (ii) M  E “ M pE P EA q; (iii) λ · M P A · A pλ P A q. pq

Proof Suppose that M P Zt pA q. Then conditions (i) and (ii) are immediate, and (iii) follows from Proposition 6.1.18 because A A “ A · A.

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Conversely, suppose that M satisfies conditions (i), (ii), and (iii). Take N P A , λ P A , and choose E P EA , say E is the weak-∗ limit of a BAI peα q in A. Since, by (iii), λ · M P A · A, it follows that limα λ · M · eα “ λ · M in A , and so lim xN, λ · M · eα y “ xN, λ · My “ xM  N, λy . α

Also lim xN, λ · M · eα y “ lim xpM · eα q  N, λy “ lim xpM · eα q l N, λy by piq α

α

α

“ xpM  Eq l N, λy “ xM l N, λy by piiq . pq

It follows that M l N “ M  N, and so M P Zt pA q. The following theorem was first proved in [314] under the additionally hypothesis that A has a sequential bounded approximate identity; the general case is given in [12]. Theorem 6.1.20 Let A be a Banach algebra that is an ideal in its bidual, is weakly sequentially complete, and has a bounded approximate identity. Then A is strongly Arens irregular. Proof Let peα q be BAI for A, and let EA be the (non-empty) set of weak-∗ accumulation points of peα q in A . Let peν q be a subnet of peα q that converges weak-∗ to E P EA , and take M P A . Since LM is weak-∗ continuous on pA , q, we have wk∗ – limν M · eν “ M  E. pq Now suppose that M P Zt pA q. Then M  E “ M l E “ M, and so the unique weak-∗ limit of a subnet of the net pM · eν q is M. Thus wk∗ – limν M · eν “ M. Since A is an ideal in A , the net pM · eν q is contained in A. Consider the case where A has a sequential BAI, say pen q. Then we have wk∗ – limnÑ8 M · en “ M. Since A is weakly sequentially complete, M P A. This pq prq shows that Zt pA q “ A. Similarly, Zt pA q “ A, and so A is SAI in this special case. pq We next consider the general case. Again take M P Zt pA q. For each λ P A , we  have λ · M P A A by Proposition 6.1.18, and so it follows from Proposition 2.3.37 that λ · M P WAPpAq. Now take a sequence pan q in Ar1s . By Proposition 2.1.27, there is a closed ideal I in A containing the sequence pan q such that I has a sequential BAI. Let F be a mixed pq identity of I  . Then M l F P I  , and we claim that M l F P Zt pI  q. Indeed, for   each N P I and λ P A , we have xpM l Fq l N, λy “ xM  pF l Nq, λy because “ xF l N, λ · My

pq

M P Zt pA q

“ xF  N, λ · My because λ · M P WAPpAq “ xN, λ · My “ xM  N, λy “ xpM l Fq  N, λy ,

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367

and so pM l Fq l N “ pM l Fq  N, giving the claim. It follows from the special case that M l F P I. We now claim that the map L M : A Ñ A is weakly compact. Indeed, again take a sequence pan q in Ar1s , and take I and F as above. Then M l F P I, and so L M l F : I Ñ I is weakly compact because I is an ideal in I  . Clearly we have M · a “ pM l Fq · a pa P Iq, and so pL M an q has a weakly convergent subsequence. The claim follows from Theorem 1.3.41. This claim implies that the net pM · eα q has a weakly convergent subnet in pq A. However, limα M · eα “ M, and so M P A. Thus Zt pA q “ A. Similarly, prq Zt pA q “ A, and so A is SAI in the general case. The following two examples were first established by Ghahramani and McClure in [132, Theorem 2.2(a)]. Example 6.1.21 Let V be the Volterra algebra of Example 4.2.3. Then V is a commutative, radical Banach algebra that has a contractive approximate identity, that is weakly sequentially complete as a Banach space, and is a compact algebra, so that it is an ideal in its bidual. It follows from Theorem 6.1.20 that V is strongly Arens irregular. Let I “ { f P V : αp f q ě 1/2}. Then I is closed ideal in V with zero product, and so I is Arens regular by Proposition 2.3.20 (but I is far from being strongly Arens irregular). For a generalization, see the paper by Pym and Saghafi [268].

Example 6.1.22 Let ω be a continuous weight on R+ , and consider the commutative Banach algebra A :“ pL1 pR+ , ωq,  q of Example 4.2.21; again A always has a sequential bounded approximate identity and is weakly sequentially complete as a Banach space by Example 1.3.3. Suppose that the weight ω is regulated at some a P R+ , and let I :“ Mαω be the (non-zero) standard ideal that was defined within Example 4.2.21, so that L f is a compact operator on A for each f P I. In particular, M · f P I whenever M P I  and f P A. A small variation of the proof of the first part of Theorem 6.1.20 shows that the closed ideal I is strongly Arens irregular. Essentially as in Example 6.1.21, the quotient algebra A/I is also strongly Arens irregular, and so, by Proposition 6.1.17, the algebra A is strongly Arens irregular.

Example 6.1.23 Consider the weakly sequentially complete Banach space L1 pIq. Then I is also a semigroup, called I∨ , where st “ s ∨ t “ max{s, t}

ps, t P Iq ;

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6 Arens regularity

the space I∨ is a compact topological semigroup. Let A “ pL1 pI∨ q,  q be the corresponding semigroup algebra, so that the product of f and g in A is given by the formula  x  x p f  gqpxq “ gpxq f ptq dt + f pxq gptq dt px P Iq . 0

0

Then A has a sequential bounded approximate identity. The centre, ZpA q, of A , is identified by Saghafi in [286], and it is shown that A  ZpA q  A , and so A is neither Arens regular nor strongly Arens irregular. This shows that we cannot drop the condition that A be an ideal in its bidual in Theorem 6.1.20. Further, the algebra A satisfies the conditions of Theorem 2.3.80, and so, by that theorem, θ E pMpAqq  ZpE l A q.

Example 6.1.24 Let ω : N Ñ r1, 8q be an increasing sequence on N, and let Mω “  1 pN∧ , ωq be as in Examples 3.2.12 and 5.5.21, so that Mω is a Tauberian Banach sequence algebra on N that is weakly sequentially complete. Suppose that lim inf nÑ8 ωpnq < 8. Then it was shown in Example 3.2.12 that Mω is not Arens regular; since Mω has a bounded approximate identity,we now see from Theorem 6.1.20 that Mω is strongly Arens irregular in this case. For p Mω is also a Tauberian Banach sequence algesuch weights ω, the algebra Mω ⊗ bra with a bounded approximate identity using Proposition 2.1.38, and we also have p Mω ∼  1 ⊗ p  1 “  1 pN × Nq. Hence it also follows from Theorem 6.1.20 that Mω ⊗ p Mω is strongly Arens irregular. Mω ⊗ When limnÑ8 ωn “ 8, the algebra Mω is Arens regular, as we noted in Example 3.2.12. The product in Mω is given by equation (2.4.4). It also follows from Theorem 6.1.20 that the maximal modular ideal M in the semigroup algebra A “ p 1 pZ∨ q,  q of Examples 2.1.26 and 3.2.13 and the algebra p M are both strongly Arens irregular; by Proposition 6.1.5, A is also strongly M⊗ Arens irregular.

We now give a result, from [313, Theorem 2.2], about the centre of the bidual of a Banach function algebra that is compact and weakly sequentially complete. It is easy to see that the Banach sequence algebras  1 and Mω (as defined in Example 3.2.12) on N satisfy the hypotheses on A in the theorem, as do the algebras ApΓq for each discrete group Γ and the algebras pL p pGq,  q for each compact abelian group G and each p with 1 ď p < 8, identified with F p pΓq, which is a Taubep Another large class of Banach rian Banach sequence algebra on Γ, where Γ “ G. function algebras to which the theorem applies is the class of closed ideals of the above-mentioned Banach function algebras. Theorem 6.1.25 Let A be a Banach function algebra such that A is a compact algebra and A is weakly sequentially complete. Then L M : A Ñ A is weakly compact for each M P ZpA q, and

6.1 Topological centres

ZpA q “ {M P A : M l A , A l M Ă A} .

369

(6.1.3)

Further, A is Arens regular if and only if A l A Ă A. In this case, pA , l q is also Arens regular and an ideal in its bidual. Proof Since A is compact, A is an ideal in its bidual by Theorem 2.3.7. Take M P A , and suppose that M l A Ă A and A l M Ă A. For each N P A , we have M l N P A and N l M P A. Since pM l Nqpϕq “ pN l Mqpϕq pϕ P ΦA q, it follows that M l N “ N l M, and so M P ZpA q. Now take M P ZpA q; we shall first show that L M : A Ñ A is weakly compact. For this, take p fn q to be a sequence in Ar1s , and set  S “ { fn pΦA q : n P N} . Since each fn is compact, it follows from Theorem 3.2.6(i) that S is countable, and so, by passing to a subsequence of p fn q, we may suppose that p fn pϕqq converges for each ϕ P ΦA . Since p fn q is a bounded sequence in A, it follows that px fn , λyq converges for each λ P LpAq, and hence, by Proposition 5.5.1, for each λ P A A. Now take μ P A : by Proposition 6.1.18 (with F “ A ), μ · M P A A, and so pxM · fn , μyq converges. Thus the sequence pL M fn q is weakly Cauchy in A. Since A is weakly sequentially complete, this sequence is weakly convergent, and so L M :  pA q Ă A, and so M l A Ă A. Since A Ñ A is weakly compact. It follows that LM   M P ZpA q, also A l M Ă A, giving equation (6.1.3). By definition, A is Arens regular if and only if ZpA q “ A . By (6.1.3), this occurs if and only if A l A Ă A. Now set B “ A , and suppose that A is Arens regular. Then L M : B Ñ B is weakly compact for each M P B, and so B is an ideal in its bidual. By Corollary 2.4.9, pB, l q is a dual Banach algebra. By Corollary 2.4.18, B l B Ă B, and so B is Arens regular by Corollary 2.4.19. Corollary 6.1.26 Let A be a Banach function algebra such that A is a compact algebra, such that A is weakly sequentially complete, and such that A has a BSE norm. Then the following are equivalent: (a) the algebra A is Arens regular; (b) the algebra C BS E pAq is Arens regular; (c) the algebra C BS E pAq is an ideal in its bidual; (d) C BS E pAq l C BS E pAq Ă C BS E pAq; (e) C BS E pAq l C BS E pAq Ă A. Proof Since A is an ideal in A , it follows from Proposition 5.2.38 that C BS E pAq is a dual Banach function algebra such that C BS E pAq – LpAq and C BS E pAq “ QpAq. (a) ⇒ (b), (c) By Theorem 6.1.25, A is Arens regular and an ideal in its bidual. Hence C BS E pAq, a quotient of A , is also Arens regular, giving (b), and, by Corollary 2.3.8, C BS E pAq is an ideal in its bidual, giving (c).

370

6 Arens regularity

(b) ⇒ (a) Suppose that C BS E pAq is Arens regular. Since A has a BSE norm, A is a closed subalgebra of C BS E pAq, and so A is Arens regular. (c) ⇔ (d) ⇒ (b) These follow from Corollary 2.4.18 and Theorem 2.4.17. (e) ⇒ (d) This is trivial. (c) ⇒ (e) Since C BS E pAq is a dual Banach algebra, and since, by hypothesis, C BS E pAq is an ideal in its bidual, it follows from Theorem 2.4.17 that C BS E pAq l C BS E pAq “ C BS E pAq2 . Take u, v P C BS E pAq. By Proposition 5.5.1, A is an ideal in C BS E pAq, and so Lu : A Ñ A is well defined. By Theorem 2.3.7, Lu : C BS E pAq Ñ C BS E pAq is weakly compact, and so Lu : A Ñ A is weakly compact because A is an ideal which is closed in C BS E pAq. By Theorem 5.2.9, there is a bounded net p fα q in A that converges pointwise to v on ΦA , and hence pLu p fα qq converges pointwise to uv on ΦA . By passing to a subnet, we may suppose that pLu p fα qq converges weakly to g P A, say. But then pLu p fα qq converges pointwise to g on ΦA , and so g “ uv. Hence uv P A. This establishes (e). Example 6.1.27 Consider the Tauberian Banach sequence algebra A “  1 . Then A is a compact algebra by Proposition 3.2.3 and, by Theorem 1.2.39(ii), A is weakly sequentially complete. As in Example 5.3.1(ii), A has a BSE norm, and, as in Example 3.2.7(ii), A is Arens regular. Thus A satisfies the hypotheses of Corollary 6.1.26 and the equivalent clauses (a)–(e) (with C BS E pAq “ A). Let Γ be a discrete group. Then the non-zero, closed ideals of ApΓq, considered as Banach function algebras, also satisfy the hypotheses of Corollary 6.1.26.

See also Theorem 6.4.4, where the above corollary will be applied to the Fourier algebra ApΓq when Γ is a discrete group. In the next result, we shall write A “ PpA q  ker P, where P is a bounded projection in MpA q, as in Theorem 2.3.84, so that A l ker P “ {0}. Corollary 6.1.28 Let A be a Tauberian Banach sequence algebra such that A is weakly sequentially complete and has a multiplier-bounded approximate identity. Take p P PpA q and q P ker P, and suppose that p + q P ZpA q. Then p, q P ZpA q. Proof Set M “ p + q, and take peα q to be an MBAI in A. The algebra A satisfies the conditions of Theorem 6.1.25, and so it follows from the theorem that M l A Ă A and A l M Ă A. Take N P A . Then N l M “ N l pp + qq “ N l p because A l ker P “ {0}, and hence N l p “ p l N + q l N P A. We have eα · pN l pq “ peα · pq l N for each α because eα · q “ 0. Since N l p P A,

6.1 Topological centres

371

it follows that limα eα · pN l pq “ N l p in pA,  · q. By equation (2.3.20), limα eα · p “ p in the space pA , σpA , A qq, and so limα peα · pq l N “ p l N in pA , σpA , A qq. This implies that N l p “ p l N. Since this equation holds for each N P A , it follows that p P ZpA q. Further, q “ M ´ p P ZpA q. We conclude this section with a calculation of the two topological centres for certain operator algebras. The account is based on [57, §6], which grew out of earlier work in [155], [220], and [253]; see also [254, §1.7.13]. For substantial generalizations of the examples and a clear summary and history, see [68] of Daws. Example 6.1.29 We shall include in this example a Banach algebra A such that A “ A · A, but A  A · A . Let E be a non-zero Banach space. For each U P BpE  q, set ηpUq “ κE ◦ U  ◦ κE  ,

E Ñ E ,

so that ηpUq P BpE  q, and then set PpUq “ ηpUq , so that PpUq P BpE  q and P is a bounded linear operator on BpE  q. Alternatively, we see that PpUq is defined from U by first restricting U to E, and then extending this latter operator by weak-∗ continuity in both its domain and range spaces. From both descriptions of P, it is clear that P is a projection on BpE  q with P “ 1 and that the range of P is exactly BpE  qa , as defined in §1.3. In particular, PpIE  q “ IE  . Let J be the kernel of P, so that J is the set of elements U P BpE  q such that U | E “ 0. Suppose that E is a reflexive Banach space and that U P BpEq. Then ηpUq “ U  and PpUq “ U. Now suppose that E is a Banach space such that E  has the approximation property and the Radon–Nikodým property; recall that E “ c 0 has these properties. Set A “ KpEq. Then it follows from Theorem 1.4.18(ii) that we have q E, A“ E⊗

p E  “ NpE  q, A “ E  ⊗

p E  q “ BpE  q A “ pE  ⊗

as Banach spaces. The canonical embedding κA : A Ñ A is the map κA : T Þ→ T  . The duality between A and A is specified by xT, μ ⊗ Λy “ xΛ, T  pμqy “ xT  pΛq, μy

pT P A, μ P E  , Λ P E  q ,

(6.1.4)

and the duality between A and A is specified by xU, μ ⊗ Λy “ xUpΛq, μy pU P A , μ P E  , Λ P E  q .

(6.1.5)

For a full description of these dualities, see [76, §16.7], for example. We now describe the canonical module actions of A on A and on A . Thus, take T P A, R P A “ NpE  q, and U P A “ BpE  q. Then:

372

6 Arens regularity

T · R “ R ◦ T ; T · U “ T  ◦ U ;

R · T “ T ◦ R ; U · T “ U ◦ T  .

 (6.1.6)

We noted in Example 2.3.54 that A “ KpEq is an ideal in its bidual BpEq when E is reflexive and has the approximation property. Now suppose that E  has the approximation property and the Radon–Nikodým property, but that E is not reflexive, so that KpEq Ă BpE  q. Assume that KpEq is a closed ideal in its bidual, BpE  q. Since the minimum non-zero, closed ideal in BpE  q is ApE  q, it follows that KpEq ⊃ ApE  q. But ApE  q contains non-zero, finite-rank operators whose restriction to E is 0, a contradiction. Thus KpEq is not an ideal in its bidual. We shall now calculate the two Arens products on A “ BpE  q, still in the case where E  has the approximation property and the Radon–Nikodým property. Let U, V P A , and take pS α q and pT β q to be nets in A such that limα S α “ U and limβ T β “ V (in the topology σpA , A q). Take μ P E  and Λ P E  . Then xpS α T β q pΛq, μy “ xT β pΛq, S α pμqy . For each α, we have limβ xT β , S α pμq ⊗ Λy “ xV, S α pμq ⊗ Λy, and so lim xT β pΛq, S α pμqy “ xVpΛq, S α pμqy “ xS α pVpΛqq, μy . β

Hence limα limβ xpS α T β q pΛq, μy “ xUpVpΛqq, μy. This shows that U l V “ lim lim S α T β “ U ◦ V α

β

pU, V P BpE  qq ,

(6.1.7)

and so pKpEq , lq is identified with pBpE  q, ◦ q; in particular, IE  is the identity of pA , lq, which implies, by Corollary 2.3.67, that A “ A · A. As a preliminary to the calculation of U  V, we make the following remark. Take T “ x ⊗ λ P E ⊗ E  Ă A, R P A , and U P A “ BpE  q. Then xT, R · Uy “ xU, T · Ry “ xU, R ◦ T  y “ xU, Rpλq ⊗ κE pxqy “ xpU ◦ κE qpxq, Rpλqy “ xx, pηpUq ◦ Rqpλqy “ xT, ηpUq ◦ Ry , and so R · U “ ηpUq ◦ R in A . Similarly, U · R “ ηpU ◦ R q. We can now calculate U  V for U, V P A . Take R “ μ ⊗ Λ P E  ⊗ E  . Then xU  V, Ry “ xV, R · Uy “ xV, ηpUq ◦ pμ ⊗ Λqy “ xV, ηpUqpμq ⊗ Λy “ xVpΛq, ηpUqpμqy “ xPpUqpVpΛqq, μy “ xPpUq ◦ V, μ ⊗ Λy “ xPpUq ◦ V, Ry . This shows that U  V “ PpUq ◦ V

pU, V P BpE  qq .

(6.1.8)

In particular, the algebra pBpE  q,  q is not unital whenever P  IBpE  q , as is the case when E “ c 0 ; essentially by Corollary 2.3.67, this implies that A  A · A , as noted in [220, Example 2.5].

6.1 Topological centres

373

The following calculation of the left and right topological centres, from [57, Example 6.2], is an elaboration of [220, Example 2.5]. Example 6.1.30 Let E be a non-zero Banach space such that E  has the approximation property and the Radon–Nikodým property. As above, set A “ KpEq. It is clear from (6.1.7) and (6.1.8) that   pq Zt pA q “ U P BpE  q : U ◦ V “ PpUq ◦ V pV P BpE  qq , and so

  pq Zt pA q “ U P BpE  q : U “ PpUq “ BpE  qa .

Similarly, we see that  prq Zt pA q “ U P BpE  q : V ◦ U “ PpVq ◦ U

 pV P BpE  qq .

Of course, this shows that A is Arens regular when E is reflexive. Now suppose, further, that E is not reflexive. For each Λ P E  \ E, there exists V P BpE  q such that V | E “ 0 and VpΛq “ Λ, and so   prq Zt pAq “ U P BpE  q : UpE  q Ă E . pq

prq

prq

Clearly IE  P Zt pAq \ Zt pAq, and so ZpA , l q  Zt pA q. On the other hand, take x0 P E • and Γ P E  with Γ | E “ 0 and Γ  0, and then define UpΛq “ xΛ, Γy x0 pΛ P E  q. Then U P BpE  q • and UpE  q Ă C x0 Ă E, but prq pq U | E “ 0, and so PpUq “ 0, whence PpUq  U. Thus U P Zt pA q \ Zt pA q. We conclude that pq

prq

Zt pA q  Zt pA q and

prq

pq

Zt pA q  Zt pA q ,

and so the two topological centres of A are different. In particular, pq

A  Zt pA q  A

and

prq

A  Zt pA q  A ,

and so A is neither Arens regular nor strongly Arens irregular. pq prq We shall identify Zt pA q ∩ Zt pA q, which we temporarily denote by Z. pq Take T P WpEq. Clearly T  P Zt pA q. Also T  pE  q Ă E, and this implies prq that T  P Zt pA q. Hence T P Z, and so WpEqaa Ă Z.  q is σpE  , E  q-compact. Now take U P Z. Since U P BpE  qa , the set UpEr1s prq

  However UpEr1s q Ă E because U P Zt pAq, and so UpEr1s q is σpE, E  q-compact in    E, and hence σpE , E q-compact in E . This shows that U P WpE  q. Set U “ R , where R P BpE  q. Then R P WpE  q.

374

6 Arens regularity

We now claim that R P BpEqa . For this, it follows from Proposition 1.2.18, that  that converges to λ it suffices to show that Rλα Ñ Rλ whenever pλα q is a net in Er1s   in pE , σpE , Eqq. Since R is weakly compact, we may suppose that pRλα q is weakly convergent, say with limit μ P E  . Take x P E, and set y “ R x P E  . Then, in fact, y P E because R pE  q Ă E. Thus xx, Rλα y “ xy, λα y Ñ xy, λy “ xx, Rλy , and so Rλ “ μ. Thus the claim is established. It follows that R P WpEqa , and hence U P WpEqaa . We conclude that pq

prq

WpEqaa “ Zt pA q ∩ Zt pA q .

(6.1.9)

For example, take E “ c0 , so that E  has the Schur property. By Proposition pq prq 1.3.56, WpEq “ KpEq, and so κA pAq “ Zt pA q ∩ Zt pA q for A “ KpEq.

6.2 Arens regularity of projective tensor products p B is Arens Here we shall discuss the question when the projective tensor product A ⊗ regular when A and B are Banach algebras that are both Arens regular. We shall start by showing that the Varopoulos algebra VpΓq is not Arens regular for each infinite, compact group Γ; eventually, in Theorem 6.2.18, we shall give conditions on compact spaces K and L that determine when VpK, Lq is Arens regular. At the p B is Arens regular when A and B are end of the section, we shall consider when A ⊗ C ∗ -algebras. Let A and B be Banach algebras that are Arens regular. In Corollary 6.2.7, we p B is Arens regular whenever BpA, B q “ KpA, B q, and in Theoshall show that A ⊗ p B to be Arens regular. rem 6.2.16 that sometimes this is a necessary condition for A ⊗ We begin this section with a striking example. Let Γ be a compact abelian group. By Theorem 4.2.24, we can regard ApΓq as a closed subalgebra of the Varopoulos p CpΓq. Since, by Proposition 4.2.5, ApΓq is not Arens regular algebra VpΓq “ CpΓq ⊗ whenever Γ is infinite, we obtain the following result. Proposition 6.2.1 Let Γ be an infinite, compact abelian group. Then VpΓq is not Arens regular. p B. We recall that the dual space Let A and B be Banach algebras, and set A “ A ⊗  A is identified with BpA, B q as in equation (1.4.6) and that, by Theorem 2.3.29, A is 

6.2 Arens regularity of projective tensor products

375

Arens regular if and only if WAPpAq “ A . For T P BpA, B q, the map RT : A Ñ A is defined by RT pa ⊗ bq “ Lb ◦ T ◦ La pa P A, b P Bq . p Bqr1s . Further, We also recall from equation (1.4.2) that copAr1s ⊗ a Br1s q “ pA ⊗ by the Krein–Šmulian Theorem, Theorem 1.2.24(iv), aco K is weakly compact in a Banach space whenever K is a weakly compact subset. Thus we obtain the following condition for the Arens regularity of A. p B is Arens regular if Proposition 6.2.2 Let A and B be Banach algebras. Then A ⊗ and only if the set {Lb ◦ T ◦ La : a P Ar1s , b P Br1s } is relatively weakly compact in BpA, B q for each T P BpA, B q. The Arens regularity of projective tensor products was studied by Ülger in [308], and the criterion in Theorem 6.2.4 is [308, Theorem 3.4]. Definition 6.2.3 Let A and B be Banach algebras. A bounded bilinear functional m : A × B Ñ C is biregular if lim lim mpa1, j a2,k , b1, j b2,k q “ lim lim mpa1, j a2,k , b1, j b2,k q

jÑ8 kÑ8

kÑ8 jÑ8

whenever pa1, j q and pa2,k q are sequences in Ar1s and pb1, j q and pb2,k q are sequences in Br1s such that both repeated limits exist. p B is Arens regular if and Theorem 6.2.4 Let A and B be Banach algebras. Then A ⊗ only if every bounded bilinear functional on A × B is biregular. Proof This follows by applying the criterion for Arens regularity given in Corollary 2.3.31 in the case where L “ Ar1s ⊗ a Br1s . The following result is contained in [228] and [308]. p B is Arens regular Proposition 6.2.5 Let A and B be Banach algebras such that A ⊗ and B 2  {0}. Then A is Arens regular. Proof There exist b1 , b2 P B such that b1 b2  0, and then there exists μ P B such that xb1 b2 , μy “ 1. Take λ P A , and define τλ on A × B by τλ pa, bq “ xa, λy xb, μy pa P A, b P Bq , pB so that τλ is a bounded bilinear functional, and hence τλ is biregular because A ⊗ is Arens regular. Since τλ ppa1 ⊗ b1 qpa2 ⊗ b2 qq “ τλ pa1 a2 ⊗ b1 b2 q “ xa1 a2 , λy pa1 , a2 P Aq ,

376

6 Arens regularity

it follows that λ satisfies the iterated=limit condition of Definition 2.3.28. Hence A is Arens regular by Corollary 2.3.31. Clearly it will be useful to have a test that shows that all bounded bilinear functionals on A × B are biregular. Let A and B be Banach algebras. Recall that a bounded bilinear functional m on p B, and hence, by equation A × B corresponds to a bounded linear functional on A ⊗ (1.4.6), to an operator T m P BpA, B q such that xb, T m ay “ mpa, bq

pa P A, b P Bq .

The following result is based on a theorem of Ülger [308, Theorem 4.5]. Theorem 6.2.6 Let A and B be two Arens regular Banach algebras, and take a bounded bilinear functional m on A × B. Suppose that the operators LN ◦ T m ◦ L M and RN ◦ T m ◦ R M are both compact operators in BpA , B q for each M P A and N P B . Then m is biregular. Proof Consider sequences pa1, j q and pa2,k q in Ar1s and pb1, j q and pb2,k q in Br1s such that the two iterated limits L1 :“ lim lim mpa1, j a2,k , b1, j b2,k q jÑ8 kÑ8

and

L2 :“ lim lim mpa1, j a2,k , b1, j b2,k q kÑ8 jÑ8

both exist. Take respective subnets pa1,α q, pa2,β q in Ar1s and pb1,α q, pb2,β q in Br1s that all converge weak-∗, say to M1 , M2 P Ar1s and N1 , N2 P Br1s , respectively. Now mpa1,α a2,β , b1,α b2,β q “ xb2,β , pLb 1,α ◦ T m ◦ La1,α qpa2,β qy . For each index α, the net ppLb 1,α ◦ T m ◦ La1,α qpa2,β qq, indexed by β, converges weak-∗ in B to pLb 1,α ◦ T m ◦ La1,α qpM2 q. Since Lb 1,α ◦ T m ◦ La1,α is compact, this net is contained in a compact subset of B ; it follows that the net converges in norm to its weak-∗ limit, and so lim mpa1,α a2,β , b1,α b2,β q “ xN2 , pLb 1,α ◦ T m ◦ La1,α qpM2 qy β

“ xb1,α , pRN2 ◦ T m ◦ R M2 qpa1,α qy . Since the operator RN2 ◦ T m ◦ R M2 is compact, the net pRN2 ◦ T m ◦ R M2 qpa1,α qq is also contained in a compact subset of B , and so it converges in norm to its weak-∗ limit, pRN2 ◦ T m ◦ R M2 qpM1 q. Thus lim lim mpa1,α a2,β , b1,α b2,β q “ xN1 l N2 , T m pM1 l M2 qy . α

β

Similarly, lim lim mpa1,α a2,β , b1,α b2,β q “ xN1  N2 , T m pM1  M2 qy . β

α

6.2 Arens regularity of projective tensor products

377

Since A and B are Arens regular, it follows that L1 “ L2 , as required. Corollary 6.2.7 Let A and B be Arens regular Banach algebras. p B is Arens regular. (i) Suppose that BpA, B q “ KpA, B q. Then A ⊗ p B is Arens regular. In par(ii) Suppose that A is a compact algebra. Then A ⊗ ticular, this holds when A is reflexive and compact. Proof By Theorem 6.2.6, in both cases each bounded bilinear functional on A × B is biregular, and so the results follow from Theorem 6.2.4. Example 6.2.8 (i) Take p with 1 ď p < 8, and set A “ p p , · q, so that A is a a Tauberian Banach sequence algebra that is a dual Banach algebra. By Corollary 3.2.4, A is Arens regular and A is compact. Let B be an Arens regular Banach p B is also Arens regular. In particular, the Banach algebra. By Corollary 6.2.7(ii), A ⊗ p  q is Arens regular whenever 1 ď p, q < 8, as noted in sequence algebra  p ⊗ Example 3.3.6(i). (ii) Let G be a compact group, and take p with 1 < p < 8, and then set A “ pL p pGq,  q. Then, using Theorem 4.1.36, A “ A is compact; certainly A is Arens p B is regular. Let B be an Arens regular Banach algebra. By Corollary 6.2.7(ii), A ⊗ also Arens regular. Corollary 6.2.9 Let A be a Banach algebra that has property pVq as a Banach space, and let B be a Banach algebra such that B has the Schur property. Then p B is Arens regular. A⊗ Proof By Theorem 2.3.48, A and B are Arens regular. Also, by Theorem 1.3.57, p B is Arens regular. BpA, B q “ KpA, B q, and so, by Corollary 6.2.7(i), A ⊗ Corollary 6.2.10 Let A and B be C ∗ -algebra such that B has the Schur property. p B is Arens regular. In particular, A ⊗ p C 0 pLq is Arens regular when L is a Then A ⊗ scattered locally compact space. Proof By Theorem 2.2.12, every C ∗ -algebra has property pVq. By Corollary 6.2.9, p B is Arens regular. By Corollary 1.3.71, C 0 pLq has the Schur property when L A⊗ is scattered. p B is Arens regular whenever A is a uniform algebra that has propSimilarly, A ⊗ erty pVq; this class of uniform algebras was mentioned on page 37, and includes the disc algebra. Corollary 6.2.11 Let K and L be non-empty, locally compact spaces such that either K or L is scattered. Then VpK, Lq is Arens regular.

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6 Arens regularity

p B is Arens regular. SupTheorem 6.2.12 Let A and B be C ∗ -algebras such that A ⊗ p B is also Arens regular. pose that A0 is a C ∗ -subalgebra of A. Then A0 ⊗ p B is Arens regular, it follows from Proposition 6.2.2 that the set Proof Since A ⊗ {Lb ◦ S ◦ La : a P Ar1s , b P Br1s } is relatively weakly compact in BpA, B q for each S P BpA, B q. Consider the map P : S Þ→ S | A0 ,

BpA, B q Ñ BpA0 , B q .

Then P is a bounded operator, and P is a surjection by Theorem 2.2.20. Now take T P BpA0 , B q, say T “ PpS q, where S P BpA, B q. Then {Lb ◦ T ◦ La : a P pA0 qr1s , b P Br1s } Ă Pp{{Lb ◦ S ◦ La : a P Ar1s , b P Br1s } , and so {Lb ◦ T ◦ La : a P pA0 qr1s , b P Br1s } is relatively weakly compact in BpA0 , B q. p B is Arens regular. By Proposition 6.2.2, the algebra A0 ⊗ One of the most useful results to give information about the Arens regularity of the projective tensor product of two Arens regular Banach algebras A and B is the p B to be Arens regular. following theorem, which gives a necessary condition for A ⊗ Recall from Definition 2.1.33 that HT “ {T a · b : a P Ar1s , b P Br1s } for T P BpA, B q. p B is Arens Theorem 6.2.13 Let A and B be non-zero Banach algebras such that A ⊗ regular and A has a bounded approximate identity. Then BpA, B q “ WpA, B q. Proof Take T P BpA, B q. By Proposition 6.2.2, the set T 0 :“ {Lb ◦ T ◦ La : a P Ar1s , b P Br1s } is relatively weakly compact in BpA, B q. By Proposition 2.3.66(ii), A has a mixed identity, say E, in A . Define τ : S Þ→ S  pEq ,

BpA, B q Ñ B .

Then τ is a bounded linear operator. Further, for each a P A and b P B, we have τpLb ◦ T ◦ La q “ pLb ◦ T ◦ La q pEq “ pLb ◦ T  qpaq “ Lb pT aq “ T a · b by equation (2.1.6). Clearly HT “ τpT 0 q, and so HT is relatively weakly compact in B . By Proposition 2.1.34(i), BpA, B q “ WpA, B q. Corollary 6.2.14 Let A and B be non-zero Banach algebras such that A and B both have a bounded approximate identity. Suppose that A and B are not reflexive and p B is not Arens regular. that either A or B contains a copy of c 0 . Then A ⊗

6.2 Arens regularity of projective tensor products

379

p B and B ⊗ p A are isomorphic, we may suppose Proof Since the Banach algebras A ⊗ that A contains a copy of c 0 . By Theorem 1.3.34, (b) ⇒ (a), A contains a complemented copy of  1 . Since  B is not reflexive, B contains a non-reflexive, separable, closed linear subspace, say E. By Proposition 1.3.15(ii), there is a surjection T P Bp 1 , Eq; by Proposition 1.3.43(ii), T is not weakly compact. Since  1 is complemented in A, there is an extension of T to an operator Tr P BpA, B q, and Tr is not weakly compact. By p B is not Arens regular. Theorem 6.2.13, A ⊗ Corollary 6.2.14 allows us to exhibit Banach algebras A and B that are both Arens p B is not Arens regular. The example is given in [307]. regular, but such that A ⊗ Example 6.2.15 Let A “  8 and B “ p 1 q , the unitization of  1 , so that both A and B are unital Banach function algebras, neither is a reflexive space, and B contains a copy of c 0 . It follows from Example 3.2.7 that both A and B are Arens regular, but p B is not Arens regular by Corollary 6.2.14. A⊗

In Corollary 6.2.7(i), we showed that a sufficient condition for the Arens regularp B (given that A and B are Arens regular) is that BpA, B q “ KpA, B q. We ity of A ⊗ shall now show that, under some extra conditions on A and B, this condition is also necessary. Theorem 6.2.16 Let A and B be non-zero Arens regular Banach algebras such that each of A and B has a bounded approximate identity and such that the operators p B is Lb ◦ T ◦ La are compact for each a P A, b P B, and T P BpA, B q. Then A ⊗   Arens regular if and only if BpA, B q “ KpA, B q. p B. By Proposition 2.1.38, A has a BAI. Proof Set A “ A ⊗ By Corollary 6.2.7(i), A is Arens regular whenever BpA, B q “ KpA, B q. Now suppose that A is Arens regular, and take T P BpA, B q, so that T corresponds to an element of A . By Theorem 2.3.44(ii), A “ A · A , and so there exist  c P A and S P BpA, B q such that T “ c · S in A , say c “ 8 i“1 ai ⊗ bi , where ai P A 8 and bi P B for i P N and i“1 ai  bi  < 8. Since    Lbi ◦ S ◦ Lai op ď ai  bi  S  pi P Nq , we have T “c · S “

8

Lb i ◦ S ◦ Lai

in BpA, B q .

i“1

Lb i

◦ S ◦ Lai is compact by hypothesis, and so T is For each i P N, the operator compact. We have shown that BpA, B q “ KpA, B q.

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6 Arens regularity

p B is Arens regular We shall now consider when the projective tensor product A ⊗ whenever A and B are C ∗ -algebras, or closed subalgebras of C ∗ -algebras. In particular, let K and L be non-empty, compact spaces. Then we shall determine when the Varopoulos algebra VpK, Lq is Arens regular. We have shown in Corollary 6.2.11 that VpK, Lq is Arens regular whenever K or L is scattered; we seek the converse, and, later, a generalization to arbitrary C ∗ -algebras. Recall from Corollary 1.3.71 that a compact space K is scattered if and only if MpKq “ CpKq has the Schur property. We first prove a special case of the result that we seek; it is contained in the thesis of Ljeskovac [228]. Theorem 6.2.17 Let K and L be two non-empty, compact, metric spaces such that VpK, Lq is Arens regular. Then K or L is scattered. Proof Assume to the contrary that neither K nor L is scattered. By Proposition 1.1.10, each of K and L contains a closed subspace that is homeomorphic to the Cantor set Δ. By the Tietze extension theorem, Theorem 2.1.15, the restriction maps f Ñ f | Δ are Banach-algebra surjections from CpKq and CpLq, respectively, onto CpΔq, and so there is a Banach-algebra surjection from VpK, Lq onto VpΔq. Since VpK, Lq is Arens regular, the algebra VpΔq is also Arens regular, a contradiction of Proposition 6.2.1 because Δ is an infinite, compact abelian group. Thus either K or L is scattered. We now give the general case of Theorem 6.2.17; this was first proved by Neufang in [250]. Theorem 6.2.18 Let K and L be two non-empty, compact spaces. Then VpK, Lq is Arens regular if and only if K or L is scattered. Proof By Corollary 6.2.11, VpK, Lq is Arens regular whenever either K or L is scattered. Now assume that VpK, Lq is Arens regular, but that neither K nor L is scattered. By Proposition 1.1.9, there are continuous surjections from K and L onto I. By Theorem 3.3.17, VpIq is isometrically Banach-algebra isomorphic to a closed subalgebra of VpK, Lq, and so VpIq is Arens regular, a contradiction of Theorem 6.2.17. Thus either K or L is scattered. p c 0 ,  · π q “ Vpc 0 q. Then A is a Tauberian Banach Example 6.2.19 Set A “ pc 0 ⊗ sequence algebra, has a bounded approximate identity, is an ideal in its bidual, and A is Arens regular by Proposition 6.2.11. By Corollary 2.4.9, A is a dual Banach algebra. By Corollary 5.5.7, LpAq “ A and  f BSE “  f π p f P Aq; see also Theorem 5.3.40. Further A is a Banach function algebra that also has a BSE norm. By Proposition 1.3.66,  8 has the bounded approximation property, and so, by Proposition 2.1.37, the algebra p Cpβ Nq p  8 “ Cpβ Nq ⊗ 8 ⊗

6.2 Arens regularity of projective tensor products

381

is a (proper) closed subalgebra of A . By Proposition 3.3.2, this algebra is a natural Banach function algebra on β N × β N. Since β N is not scattered, it follows from Theorem 6.2.18 that Vpβ Nq is not Arens regular, and so A is not Arens regular.

The following, more general, theorem is proved by Neufang in [250] by a very different approach; it uses the language of various Haagerup tensor products. These tensor products that are defined for general ‘operator spaces’ and, in particular, for C ∗ -algebras and for their dual spaces. For a full description of operator spaces and these tensor products, see the texts [20, §§1.5, 1.6], [93, Chapter 9], [263], and [265]. Theorem 6.2.20 Let K and L be two non-empty, locally compact spaces, and set V “ VpK, Lq. Then V is Arens regular if and only if either K or L is scattered. Further, the centre ZpV  q of V  is Banach-algebra isomorphic to r ⊗w∗ h CpLq r “ CpKq r ⊗eh CpLq r . CpKq ⊗w∗ h CpLq “ CpKq Note that the two uniform algebras ApDq and H 8 pDq each satisfy at least one of the conditions that A or B needs to satisfy in the following theorem. Theorem 6.2.21 Let K and L be two non-empty, compact spaces, and let A and B be uniform algebras on K and L, respectively. Suppose that each of K and L is either uncountable and metrizable or contains an interpolating sequence for A or p B is not Arens regular. B, respectively. Then A ⊗ Proof Suppose that K is uncountable and metrizable. Then it follows from Theorem 3.6.3 that there is a continuous surjection R : A Ñ CpΔq. Suppose that K contains an interpolating sequence for A. Then there is a continuous surjection R : A Ñ Cpβ Nq. A similar remark applies to the space L, with a continuous surjection S . p B Ñ VpK0 , L0 q is a continuous surjection, where K0 and L0 are Then R ⊗ S : A ⊗ compact spaces that are not scattered. By Theorem 6.2.18, VpK0 , L0 q is not Arens p B is not Arens regular. regular, and so A ⊗ We shall conclude this section by considering analogous results that determine p B is Arens regular when A and B are general C ∗ -algebras. For a recent when A ⊗ monograph on the projective tensor product of C ∗ -algebras, see [265]. p H  q “ BpHq, as in §2.2, and We recall that, for a Hilbert space H, we have pH ⊗ that a closed linear subspace B of BpHq is norming if



p Hq ; z “ sup{

xz, T y

: T P Br1s } pz P H ⊗ p H  in B is isometric by Corollary 1.3.19. in this latter case, the embedding of H ⊗ Theorem 6.2.22 Let A be a C ∗ -algebra, and let B be a C ∗ -subalgebra of BpHq for p B is Arens an infinite-dimensional Hilbert space H such that B is norming. Then A ⊗ regular if and only if A has the Schur property.

382

6 Arens regularity

p B is Arens regular by Proof Suppose that A has the Schur property. Then A ⊗ Corollary 6.2.10. p B is Arens regular, and assume towards a contradiction Now suppose that A ⊗ that A does not have the Schur property. By Theorem 2.2.25, there is a quotient 2 . map θ : A Ñ  2 ; we may suppose that θpAr1s q ⊃ r1s Since H is infinite dimensional, we can regard  2 as a closed, complemented p H  . Since B p H  is a closed linear subspace of H ⊗ linear subspace of H, and so  2 ⊗ 2p    is norming,  ⊗ H is also a closed linear subspace of B . Take λ P H with λ “ 1, and define the map T : a Þ→ θpaq ⊗ λ ,

p H  Ă B , A Ñ 2 ⊗

so that T P BpA, B q. For each R, S P B, we have xS , T a · Ry “ xR ◦ S , T ay “ xR ◦ S , θpaq ⊗ λy “ xS  pR pλqq, θpaqy “ xS , θpaq ⊗ R pλqy , and so T a · R “ θpaq ⊗ R pλq for each R P B. Hence rr1s pλq , HT “ {θpaq ⊗ R pλq : a P Ar1s , R P Br1s } “ θpAr1s q ⊗a B where we are using the notation of Definitions 1.4.6 and 2.1.33. By Proposition rr1s pλq is dense in H  , and so the closure of HT in H ⊗ p H  contains the set 1.4.7, B r1s p a H  . But the space  2 ⊗ p H  is not a reflexive space, so that HT is not relatively 2 ⊗ r1s

r1s

p H  , and hence HT is not relatively weakly compact in B , a weakly compact in  2 ⊗ contradiction of Theorem 6.2.13. Thus A has the Schur property. In fact, the argument of the above theorem also shows the following; we use the fact that  2 is a complemented linear subspace of L p pIq. Proposition 6.2.23 Let A be a C ∗ -algebra, and let B be a closed subalgebra of BpL p pIqq, where 1 < p < 8, such that B is norming and B is weakly sequentially p B is Arens regular if and only if A has the Schur property. complete. Then A ⊗ p B is Arens Theorem 6.2.24 Let A and B be two unital C ∗ -algebras such that A ⊗   regular. Then A or B has the Schur property. Proof First, assume towards a contradiction that neither A nor B is scattered. By Theorem 2.2.22, (b) ⇒ (a), there are separable, commutative C ∗ -subalgebras A0 and B0 of A and B, respectively, such that neither A0 nor B0 is scattered. By Theorem p B0 is also Arens regular, a contradiction of Theorem 6.2.18. Thus either 6.2.12, A0 ⊗ A or B is a scattered C ∗ -algebra, say B is scattered. Suppose that B does not have the Schur property. Then, by Theorem 2.2.24, B does not have the Dunford–Pettis property. By Proposition 2.2.29, B has an infinite dimensional, irreducible representation, say pπ, Hq. By Proposition 2.2.28, B is

6.3 Biduals of semigroup algebras

383

p B is Arens regupostliminal, and so πpBq ⊃ KpHq and πpBq is norming. Since A ⊗ p πpBq is Arens regular. By Theorem 6.2.22, A has the Schur property, lar, also A ⊗ giving the result. The equivalence of at least conditions (a), (b), and (c) in the following theorem was first given by Neufang in [250], building on a theorem of Kumar and Rajpal [205]. Theorem 6.2.25 Let A and B be unital C ∗ -algebras. Then the following conditions are equivalent: p B is Arens regular; (a) A ⊗ (b) A or B is scattered and has the Dunford–Pettis property; (c) A or B has the Schur property; (d) for each T P BpA, B q, the set HT is relatively compact in B ; (e) BpA, B q “ KpA, B q ; p Bq is weakly sequentially complete; (f) pA ⊗ p B has property (V). (g) the Banach space A ⊗ Proof First, (b) ⇔ (c) by Theorem 2.2.24. We have (c) ⇒ (a) by Corollary 6.2.10, and (a) ⇒ (c) by Theorem 6.2.24. Further, (c) ⇒ (e) by Theorem 1.3.57, and then (c) ⇒ (d) by Proposition 2.1.34(iii). We have (d) ⇒ (e) by Proposition 2.1.34(ii), (e) ⇒ (a) by Corollary 6.2.7(i), (c) ⇒ (f) by Proposition 1.4.8, and (f) ⇒ (c) by Corollary 2.2.26. Finally, (g) ⇒ (a) by Theorem 2.3.48, and (e) ⇒ (g) by Theorem 1.4.9, noting that, because each C ∗ -algebra has property pVq by Theorem 2.2.12, we have BpA, B q “ WpA, B q by Corollary 1.3.52.

6.3 Biduals of semigroup algebras In this section, we shall consider the semigroup algebra p 1 pS q,  · 1 ,  q associated with a semigroup S , and the related weighted semigroup algebra  1 pS , ωq, where ω is a weight on S . These algebras were introduced in Example 2.1.13, (iv) and (v), and were mentioned in Examples 2.1.19, 2.1.25, 3.1.7, 3.2.12, and 3.5.5. In certain cases, we shall determine the biduals of these algebras, and consider when the algebras are Arens regular or strongly Arens irregular or neither, and sometimes show that there are small DLTC sets; we shall give various examples. The results are mainly based on [52, 57, 58, 62]. Let S be a semigroup. We start by defining two products l and  on the Stone– ˇ Cech compactification β S of S such that pβ S , l q and pβ S ,  q are also semigroups, and relating these products to those in the bidual of  1 pS q. Of course, the semigroups

384

6 Arens regularity

pβ S , l q and pβ S ,  q are very well-known; they are the main topic of the excellent monograph of Hindman and Strauss [172]. We identify the semigroup S as a subset of β S . For each s P S , the map L s : t Þ→ st ,

S Ñ S Ă βS ,

has a continuous extension to a map L s : βS Ñ βS . For each v P β S , define s l v “ L s pvq. Next, each map Rv : s Þ→ s l v, S Ñ βS , has a continuous extension to a map Rv : β S Ñ β S . For u, v P β S , set u l v “ Rv puq . Then l is a binary operation on β S × β S , and the restriction of l to S × S is the original semigroup product. Further, for each v P β S , the map Rv : β S Ñ β S is continuous, and, for each s P S , the map L s : β S Ñ β S is continuous. Similarly, we can define a binary operation  on β S . For u, v P β S , we see that u

l

v “ lim lim sα tβ , α

β

u  v “ lim lim sα tβ β

α

(6.3.1)

whenever psα q and ptβ q are nets in S with limα sα “ u and limβ tβ “ v. We shall soon see that the notations l and  for products on β S are consistent with our previous usage of these symbols. In the case where S is an abelian semigroup, we have u

l

v “ vu

pu, v P βS q .

It is immediately checked that both l and  are associative operations on β S . Thus we obtain the following fundamental result; we recall that topological semigroups were defined on page 12. Theorem 6.3.1 Let S be a semigroup. Then pβ S , containing S as a subsemigroup. Further:

lq

and pβ S ,  q are semigroups

(i) for each v P β S , the map Rv : u Þ→ u l v is continuous, and pβ S , compact, right topological semigroup;

lq

is a

(ii) for each s P S , the map L s : v Þ→ s l v is continuous. The semigroup maps l and  that we have defined here agree with the semigroup maps defined in several different ways in [172]. From now on, we shall generally discuss the semigroup pβ S , l q; of course, analogous results hold for the compact, left topological semigroup pβ S ,  q. In fact, pβ S , l q is the largest compactification of S which is a compact, right topological semigroup, in the sense that any other such compactification is a continuous homomorphic image of pβ S , l q [172, Theorem 4.8]. The points of β S have been identified with ultrafilters on S , and so the product l can be defined in these terms; see [172, Theorem 4.15]. Note that, in the case where S “ pN, + q and u, v P β N, the element u l v is denoted by u + v in [172]; however, usually u + v  v + u.

6.3 Biduals of semigroup algebras

385

We now consider the analogues for semigroups of the topological centres for biduals of Banach algebras that were defined in Definition 6.1.1. The following concept is well-known; see [172] and [218], for example. Definition 6.3.2 Let S be a semigroup. The left and right topological centres, pq prq Zt pβ S q and Zt pβ S q, of β S are pq

Zt pβ S q “ {u P β S : u l v “ u  v pv P β S q} and

prq

Zt pβ S q “ {u P β S : v l u “ v  u pv P β S q} , pq

prq

respectively. The semigroup S is Arens regular if Zt pβ S q “ Zt pβ S q “ β S ; the pq semigroup S is left strongly Arens irregular if Zt pβ S q “ S , right strongly Arens prq irregular if Zt pβ S q “ S , and strongly Arens irregular if it is both left and right strongly Arens irregular. A subset V of β S is determining for the left topological centre (a DLTC set) of β S if u P S whenever u l v “ u  v pv P Vq. pq

Our set Zt pβ S q is equal to the topological centre Λpβ S q “ {u P β S : Lu is continuous on β S } , as defined in the book of Hindman and Strauss [172, Definition 2.4]. In the case where S is an abelian semigroup, we have pq

prq

Zt pβ S q “ Zt pβ S q “ Zpβ S q , the centre of both the semigroups pS , l q and pS ,  q. Again, we refer to a ‘DTC set’ when the semigroup S is abelian. Let S be a semigroup. We now consider the bidual of the semigroup algebra p 1 pS q,  q. Recall that the Banach space  1 pS q has been identified with Mpβ S q, the space of complex, regular Borel measures on β S , and that Mpβ S q “  1 pS q ‘1 MpS ∗ q . We identify a point u P β S with the corresponding point mass δu P Mpβ S q. Definition 6.3.3 Let S be a semigroup. Then pMpβ S q, l q and pMpβ S q,  q denote the space Mpβ S q taken with the two products l and  that are defined by identifying Mpβ S q with  1 pS q . Thus we have definitions of μ

l

ν

and

μν

for

μ, ν P Mpβ S q .

We shall usually write u l μ for δu l μ, etc., in the case where u P β S and μ P Mpβ S q.

386

6 Arens regularity

Note that the augmentation character on  1 pS q corresponds to the constant function 1S in  8 pS q, and the bidual of this character on Mpβ S q corresponds to the constant function 1 in Cpβ S q; indeed, this latter character is the map ϕS : μ Þ→ xμ, 1S y “ μpβ S q,

Mpβ S q Ñ C .

Take u, v P β S . Then there are nets psα q and ptβ q in S with limα sα “ u and limβ tβ “ v; by (6.3.1), u l v “ limα limβ sα tβ . For each f P Cpβ S q, we have limα f psα q “ f puq, and so wk∗ – limα δ sα “ δu in Mpβ S q, etc. Hence δu

l δv

“ lim lim δ sα  δtβ “ lim lim δ sα tβ “ δu l v α

α

β

β

where the limits are in the weak-∗ topology. Thus it is consistent to identify δu l δv with δu l v . Clearly pq

pq

pZt pMpβ S qq ∩ β S q Ă Zt pβ S q “ Λpβ S q ; we do not know an example where this inclusion is proper. Suppose that a subset V of β S is a DLTC set for Mpβ S q. Then V is also a DLTC set of β S . Suppose that S is Arens regular. Then also  1 pS q is Arens regular because  1 pβ S q is weak-∗ dense in Mpβ S q by Corollary 1.2.36. Let S be a weakly cancellative semigroup, set A “  1 pS q and E “ c0 pS q. Then, as in Example 2.4.30, we have

and also

pMpβ S q, l q “ pA , l q “ A  E K “  1 pS q  MpS ∗ q ,

(6.3.2)

p 1 pβ S q, l q “  1 pS q   1 pS ∗ q .

(6.3.3)

We shall now discuss the topological centres and DLTC sets of some examples. Let S be an infinite semigroup, and suppose that a subset V of S ∗ is such that μ “ 0 whenever μ P MpS ∗ q and μ l v “ μ  v for each v P V. Each element of Mpβ S q has the form f + μ, where f P  1 pS q and μ P MpS ∗ q. Suppose that p f + μq l v “ p f + μq  v for each v P V. Then μ “ 0 and so f + μ P  1 pS q, whence V is a DLTC set. A generalization of the following result was given by Young in [329]. Proposition 6.3.4 Let S be a semigroup. Then the semigroup algebra  1 pS q is not Arens regular if and only if there exist two sequences ps j q and ptk q in S such that {s j tk : j < k, j, k P N} ∩ {s j tk : k < j, j, k P N} “ ∅ ;

(6.3.4)

Proof Suppose that there are sequences ps j q and ptk q in S such that (6.3.4) holds. Then the argument of Example 2.3.52 shows that  1 pS q is not Arens regular. Conversely, suppose that  1 pS q is not Arens regular. Then there are sequences ps j q and ptk q in S , λ P  R8 pS q, and α < β in R such that lim jÑ8 limkÑ8 λps j tk q “ α and limkÑ8 lim jÑ8 λps j tk q “ β. Choose γ and δ with α < γ < δ < β, and set

6.3 Biduals of semigroup algebras

387

U “ {s P S : λpsq < γ} and V “ {s P S : λpsq > δ}. Then we may suppose that, for each j P N, we have s j tk P U for all but finitely-many k P N, and, for each k P N, we have s j tk P V for all but finitely-many j P N. By an easy induction, we can find subsequences ps j q and ptk q such that (6.3.4) holds. Example 6.3.5 (i) Let U be a non-empty set, and take an element o  U. Then set S “ U ∪ {o}, and define st “ o ps, t P S q, so that S is an abelian semigroup. It is easy to see from equation (2.3.1) that  1 pS q and S are Arens regular. (ii) Let S be a set such that |S | ě 2, and consider the right-zero semigroup on S . Then u l v “ uv “ v pu, v P β S q, and so S is Arens regular. Also we have μ

l

ν “ μ  ν “ xμ, 1y ν

pμ, ν P Mpβ S qq ,

and so  1 pS q is Arens regular, as first noted by Lau in [212]. The following two theorems are from [58, Chapter 12], where the results are given in greater generality, and some historical antecedents are recorded. Theorem 6.3.6 Let S be an infinite semigroup. Suppose that there exist subsets U and V of S and right cancellable elements a P U ∗ and b P V ∗ such that the following conditions are satisfied: (i) U ∩ V “ ∅; (ii) S l a Ă U and S l b Ă V; (iii) for each x P S ∗ , either px  S q ∩ U “ ∅ or px  S q ∩ V “ ∅. Let μ P MpS ∗ q with μ l a “ μ  a and μ l b “ μ  b. Then μ “ 0. Proof Assume towards a contradiction that μ  0. Then we may suppose that μ P M R pS ∗ q and that μ “ 1. Take ε P p0, 1/2q.   Take μ+ , μ´ P MpS ∗ q+ such that μ “ μ+ ´ μ´ and μ+  + μ´  “ 1. There are compact subsets K and L of S ∗ such that μ+ pKq > μ+  ´ ε and μ´ pLq > disjoint,  μ´  ´ ε. The sets K l a and L l a are disjoint because a is right cancellable, and so   μ l a “ μ+ l a ´ μ´ l a ě pμ+ l aqpK l aq + pμ´ l aqpL l aq “ μ+ pKq + μ´ pLq ě 1 ´ 2ε . Thus μ l a “ 1, and so there exists λa P Cpβ S qr1s with xμ Similarly, there exists λb P Cpβ S qr1s with xμ l b, λb y > 1 ´ ε. By (ii), β S l a Ă U and β S l b Ă V, and so supp pμ

l

aq Ă U

and

supp pμ

l

l

a, λa y > 1 ´ ε.

bq Ă V .

By (i), U ∩ V “ ∅. Thus, by replacing λa by λa · χU and λb by λb · χV , we may suppose that λa and λb vanish outside U and V, respectively.

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6 Arens regularity

 ∗ Take m P N, α1 , . . . , αm P C with m i“1 |αi | “ 1, and x1 , . . . , xm P S , and consider m 1 ∗ f “ i“1 αi xi P  pS qr1s . For each i P Nm , it follows from (iii) that either

xxi  s, λa y “ 0 ps P S q or

xxi  s, λb y “ 0

ps P S q .

Thus either



1

x f  s, λa y

ď 2

ps P S q or



1

x f  s, λb y

ď 2

ps P S q .

By Corollary 1.2.36,  1 pS ∗ qr1s is weak-∗ dense in MpS ∗ qr1s , and so either

1



xμ  s, λa y

ď 2

ps P S q or



1

xμ  s, λb y

ď 2

ps P S q .

x Þ→ μ  x, β S Ñ MpS ∗ q, is continuous, we may suppose that



Since the map

xμ  a, λa y

ď 1/2. Hence 1 ´ ε < xμ l a, λa y “ xμ  a, λa y ď 1/2, the required contradiction. Thus μ “ 0, as required. The following theorem is given for (more general) semigroups that are ‘weakly cancellative and nearly right cancellative’ in [58]; see also [115, Theorem 18]. Let S be an infinite semigroup with |S | “ κ; we may suppose for convenience that S has an identity eS . We enumerate S as a sequence psα : α < κq, where s0 “ eS . For s “ sα and t “ sβ in S , set s  t if α ď β in κ and set s ≺ t if α < β in κ. Thus pS , q is a totally ordered set. For t P S and a subset F of S , set  rts “ {s P S : s  t} , rFs “ {rts : t P F} . Theorem 6.3.7 Let S be an infinite, countable, cancellative semigroup. Then there exist a and b in S ∗ such that the two-point set {a, b} is determining for the left topological centre of Mpβ S q, and so the semigroup algebra  1 pS q and the semigroup S are strongly Arens irregular. Further, let P and Q be infinite subsets of S . Then we can require that a P P∗ and b P Q∗ . Proof We may suppose that S has an identity eS . We enumerate S as a sequence psn q, where s0 “ eS . We shall construct a sequence ptn q in S by induction. Set t0 “ s0 . Once t0 , . . . , tn have been defined for n P Z+ , set T n “ {s0 , . . . , sn , t1 , . . . , tn } . The sequence ptn q will satisfy the following three conditions for each n P Z+ : (i) sT n ∩ rT n s “ ∅ whenever s P S with tn+1  s; (ii) stn+1  ttn+1 whenever s, t P T n with s  t; (iii) rs ≺ ttn+1 whenever r, s, t P T n .

6.3 Biduals of semigroup algebras

389

Take n P Z+ , and assume that t0 , . . . , tn have been specified. Since S is cancellative,  the sets T n´1 rT n2 s and {{x P S : sx “ tx} : s, t P T n , s  t} are finite. We choose tn+1 P S so that tn+1 is strictly greater than the maximum of the union of these sets; the element tn+1 is such that the above conditions (i)–(iii) are satisfied. The inductive construction continues.  Note that sn  tn pn P Z+ q, and so {rtn s : n P Z+ } “ S . Define N : S Ñ Z+ by setting Npsq “ min{n P Z+ : s P rtn s}

ps P S q .

Suppose that Npsq “ m P N. It follows from (iii) that tn´1 ≺ stn ≺ tn+1 whenever n > m, and so (6.3.5) Npstn q P {n, n + 1} pn > mq . Now suppose that m ě 2 and that k ď m ´ 2. Then tm´2 ≺ ssk by (i) because sT m´2 ∩ rT m´2 s “ ∅. Further, ssk ≺ tm+1 by (iii) because s P T m . Thus Npssk q P {m ´ 1, m, m + 1}

pm ě k + 2q .

(6.3.6)

For s P S , set γpsq ≡ Npsq pmod 8q. Then γ : S Ñ Z 8 has a continuous extension, also denoted by γ, to a map γ : β S Ñ Z 8 . It follows from (6.3.6) that γpx

l

sq “ γpx  sq P {γpxq ´ 1, γpxq, γpxq + 1} Ă Z 8

for x P S ∗ and s P S . Set A “ {tn : γptn q “ 1} ,

(6.3.7)

B “ {tn : γptn q “ 5} ,

so that A and B are infinite subsets of S . Then set U “ {s P S : γpsq P {1, 2}} ,

V “ {s P S : γpsq P {5, 6}} ,

so that A Ă U, B Ă V, and U ∩ V “ ∅. Choose a P A∗ and b P B∗ , so that a, b P S ∗ . We claim that a is right cancellable in β S . Indeed, let u1 and u2 be distinct points of β S , and take disjoint subsets N1 and N2 of S such that u j P N j p j “ 1, 2q. For j “ 1, 2, set Y j “ {sm tn : sm P N j , tn P A, m < n} , so that u j l a P Y j . Take m1 , m2 , n1 , n2 P N with m1 < n1 , m2 < n2 , and m1  m2 . Then sm1 tn1  sm2 tn2 : this holds for n1 < n2 by (iii) and for n1 “ n2 by (ii). It follows that Y1 ∩ Y2 “ ∅, and so u1 l a  u2 l a, as required. Similarly, b is right cancellable in β S . It follows from (6.3.5) that S l a Ă U and S l b Ă V. Let x P S ∗ , and take s P S . Suppose that x  s P U. Then it follows from (6.3.7) that γpxq P {0, 1, 2, 3}. Suppose that x  s P V. Then, similarly, γpxq P {4, 5, 6, 7}. Thus either px  S q ∩ U “ ∅ or px  S q ∩ V “ ∅.

390

6 Arens regularity

We have shown that the hypotheses of Theorem 6.3.6 are satisfied, and so {a, b} is determining for the left topological centre of Mpβ S q, and hence  1 pS q is LSAI. In the case where P and Q are infinite subsets of S , we can choose the above sequence ptn q to have the extra property that tn P P when n ≡ 1 pmod 8q and tn P Q when n ≡ 5 pmod 8q. This ensures that a P P∗ and b P Q∗ . The semigroup S op satisfies the same conditions as S , and so  1 pS op q is LSAI. Hence  1 pS q is RSAI, and so is SAI. In particular, each subsemigroup of the group pR, + q has a two-point DTC set, and so is strongly Arens irregular. A small extension of the argument in the above proof shows that the above theorem applies without the requirement that S be countable; see [58, Theorem 12.15]. In particular, for each cancellative semigroup S , the semigroup algebra  1 pS q and S itself are strongly Arens irregular, so that each group is strongly Arens irregular. It is also shown in [58, Chapter 12] that the elements of a DLTC set for a semigroup algebra can be chosen to have some special algebraic properties. Example 6.3.8 We give an example, from [58, Example 6.12], of a semigroup S that is right strongly Arens irregular, but not left strongly Arens irregular. Let G be an infinite group with identity e, and set S “ G × G, with the product specified by the formula pa, xq · pb, yq “ pab, ayq

pa, b, x, y P Gq ,

so that S is a semigroup; it is left cancellative, but not weakly right cancellative. We claim that S is not left strongly Arens irregular. Indeed, take u P G∗ , say u “ limα sα , where psα q is a net in G. Then ppe, sα qq is a net in S , and it is convergent to an element of β S that we may denote by pe, uq. Since pe, sα q · pb, yq “ pb, yq for pq each α and each pb, yq P S , we see that pe, uq P Zt pβ S q. This gives the claim. We also claim that S is right strongly Arens irregular. For this we use the continuous surjection π : u Þ→ pπ1 puq, π2 puqq, β pG × Gq Ñ β G × β G, of equaprq tion (1.1.6). Take Q P Zt pβ S q. For each u P β G, we have pu, eq P β S , and so pu, eq l Q “ pu, eS q  Q. Thus u l π1 pQq “ u  π1 pQq and u l π2 pQq “ u  π2 pQq. prq It follows that both π1 pQq, π2 pQq P Zt pβGq. But G is strongly Arens irregular, and prq so Q P G × G “ S . Thus Zt pβ S q “ S , and hence S is right strongly Arens irregular, as claimed.

As noted in [58, Theorem 12.20], when one is looking for DLTC sets just for semigroups, it is sufficient to suppose that S is weakly cancellative, rather than cancellative, so we have the following theorem; the result also follows from the short argument by Pachl and Stepr¯ans in [252, Theorem 2.2]. Theorem 6.3.9 Let S be an infinite, weakly cancellative semigroup. Then there is a two-point subset of S ∗ that is determining for the left topological centre of β S , and so S is strongly Arens irregular.

6.3 Biduals of semigroup algebras

391

In the case where S is a weakly cancellative semigroup, the map Lv is continuous for v P β S if and only if v P S . However, Example 6.3.5 (ii) shows that this is not true for every semigroup. The following example shows that we cannot replace ‘cancellative’ by ‘weakly cancellative’ in the hypotheses of Theorem 6.3.7. Example 6.3.10 Let S “ N × {0, 1}, with the operation pm, iq · pn, jq “ pm + n, 0q

pm, n P N, i, j P {0, 1}q .

Then S is an abelian, countable, weakly cancellative semigroup. However S is not cancellative because p1, 0q  p1, 1q, but p1, 0q · x “ p1, 1q · x for each x P S . We see that β S is identified as a set with β N × {0, 1} and pu, iq

l

pv, jq “ pu

l

v, 0q, pu, iq  pv, jq “ pu  v, 0q ,

(6.3.8)

for each u, v P β N and i, j P {0, 1}. Further, Mpβ S q can be identified with the space Mpβ Nq × {0, 1}, and pμ, iq

l

pν, jq “ pμ

l

ν, 0q, pμ, iq  pν, jq “ pu  v, 0q

for each μ, ν P Mpβ Nq and i, j P {0, 1}. Now choose u P N∗ . For each pν, jq P β S , we have ppu, 0q ´ pu, 1qq

l

pν, jq “ pu

l

ν, 0q ´ pu

l

ν, 0q “ p0, 0q pq

and, similarly, ppu, 0q´pu, 1qq  pν, jq “ p0, 0q. Thus pu, 0q´pu, 1q P Zt pMpβ S qq. Clearly pu, 0q ´ pu, 1q   1 pS q, and so  1 pS q is not strongly Arens irregular. Since pN, + q is a subsemigroup of S , so that  1 pNq is a closed subalgebra of  1 pS q, it is clear that  1 pS q is not Arens regular. Since the semigroup pN, + q is strongly Arens irregular, it is immediate from equation (6.3.8) that the semigroup S is also strongly Arens irregular.

We now turn to some results concerning totally ordered semigroups, based on [62]. Now T will denote an infinite, totally ordered space. As in equation (1.1.1), we set s ∧ t “ min{s, t} and s ∨ t “ max{s, t} for s, t P T , so that T is a lattice and a semigroup with respect to the operations ∧ and ∨. Thus T is an abelian, idempotent semigroup with respect to both these operations; in the present theory, we shall just consider the operation ∧. We further suppose that T has a minimum element, called 0, and a maximum element, called 8, and that T is complete, so that every nonempty subset of T has a supremum and an infimum. We give T its interval topology, so that the closed intervals provide a subbase for the closed sets, and the intervals of the form pa, bq, r0, aq, pa, 8s, and r0, 8s are a subbase for the open sets of T , and we shall refer to them as open intervals. The space T is then a compact topological semigroup.

392

6 Arens regularity

We shall denote by S an arbitrary, infinite subset of T , so that S is a subsemigroup of pT, ∧q that is also an abelian, idempotent semigroup. As in Example 2.1.19, we identify p 1 pS q,  q with the Banach function algebra ApΦS q. For subsets A and B of S , we shall write A ď B if s ď t ps P A, t P Bq. The set S with the discrete topology is denoted by S d , and we set X “ β S d , so that X ∗ “ X\S . The closures of a subset A of S in T and X are clT A and clX A, respectively. The map π : X Ñ T denotes the continuous extension of the inclusion map of S into T , so that πpXq “ clT S . We shall write Ft for the fibre {x P X : πpxq “ t} for t P clT S . We set Ft∗ “ Ft ∩ X ∗ , so that Ft∗ “ Ft

pt P T \ S q and

Ft∗ “ Ft \ {t} pt P S q .

We recall that, for every subset A of S , the set clX A is clopen in X, and that, for every subsets A and B of S that are disjoint, the two sets clX A and clX B are disjoint in X. We let E denote the set of accumulation points of S in T , so that E  ∅. Take t P T . Then Ft∗ is a closed, and hence compact, subset of X, and clearly Ft∗ Ø if and only if t P E. The augmentation character on pMpβ S q, l q is the map  dμ . ϕ0 : μ Þ→ x1β S , μy “ μpβ S q “ βS

Clearly ϕ0 pμ l νq “ ϕ0 pμqϕ0 pνq pμ, ν P Mpβ S qq, so ϕ0 is indeed a character on pMpβ S q, l q, and ϕ0 is weak-∗ continuous. Take t P E. Throughout we shall write At “ S ∩ r0, tq

and

Bt “ S ∩ pt, 8s ;

at least one of these sets is non-empty. Hence the sets Ft∗ ∩ clX At and Ft∗ ∩ clX Bt are disjoint, compact subsets of X whose union is Ft∗ . Lemma 6.3.11 (i) Take a subset A of S , and suppose that μ P MpXqr1s with supp μ Ă clX A. Then the measure μ belongs to the weak-∗ closure of aco{δ s : s P A}. (ii) Suppose that A and B are subsets of S such that A ď B. Take μ, ν P MpXq with supp μ Ă clX A and supp ν Ă clX B. Then μ l ν “ ν l μ “ ϕ0 pνqμ . (iii) Take p, q P X with πppq < πpqq. Then p l q “ q l p “ p. Proof (i) Certainly μ is in the weak-∗ closure of the set aco{δu : u P clX A}, and each δu for u P clX A is the weak-∗ limit of a net in {δ s : s P A}. (ii) We may suppose that both μ, ν P MpXqr1s . Take σ P aco{δ s : s P A} and τ P aco{δt : t P B}. We have δ s  δt “ δt  δ s “ δ s ps P A, t P Bq, and so σ  τ “ τ  σ “ ϕ0 pτqσ. Using (i), we can take weak-∗ limits to see that

6.3 Biduals of semigroup algebras

393

μ l ν “ lim lim σ  τ “ lim lim ϕ0 pτqσ “ lim ϕ0 pνqσ “ ϕ0 pνqμ . σÑμ τÑν

σÑμ τÑν

σÑμ

Similarly, ν l μ “ ϕ0 pνqμ, and so μ l ν “ ν l μ. (iii) We may suppose that there exists t P rπppq, πpqqs such that p P clX At and q P clX Bt , and so this follows from clause (ii). Lemma 6.3.12 Take p, q P X ∗ , and t P E. (i) Suppose that p, q P Ft∗ ∩ clX At . Then p l q “ p. (ii) Suppose that p, q P Ft∗ ∩ clX Bt . Then p l q “ q. (iii) Suppose that p P Ft∗ ∩ clX At and q P Ft∗ ∩ clX Bt . Then p l q “ q l p “ p. Proof (i) and (ii) follow because p l q “ lim sÑp limtÑq s ∧ t, and (iii) is a special case of Lemma 6.3.11(ii). Lemma 6.3.13 Take t P E, and suppose that Ft∗ ∩ clX At  ∅. Then Ft∗ ∩ clX At is infinite. Proof We first note that, since T is totally ordered, there is an infinite limit ordinal τ and a strictly increasing net psα : α < τq in At that converges to t in T . Let pNk q be a family of pairwise disjoint, infinite subsets of N. For each k P N, take Ek to be the set of sα such that α “ λ + n, where λ is 0 or a limit ordinal and n P Nk . The family {Ek : k P N} partitions the set {sα : α < τ} into an infinite number of disjoint subnets. The sets clX Ek are pairwise disjoint in X and pclX Ek q ∩ Ft  ∅ pk P Nq. Thus Ft is infinite.



In fact, since Ft∗ is an infinite, compact subset of X, we have

Ft∗

ě 2c by [172, Theorem 3.59]. Theorem 6.3.14 The semigroup pS , ∧q is strongly Arens irregular, and the semigroup algebra p 1 pS q,  q is not Arens regular. Proof Consider a point p P X ∗ , say p P Ft∗ , where t P E. We may suppose that p P clX At , and so, by Lemma 6.3.13, there exists q P Ft∗ ∩ clX At with q  p. By Lemma 6.3.12(i), it follows that p l q  q l p, and so p  ZpXq. Hence ZpXq “ S , showing that S is strongly Arens irregular. Clearly δ p  ZpMpXqq, and so  1 pS q is not Arens regular. The two examples that follow are [58, Examples 7.32 and 7.33]. Example 6.3.15 (i) Let S “ N∨ , as in Example 1.1.3(ii), so that S is weakly cancellative, and hence satisfies equation (6.3.2). Let u, v P S ∗ . Then u l v “ v and u  v “ u, so that pS ∗ , l q is a right-zero semigroup. Clearly Zpβ S q “ S , and so S is strongly Arens irregular, as in Theorem 6.3.9. We have

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6 Arens regularity

ν “ xμ, 1y ν

pμ P Mpβ S q, ν P MpS ∗ qq ,

μ  ν “ xν, 1y μ

pμ P MpS ∗ q, ν P Mpβ S qq ,

μ

l

and so both pMpS ∗ q, l q and pMpS ∗ q,  q factor. Let a and b be two distinct points of S ∗ , and suppose that μ P MpS ∗ q is such that μ l a “ μ  a and μ l b “ μ  b. Then clearly μ “ 0, and so {a, b} is a two-point DTC set for  1 pS q , and hence for β S , showing that  1 pS q is strongly Arens irregular. (ii) Let T “ N ∪ {8}, and let S be the semigroup N∧ , also first mentioned in Example 1.1.3(ii); here E “ {8}. The semigroup S not weakly cancellative, and hence MpS ∗ q is not an ideal in Mpβ S q. The semigroup algebra  1 pS q was identified with the Banach sequence algebra bv0 in Example 3.2.12; as in Example 6.1.24, bv0 is strongly Arens irregular, and so  1 pN∧ q is strongly Arens irregular. Let u, v P S ∗ . Then u l v “ u and u  v “ v, so that pS ∗ , l q is a left-zero semigroup. By Theorem 6.3.14, S is strongly Arens irregular. We have μ l ν “ xν, 1y μ pμ P Mpβ S q, ν P MpS ∗ qq , μ  ν “ xμ, 1y ν pμ P MpS ∗ q, ν P Mpβ S qq , so that MpS ∗ q is a closed subalgebra of Mpβ S q, and so both pMpS ∗ q, l q and pMpS ∗ q,  q factor. Also, f · μ “ μ · f “ xμ, 1y f

p f P  1 pS q, μ P MpS ∗ qq .

It follows that  1 pS q is a closed ideal in Mpβ S q, and so we have the decomposition Mpβ S q “ MpS ∗ q   1 pS q . Let a and b be two distinct points of S ∗ , and suppose that μ P MpS ∗ q is such that μ l a “ μ  a and μ l b “ μ  b. Then again μ “ 0, and so {a, b} is a two-point DTC set for  1 pS q , again showing that  1 pS q is strongly Arens irregular. We shall now consider when the semigroup algebra p 1 pS q,  q is strongly Arens irregular, with the semigroups S and T as above. Lemma 6.3.16 Let μ P MpXq, and take t P E. (i) Suppose that μ | pFt∗ ∩ clX At q “ 0. Then μ l p “ p l μ pp P Ft∗ ∩ clX At q. (ii) Suppose that p P Ft∗ ∩ clX At and that μ l p “ p l μ. Then there exists z P C such that μ | pFt∗ ∩ clX At q “ zp. Suppose also that q P Ft∗ ∩ clX At with q  p and that μ l q “ q l μ. Then μ | pFt∗ ∩ clX At q “ 0. Proof Set U “ Ft∗ ∩ clX At . (i) Take p P U, and choose ε > 0. Since μ | U “ 0, there exists u P At such that |μ| pclX pS ∩ ru, tqqq < ε. Set μ1 “ μ | clX Au and μ2 “ μ | clX Bt . By Lemma 6.3.11(ii), μ1 l p “ p l μ1 and μ2 l p “ p l μ2 . Since μ ´ pμ1 + μ2 q < ε, it follows that μ l p ´ p l μ < ε. This holds for each ε > 0, and so μ l p “ p l μ. (ii) We may suppose that μ P MpXqr1s .

6.3 Biduals of semigroup algebras

395

Set ν “ μ | U. Since pμ ´ νq | U “ 0, we have pμ ´ νq l p “ p l pμ ´ νq by (i). By hypothesis, μ l p “ p l μ, and so ν l p “ p l ν. Take s P r0, tq. By Lemma 6.3.11(i), the measure ν belongs to the weak-∗ closure of C s :“ aco{δr : r P S ∩ ps, tq}. Each σ P C s has the form σ “ ζ1 δ s1 + · · · + ζm δ sm , where m P N, ζ1 , . . . , ζm P C with |ζ1 | + · · · + |ζm | ď 1, and s1 , . . . , sm P ps, tq. We have s  σ “ ϕ0 pσqs, and so p l ν “ lim lim s  σ “ lim ϕ0 pνqs “ zp , sÑp σÑν

sÑp

where z “ ϕ0 pνq P D. On the other hand, take σ P aco{δr : r P At } of the above form. Then σ  s “ σ for s > si pi P Nm q, and so ν l p “ lim lim σ  s “ lim σ “ ν . σÑν sÑp

σÑν

We conclude that ν “ ν l p “ p l ν “ zp, as required. Now suppose also that q P U with q  p and that μ l q “ q l μ. Then there exist z, w P D such that ν “ zp and ν “ wq. Since q  p, we have zw “ 0, and so ν “ 0. Theorem 6.3.17 Let μ P MpXq. Then μ P ZpMpXqq if and only if μ | Ft∗ “ 0 for all t P E. Proof Suppose that μ | Ft∗ “ 0 pt P Eq. Since  1 pS q Ă ZpMpXqq, we may suppose that μ | S “ 0. Take ν P MpXqr1s and ε > 0. Each t P T is contained in an open interval, say Ut , of T such that |μ| pπ´1 pUt qq < ε, and, since T is compact, we can suppose that the union of finitely many sets of the form Ut is T . Hence there exist n P N and t1 , . . . , tn+1 P T with 0 “ t1 < t2 < · · · < tn < tn+1 “ 8 such that |μ| pπ´1 pIi qq < ε for i “ 1, 2, . . . , n, where Ii :“ rti , ti+1 s. Set μi “ μ | clX pS ∩ Ii q and νi “ ν | clX pS ∩ Ii q for i “ 1, 2, . . . , n, so that μ “ n n i“1 μi and ν “ i“1 νi . It follows from Lemma 6.3.11(ii) that μi l ν j “ ν j l μi whenever i, j P {1, 2, . . . , n} and i  j. Take i P {1, 2, . . . , n}. Since |μ| pπ´1 pIi qq < ε and |μ| pπ´1 pIi qq < ε, we have μi l νi  < ε νi  and νi l μi  < ε νi , and so     n n      μi l νi  < ε and  νi l μi  < ε .    i“1  i“1 Hence μ l ν ´ ν l μ < 2ε. This holds for each ε > 0, and so μ l ν “ ν l μ. We conclude that μ P ZpMpXqq. Conversely, suppose that there exists t P E such that μ | Ft∗  0. We again set U “ Ft∗ ∩ clX At , and we may suppose that μ | U  0. By Lemma 6.3.13, there are two distinct points, say p and q, in U. It follows from Lemma 6.3.16(ii) that either μ l p  p l μ or μ l q  q l μ, and hence μ  ZpMpXqq. Theorem 6.3.18 The semigroup algebra p 1 pS q,  q is strongly Arens irregular if and only if clT S is scattered.

396

6 Arens regularity

Proof First, suppose that clT S is scattered. Take μ P ZpMpXqq, so that, by Theorem ´1 1 6.3.17, μ | Ft∗ “ 0 pt P Eq. By Theorem  1.3.39, we see that |μ| ◦ π P  pclT S q, ´1 1 ´1 and hence |μ| ◦ π P  pS q. Since  |μ| ◦ π 1 “  |μ| 1 , it follows that |μ| P  1 pS q, and so μ P  1 pS q. Hence ZpMpXqq “  1 pS q, showing that  1 pS q is strongly Arens irregular. Conversely, suppose that clT S is not scattered. Choose a continuous probability measure, say μ, on clT S , and set F “ { f ◦ π : f P CpclT S q} , so that F is a closed linear subspace of CpXq. We define a continuous linear functional ν on F by setting νp f ◦ πq “ μp f q p f P CpclT S qq , so that ν “ νp1clT S q “ 1. By the Hahn–Banach theorem, we can extend ν to be a continuous linear functional on CpXq, still with ν “ νp1clT S q “ 1; we regard ν as a probability measure on X. Since νpFt∗ q “ 0 pt P Eq, it follows from Theorem 6.3.17 that ν P ZpMpXqq. Since ν   1 pS q, the Banach algebra  1 pS q is not strongly Arens irregular. Let S , T , X, and E be as above. We can obtain a DTC set V for MpXq that is contained in X ∗ as follows. For each t P E, we choose two distinct points in Ft∗ ∩clX At and two distinct points in Ft∗ ∩ clX Bt whenever the respective sets are non-empty. The collection of these points is called V. Now take μ P MpXq, and suppose that μ l p “ p l μ for each p P V. It follows from Lemma 6.3.16(ii) that μ | Ft∗ ∩clX At “ μ | Ft∗ ∩clX Bt “ 0 for each t P E. Thus μ | Ft∗ “ 0. By Theorem 6.3.17, this implies that μ P ZpMpXqq, and hence V is a DTC set for MpXq. It follows that MpXq has a DTC set consisting of κ at most 4κ points of X, where κ “ |E|; this is a small subset of X because |X| “ 22 . Suppose that E is finite. Then the above DTC set V is also finite. In the case where E is infinite, it follows that |V| “ |E|. Suppose that E is infinite. Then there cannot be a finite DTC set for the semigroup S . For suppose that V is a finite subset in X, and choose t P E\πpVq, and then choose p P Ft∗ . We have p l v “ v l p pv P Vq by Lemma 6.3.11(iii), but p  S . Since S is strongly Arens irregular by Theorem 6.3.14, this shows that V is not a DTC set for S . Similarly, there is no countable DTC set for S when E is uncountable. Thus we obtain the following result. Theorem 6.3.19 Suppose that the set E is finite. Then there is a finite DTC set for the semigroup S . Suppose that the set E is infinite or uncountable. Then there is no finite or countable DTC set for the semigroup S , respectively. The following theorem is also proved in [62, Theorem 3.2].

6.3 Biduals of semigroup algebras

397

Theorem 6.3.20 Suppose that the set E is countable. Then the semigroup algebra  1 pS q is strongly Arens irregular and has a DTC set consisting of at most four measures in MpX ∗ q+ . Proposition 6.3.21 Suppose that |E| “ κ, where κ ě ℵ1 . Then there is no DTC set for MpXq with cardinality less than κ. Proof Assume that V is a DTC set for MpXq with |V| < κ. For each ν P V, the set {t P E : ν | Ft∗  0} is countable because {Ft∗ : t P E} is a family of pairwise disjoint, non-empty, compact sets. Hence there exists t P E such that ν | Ft∗ “ 0 for each ν P V. Choose p P Ft∗ . It follows from Lemma 6.3.16(i) that p l ν “ ν l p pν P Vq. By the assumption, p P ZpMpXqq, a contradiction of Theorem 6.3.17 because δ p | Ft∗  0. It follows that there is no DTC set for MpXq with |V| < κ. Example 6.3.22 (i) Set T “ {´8}∪R∪{8} and S “ Q, so that E “ T . By Theorem 6.3.14, the semigroup pQ, ∧q is strongly Arens irregular and  1 pQ, ∧q is not Arens regular. By Theorem 6.3.19, there is no countable DTC set for this semigroup. Since E is not scattered, it follows from Theorem 6.3.18 that  1 pQ, ∧q is not strongly Arens irregular, and the centre of MpXq is characterized in Theorem 6.3.17. By Proposition 6.3.21, there is no countable DTC set for MpXq. (ii) Consider the subset S of T :“ {´8} ∪ R ∪ {8} that consists of numbers of the form n ´ x, where n P Z and x P {1/2, 1/4, 1/8, . . . }. Then the corresponding set E is equal to {´8} ∪ Z ∪ {8}, a countable set, and so, by Theorem 6.3.18, the semigroup algebra  1 pS q is strongly Arens irregular. By Theorem 6.3.19, there is no finite DTC set for the semigroup S , but, essentially as in Theorem 6.3.20, MpXq has a two-element DTC set in MpX ∗ q+ . (iii) Consider the semigroup T “ pr0, κs, ∧q, where κ is a cardinal with |κ| ě ℵ1 , and set S “ r0, κq, so that the corresponding set E has cardinality κ. Since T is scattered, the algebra  1 pS q is strongly Arens irregular by Theorem 6.3.18. By Proposition 6.3.21, there is no DTC set for MpXq with cardinality strictly less than κ. This shows that the cardinality of a DTC set can be arbitrarily large, even when  1 pS q is strongly Arens irregular. Let S be a semigroup, and let ω : S Ñ R +• be a weight on S . We now consider when the weighted semigroup algebra p 1 pS , ωq,  · ω ,  q is Arens regular or strongly Arens irregular or neither. Throughout the remainder of this section, we now set Aω “  1 pS , ωq

and

Eω “ c 0 pS , 1/ωq ,

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6 Arens regularity

so that Eω is a concrete Banach-space predual of Aω . As in equation (1.3.7), we have Aω “ Aω ‘ EωK . In the case where S is weakly cancellative, Proposition 2.4.28 shows that Aω is a dual Banach algebra with isometric Banach-algebra predual Eω , and so, as before, we have (6.3.9) Aω “ Aω  EωK .   For s P S , the normalized point mass at s is δr “ δ /ωpsq, so that δr  “ 1. For s

s

s ω

a subset T of S , we denote by β T ω the weak-∗ closure of {δrt : t P T } in Aω , and set T ω∗ “ β T ω \ T , the growth of T in β S ω . Following [48], we shall also use the following standard notation: we set Ωps, tq “

ωpstq ωpsqωptq

ps, t P S q .

(6.3.10)

Thus 0 < Ωps, tq ď 1 ps, t P S q. Let G be a group. Then we see that Ωps, tq ě

1 ωptqωpt´1 q

ps, t P Gq

(6.3.11)

because ωpsq “ ωpstt´1 q ď ωpstqωpt´1 q ps, t P Gq. Let S be an infinite semigroup, and let ω be a weight on S . For each s P S , the function t Þ→ Ωps, tq, S Ñ I, has a continuous extension to a function β S Ñ I ; the value of this function at v P β S is denoted by Ωps, vq. Next, the function s Þ→ Ωps, vq, S Ñ I, has a continuous extension to a function β S Ñ I; the value of this function at u P β S is denoted by Ω l pu, vq. Let u, v P β S , say u “ limα sα and v “ limβ tβ , where psα q and pβt q are nets in S . Then we express Ω l pu, vq by the repeated limit (6.3.12) Ω l pu, vq “ lim lim Ωpsα , tβ q ; α

β

of course the limit is independent of the nets psα q and ptβ q. Similarly, we define the function Ω by Ω pu, vq “ limβ limα Ωpsα , tβ q. By Proposition 1.1.2, there are sequences psm q and ptn q, which consist of distinct terms, in S such that Ω l pu, vq “ lim lim Ωpsm , tn q , mÑ8 nÑ8

Ω pu, vq “ lim lim Ωpsm , tn q . nÑ8 mÑ8

The following theorem is taken from Dales and Lau [57, Theorem 8.8]. Theorem 6.3.23 Let ω be a weight on a semigroup S , and let U and V be infinite subsemigroups of S . Suppose that Ω 0-clusters on U×V. Then M l N “ M  N “ 0 whenever M P  1 pU, ωq ∩ EωK and N P  1 pV, ωq ∩ EωK . Proof We identify BU :“  1 pU, ωq ∩ EωK and BV :“  1 pV, ωq ∩ EωK with weak∗-closed subalgebras of pAω , l q and pAω ,  q. Take M P pBU qr1s and N P pBV qr1s , and take λ P  8 pS q. It follows from equation (2.3.2) that there exist sequences p fm q in  1 pUqr1s and pgn q in  1 pVqr1s , respectively, such that

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399

xM l N, λωy “ lim lim

mÑ8 nÑ8



fm psqgn ptqλpstqΩps, tq .

sPU, tPV

Set Km “ supp fm pm P Nq and Ln “ supp gn pn P Nq. Then we may suppose that each Km and Ln is finite. Also limmÑ8 fm psq “ limnÑ8 gn psq “ 0 ps P S q, and so we may suppose that Kn+1 ∩ pK1 ∪ · · · ∪ Kn q “ Ln+1 ∩ pL1 ∪ · · · ∪ Ln q “ ∅ pn P Nq .   Set H “ {Km : m P N} ∪ {Ln : n P N}, a countable set. By replacing U and V by U ∩ H and V ∩ H, respectively, we may suppose that U and V are countable. and Set Ωt psq “ Ωps, tq ps P U, t P Vq,

regard F :“ {Ωt : t P V} as a subset of 

Cpβ Uq. For n P N, define hn “ {

gn ptq

Ωt : t P Ln }, so that hn P aco F . Since Ω clusters on U ×V, it follows from Theorem 1.2.32, (d) ⇒ (c), that aco F is relatively weakly sequentially compact, and so we may suppose that hn Ñ h P Cpβ Uq weakly as n Ñ 8. on U ×V, it follows from Corollary 1.2.33 Fix x P U ∗ . Since Ω

0-clusters

that,

for each ε > 0, we have

hn pxq

< ε for all but finitely-many n P N, and so

hpxq

ď ε. This proves that hpxq “ 0, and so h | U ∗ “ 0. Again fix ε > 0. Then there exists m0 P N with |h|Km < ε pm ě m0 q, and hence

{

fm psqgn ptqλpstq

Ωps, tq : s P Km , t P Ln } lim lim mÑ8 nÑ8

f psqhptq

λ ď ε λ . “ lim m 8 8 mÑ8

sPKm

This holds for each ε > 0, and so xM l N, λωy “ 0. But this holds for each λ P  8 pS q, and so M l N “ 0. Similarly, M  N “ 0, as required. The following result is due to Craw and Young [48, Theorem 1]. Corollary 6.3.24 Let ω be a weight on a semigroup S . Suppose that the function Ω 0-clusters on S × S . Then  1 pS , ωq is Arens regular Proof By Theorem 6.3.23, M l N “ M  N “ 0 whenever M, N P EωK , and so it follows from Proposition 2.3.24 that  1 pS , ωq is Arens regular. Corollary 6.3.25 Let ω be a weight on an abelian semigroup S , and let U be an infinite subset of S . Suppose that Ω 0-clusters on U × S . Then  1 pS , ωq is not strongly Arens irregular. Proof Take M P BU ∩ EωK , so that M  Aω , and take N P EωK . By the theorem, M l N “ M  N “ 0. Take f P Aω . Then M · f “ f · M. Since Aω “ Aω ‘ EωK , it follows that M P ZpAω q, and so Aω is not SAI.

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6 Arens regularity

The following characterization of Arens regularity extends [57, Theorem 8.11] from groups to weakly cancellative semigroups; a condition extending (b) is shown to be equivalent to (a) for arbitrary semigroups by Baker and Rejali in [13, Theorem (3.3)]. See also [66, Theorem 5.3] Theorem 6.3.26 Let ω be a weight on a weakly cancellative semigroup S . Then the following conditions are equivalent: (a) the algebra  1 pS , ωq is Arens regular; (b) the function Ω 0-clusters on S × S ; (c) M l N “ M  N “ 0 pM, N P EωK q. Proof (b) ⇒ (c) ⇒ (a) This follows as in Corollary 6.3.24. (a) ⇒ (b) Assume towards a contradiction that (b) fails. Then there exist sequences psm q and ptn q in S , each consisting of distinct points, such that lim lim Ωpsm , tn q “ 2δ

mÑ8 nÑ8

for some δ > 0. By passing to a subsequence of psm q, we may suppose that limnÑ8 Ωpsm , tn q > δ for all m P N. We now choose subsequences of psm q and ptn q inductively. Indeed, set u1 “ s1 , and let v1 be the first element tn with Ωpu1 , tn q > δ. Having chosen u1 , . . . , un and v1 , . . . , vn , choose un+1 to be the first element in the sequence psm q such that un+1 vk  ui v j for all i, j, k P Nn , and then choose vn+1 to be the first element in the sequence ptn q such that ui vn+1  u j vk for all i, j P Nn+1 and k P Nn and such that Ωpui , vn+1 q > δ pi P Nn+1 q. (The elements un+1 and vn+1 exist because S is weakly cancellative.) The sequences pum q and pvn q are such that the elements um vn are all distinct for m, n P N and such that Ωpum , vn q > δ whenever m ď n. Set a j “ δu j /ωpu j q and bk “ δvk /ωpvk q for j, k P N to obtain a contradiction as in Proposition 6.3.4. In contrast to the above results, we state the following result from [52, Theorem 8.1]. Theorem 6.3.27 Let ω be a continuous weight on pR+ , + q. Then the semigroup algebra  1 pR+• , ωq is strongly Arens irregular. Examples (i) Consider the group pZ, + q, and take α > 0. Define ωα pnq “ p1 + |n|qα

pn P Zq .

Then ωα is a semisimple weight on Z, and the corresponding function Ωα 0-clusters on Z × Z, so that  1 pZ, ωα q is Arens regular. (ii) Let ω : N Ñ r1, 8q be any function, and again consider the weighted semigroup algebra Aω “ p 1 pN∧ , ωq,  q.

6.3 Biduals of semigroup algebras

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Suppose that limnÑ8 ωn “ 8. Then we noted in Example 3.2.12 that Aω is Arens regular. In fact, in this case, the corresponding function Ω is such that Ωpxn , ym q “ 1/ωpym q when pxn q and pym q are sequences of distinct points in N and ym ě xn , and so it follows that Ω 0-clusters. Hence Aω is Arens regular by Corollary 6.3.24. Similarly, the weighted semigroup algebra p 1 pN∨ , ωq,  q is Arens regular whenever limnÑ8 ωpnq “ 8. (iii) Consider the semigroup pN, + q. Define ωpnq “ expp´n2 q

pn P Nq ,

so that ω is a radical weight on N. The corresponding function is Ω, and clearly Ωpm, nq “ expp´2mnq pm, n P Nq, so that Ω 0-clusters on N × N, and hence  1 pN, ωq is Arens regular. The following example is given in [57, Example 9.2]. Example 6.3.29 We claim that there is a sequence pAk q of Arens regular Banach algebras such that  8 pAk q is not Arens regular. Indeed, for each k P N, set Ak “  1 pZ, ω1/k q, where ω1/k was defined in Example 6.3.28 (i), so that Ak is an Arens regular Banach algebra, and then set A “  8 pAk q, so that A is a commutative Banach algebra. For each k P N, we define λk P pAk qr1s to be the map λk :

8 j“´8

α j δ j Þ→

8

α j ω1/k p jq .

j“0

Let U be an ultrafilter on N, and define λ on A by xa, λy “ limU xak , λk y; the limit always exists, and λ P A with λ “ 1. For each m, n P N, define am “ pam,k q and bn “ pbn,k q by setting am,k “ δm /ω1/k pmq,

bn,k “ δ´n /ω1/k pnq pk P Nq .     For each m, n, k P N, we have am,k , bn,k P Ak with am,k ω “ bn,k ω “ 1, and so 1/k 1/k am , bn P A. For n > m, we have xam,k  bn,k , λk y “ 0 pk P Nq because m ´ n  Z+ , and so limmÑ8 limnÑ8 xam bn , λy “ 0. For m > n and k P N, we have xam,k  bn,k , λk y “



1/k ω1/k pm ´ nq 1 + |m ´ n| “ Ñ1 ω1/k pmqω1/k pnq p1 + |m|qp1 + |n|q

as k Ñ 8, and so limnÑ8 limmÑ8 xam bn , λy “ 1. It follows from Corollary 2.3.31 that A is not Arens regular. Set B “ c0 pAk q. By Proposition 2.3.40, the Banach algebra B is Arens regular. But B “  8 pAk q, which contains A as a closed subalgebra, and so B is not Arens regular.

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6 Arens regularity

We now consider when a weighted semigroup algebra is strongly Arens irregular, following the memoir of Dales and Dedania [52]. Definition 6.3.30 Let S be a semigroup, let ω be a weight on S , and let T be a subset of S . Then: (i) ω is diagonally bounded on T , with bound dT > 0, if 1 ď Ωps, tq ď 1 ps P S , t P T q ; dT (ii) ω is weakly diagonally bounded on T , with bound cT > 0, if 1 ď Ω l pu, vq ď 1 pu P S ∗ , v P T ∗ q . cT Thus ω is weakly diagonally bounded on T , with bound cT , if and only if, for each ε > 0, there is a cofinite subset S 0 of S such that, for each s P S 0 , there is a cofinite subset T s of T with 1 p1 ´ εq ď Ωps, tq ď 1 cT

pt P T s q .

Let ω be a weight on a group G, and take a subset T of G. Then ω is diagonally bounded on T , with bound dT , if and only if sup{ωptqωpt´1 q : t P T } ď dT .

(6.3.13)

The following is the weighted version of Theorem 6.3.7; it is [52, Theorem 5.6], and it is proved by a small modification of the argument in the proof of Theorem 6.3.7. A similar theorem is proved by Filali and Salmi in [116, Theorem 14]. Earlier versions that show that Aω is strongly Arens irregular when S is a group and ω is diagonally bounded on an infinite set are given in [57, Corollary 11.10] and [248, Theorem 1.2]. Theorem 6.3.31 Let S be an infinite, countable, cancellative semigroup, and let ω be a weight on S that is weakly diagonally bounded on an infinite subset T of S , with bound cT . Take n P N with n > cT . Then there is a subset V of T ω∗ with |V| “ n such that V is determining for the left topological centre of Aω . Further, Aω is left strongly Arens irregular. In the case where S is abelian, Aω is strongly Arens irregular. Corollary 6.3.32 Let ω be a weight on pZ, + q. Suppose that there is strictly increasing sequence pnk q in N such that   sup ωpnk qωp´nk q : k P N < 8 . Then  1 pZ, ωq is strongly Arens irregular.

6.3 Biduals of semigroup algebras

403

Proof The condition implies that ω is diagonally bounded on the group pZ, + q by equation (6.3.13). It seems to be open whether  1 pZ, ωq is necessarily strongly Arens irregular for each weight ω on Z such that lim inf nÑ8 ωpnq < 8 and lim inf nÑ8 ωp´nq < 8. The analogue of Theorem 6.3.31 when S is not necessarily countable is the following [52, Theorem 5.10]. Theorem 6.3.33 Let S be an infinite, cancellative semigroup, and let ω be a weight on S that is weakly diagonally bounded on a subset T of S with |T | “ |S |. Then  1 pS , ωq is strongly Arens irregular. Example 6.3.34 We give an example [57, Example 9.17] of a symmetric weight ω : Z Ñ r1, 8q that is unbounded, but such that  1 pZ, ωq is strongly Arens irregular and has a two-point DTC set. Each n P Z• can be written (in many ways) in the form n“

r

ε j 2a j ,

(6.3.14)

j“1

where ε j P {´1, 1} and a j P Z+ for j P Nr and where a1 > a2 > · · · > ar . For each n P Z• , we define ηpnq to be the minimum value of r P N that can arise in equation (6.3.14); we also set ηp0q “ 0. It is clear that ηpnq P Z+ pn P Zq, that ηpnq “ 0 if and only if n “ 0, that √ ηp´nq “ ηpnq pn P Zq, and that η is subadditive on Z. Take c with 1 < c < 2, and set ωpnq “ c ηpnq pn P Zq, so that ω : Z Ñ r1, 8q is a symmetric weight. Set T “ {2k , ´2k : k P N}. Clearly ηp2k q “ 1 and ωp2k q “ c for each k P N, and so Ω is diagonally bounded on T with bound 2. Thus  1 pZ, ωq is strongly Arens irregular and has a two-point DTC set by Theorem 6.3.31. Finally, we claim that η, and hence ω, is unbounded in Z. To see this, consider numbers of the form nk “ 22k + 22k´2 + · · · + 24 + 22 + 1, where k P N. We shall prove by induction on k that ηpnk q “ k + 1. Certainly ηpnk q ď k + 1. The result is immediate for k “ 1; assume that the result holds for k. Suppose that r ε j 2a j , (6.3.15) nk+1 “ j“1

where ε j P {´1, 1} , a j P Z+ , and a1 > a2 > · · · > ar ě 0. Since nk+1 is an odd number and ar´1 ě 1, we have ar “ 0. Suppose that εr “ 1. Then we subtract 1 from each side of (6.3.15), and note that necessarily ar´1 ě 2 because the new right a j ´2 with ar´1 ´ 2 ě 0. Suppose that hand side is divisible by 4. Then nk “ r´1 j“1 ε j 2 εr “ ´1. Then necessarily ar´1 “ 1 and εr´1 “ ´1. Further, we have

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6 Arens regularity

nk+1

⎞ ⎛ r´2 ⎟⎟⎟ ⎜⎜⎜ a ´2 ⎟⎟⎟ ´ 2 ´ 1 , ⎜ j “ 1 + 4nk “ 4 ⎜⎝⎜ ε j 2 ⎠ j“1

 a j ´2 ´ 1. In both these two cases, the inductive hypothesis and so nk “ r´2 j“1 ε j 2 shows that r ´ 1 ě k + 1. Thus ηpnk+1 q “ k + 2, continuing the induction.

Let ω be a weight on a semigroup S . There are various results on the radical, rad Aω , of Aω in the memoir of Dales and Lau [57, Chapter 8]. It is immediate from Proposition 2.3.3 that prad Aω q ∩ Aω “ {0} whenever Aω is semisimple, and so rad Aω Ă EωK in this case. Suppose that S is weakly cancellative, that Aω is semisimple, and that Ω 0-clusters on S × S . Then, by Theorem 6.3.26, EωK is a nilpotent ideal in Aω , and so rad Aω “ EωK . However, as in [57, Theorem 8.20], there are easy examples of weights ω on Z such that rad Aω  EωK . Indeed, it is shown in [57, Example 9.17] that the weight on Z of the above Example 6.3.34 is such that rad Aω  EωK . It is proved by White in [322] that  1 pZ, ωq is never semisimple for any weight ω on Z and that there are weights ω on Z such that the radical of  1 pZ, ωq is not a nilpotent ideal. Example 6.3.35 We give an example [52, Example 6.4] of a weight ω on the abelian group pQ, + q. Indeed, set ωpp/qq “ 1 + |p| + q pp/q P Q • q , and set ωp0q “ 1. Then ω is a weight on Q. The weight ω is not bounded, and hence not diagonally bounded, on any infinite subset of Q. However, we claim that the weight ω is weakly diagonally bounded on the infinite set T “ {1/r : r P P}, with bound cT “ 1, where P denotes the set of prime numbers. Indeed, given ε > 0, we see that |p| + q >1´ε 1 + |p| + q for all save finitely-many p/q P Q, say for p/q P S 0 , where S 0 is a certain cofinite subset of Q. Given s “ p/q P S 0 , take t “ 1/r, where r P P with r > q. Then Ωps, tq “

1 + |pr + q| + qr |p| + q Ñ 1 + |p| + q p1 + |p| + qqp2 + rq

as

rÑ8

(where we note that pr + q and qr are coprime), and so there is a cofinite subset T s of T such that Ωps, tq > 1 ´ ε ps P S 0 , t P T s q, giving the claim. It follows that the semigroup algebra  1 pQ, ωq is strongly Arens irregular and has a two-point DTC set.

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405

We now give an example [57, Example 9.7] of a weighted semigroup algebra on Z that is neither Arens regular nor strongly Arens irregular. No such example on N seems to have been constructed. Example 6.3.36 We define ω : Z Ñ r1, 8q by setting ωp´ jq “ exp j ,

ωp jq “ 1 + j p j P Z+ q .

It is easily checked that ω is a weight on the group pZ, + q. We first claim that Aω “  1 pZ, ωq is not Arens regular. Indeed, we have Ωp´ j, ´kq “ 1 p j, k P Nq, and so Ω does not 0-cluster on Z × Z; by Theorem 6.3.26, Aω is not Arens regular, as claimed. We next claim that Ω 0-clusters on N × Z; by Corollary 6.3.25, this is sufficient to show that Aω is not strongly Arens irregular. Take j, k P N. Then lim jÑ8 limkÑ8 Ωp j, kq “ lim jÑ8 1/p1 + jq “ 0. Similarly, limkÑ8 lim jÑ8 Ωp j, kq “ 0. For k > j, we have Ωp j, ´kq “ e´ j /p1 + jq, and so lim jÑ8 limkÑ8 Ωp j, ´kq “ 0. For j > k, we have Ωp j, ´kq “ p1 + j ´ kq/p1 + jqek , and so limkÑ8 lim jÑ8 Ωp j, ´kq “ 0. Thus Ω 0-clusters on N × Z, as claimed.

Example 6.3.37 It is clear that a radical weight ω on N cannot be diagonally bounded on any infinite subset of N. For let ω be a radical weight on N, and assume that Ωp1, nq ě δ > 0 for infinitely-many values of n P N. Then νω “ lim ωpnq1/n ě lim inf nÑ8

nÑ8

ωpn + 1q ě ωp1qδ > 0 , ωpnq

a contradiction of the fact that ω is radical. However, we shall now exhibit, from [52, Example 6.5], a radical weight ω on N that is weakly diagonally bounded on an infinite subset of N. We define inductively a sequence pmk : k P Nq in N by setting m1 “ 1 and mk+1 “ 2mk + 1 pk P Nq, and then we define a function η : N Ñ N by setting ηp1q “ 1 and by using the inductive formula ηpmk + 1q “ max{2ηpmk q, p2mk + 1qk} , ηpmk + 1 + rq “ ηpmk + 1q + ηprq pr P Nmk q , for k P N. Clearly η is an increasing function on N. We claim that ηpm + nq ě ηpmq + ηpnq pm, n P Nq. This is trivially true for m “ n “ 1. Assume that it is true whenever m + n ď mk , where k P N, and take m, n P N with m + n ď mk+1 ; we may suppose that m + n ě mk + 1. First, suppose that m, n ď mk . Then ηpm + nq ě ηpmk + 1q ě 2ηpmk q ě ηpmq + ηpnq .

406

6 Arens regularity

Second, suppose that m ě mk + 1, say m “ mk + 1 + r, where r P Z+mk . Then n ď mk and ηpm + nq “ ηpmk + 1q + ηpr + nq ě ηpmk + 1q + ηprq + ηpnq “ ηpmq + ηpnq . This establishes the claim. Set ω “ expp´ηq. Then, by the claim, ω is a weight on N. Take k P N. For each n P N with mk + 1 ď n ď mk+1 , we have ηpnq ηpmk + 1q ηpmk + 1q ě ě k, “ n mk+1 2mk + 1 and so ωpnq1/n ď expp´kq. Thus limnÑ8 ωpnq1/n “ 0, and ω is radical. Set T “ {mk + 1 : k P N}, so that T is an infinite subset of N. For each m P N, set T m “ {n P T : n > m}, a cofinite subset of T . For m P N and n “ mk + 1 P T m , we have Ωpm, nq “ exppηpmq + ηpmk + 1q ´ ηpmk + 1 + mqq “ 1 , and so ω is weakly diagonally bounded on T , with cT “ 1. By Theorem 6.3.31, the algebra  1 pN, ωq is strongly Arens irregular and there is a two-element DTC set.

We conclude this section with an example, without proof, that shows that biduals of weighted group algebras may have strange properties; in particular, the left and right topological centres need not be equal. See the memoir of Dales and Lau [57, Chapter 10]. Example 6.3.38 There is a weight on ω on the group F2 such that the following properties hold, where Aω “  1 pF2 , ωq: (i) as a subset of pAω , l q, the ideal EωK is nilpotent of degree 2; (ii) as a subset of pAω ,  q, the ideal EωK is nilpotent of degree pq prq prq pq 3; (iii) Zt pAω q  Zt pAω q and Zt pAω q  Zt pAω q.

6.4 Arens regularity for algebras on locally compact groups In this final section, we shall investigate the Arens regularity of group algebras, Fourier algebras, and Figà-Talamanca–Herz algebras and some of their relatives. There are many impressive results in this area, and we shall prove only a selection. We shall begin this section by considering the Arens regularity of the Fourier algebra ApΓq of §4.3 and the Figà-Talamanca–Herz algebras A p pΓq of §4.4. The algebra A p pΓq can be Arens regular only if Γ is discrete and every abelian subgroup

6.4 Arens regularity for algebras on locally compact groups

407

of Γ is finite, but we do not know of an infinite group Γ such that A p pΓq is Arens regular. The Fourier algebra ApΓq is strongly Arens irregular whenever Γ is amenable and discrete, and in some other cases. Let G be a locally compact group. We shall show in Theorem 6.4.15 that Zt pLUCpGq q “ MpGq when G is non-compact, an important step to later results. We shall then show as a main theorem, Theorem 6.4.18, that each group algebra pL1 pGq,  q is strongly Arens irregular, and that there is a two-point DLTC set for this algebra whenever G is not compact. Our final theorem, not proved, is that pMpGq,  q is strongly Arens irregular for every locally compact group G. Our first theorem in this section is the following. The original proof of this result was given by Lau and Wong in [223, Proposition 5.3]; see also [313, Corollary 3.7]. Theorem 6.4.1 Let Γ be an amenable locally compact group. Then ApΓq is Arens regular if and only if Γ is finite. Proof It follows from Theorem 4.3.12 that, for each locally compact group Γ, the Banach function algebra ApΓq is strongly regular, has the strong separating ball property, and is weakly sequentially complete. In the case where Γ is also amenable, ApΓq has a CAI by Theorem 4.3.29. Hence the result is an application of Corollary 3.4.18. Let Γ be an amenable locally compact group, and take p with 1 < p < 8. Then the same theorem and proof as the above would cover the algebra A p pΓq if we knew that A p pΓq were weakly sequentially complete. We now seek a more general result than the above theorem. It is a conjecture that a locally compact group Γ must be finite whenever 1 < p < 8 and A p pΓq is Arens regular; we give some results towards this. The first extends a theorem of Forrest [120, Theorem 3.2]; our proof is different. Theorem 6.4.2 Let Γ be a locally compact group, and take p with 1 < p < 8. (i) Suppose that Γ is metrizable and that there is a non-zero, closed ideal in A p pΓq that is Arens regular. Then Γ is discrete. (ii) Suppose that Γ is metrizable and that there is a weak Segal algebra with respect to A p pΓq that is Arens regular. Then Γ is discrete. (iii) Suppose that A p pΓq is Arens regular. Then Γ is discrete. Proof Set A “ A p pΓq, so that, by Theorem 4.4.2, A is a natural Banach function algebra on Γ. (i) Let I be a non-zero, closed ideal and take γ P ΦI .

in A that is

Arens regular,

ď 1, and so TIMpγ, Aq

ď 1 by Proposition By Proposition 3.4.14(i),

TIMpγ, Iq



3.4.13. By Proposition 4.4.4,

TIMpγ, Aq

“ 1. Since Γ is metrizable, it follows from Theorem 4.4.8, (b) ⇒ (c), that Γ is discrete. (ii) This follows from Theorem 3.4.20.

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(iii) Now Γ is a general locally compact group. Take V to be an open, symmetric,  relatively compact neighbourhood of eΓ , and define Γ0 “ {V n : n P N}, so that Γ0 is an open, σ-compact subgroup of Γ and, further, Γ0 is closed. By Theorem 4.4.6(i), A p pΓ0 q is a quotient algebra of A p pΓq, and so A p pΓ0 q is Arens regular. Assume that mΓ0 p{eΓ0 }q “ 0. By Theorem 4.1.2, there is a compact, normal subgroup N of Γ0 such that mG0 pNq “ 0 and Γ0 /N is metrizable. By Theorem 4.4.6(ii), A p pΓ0 /Nq is identified as a closed subalgebra of A p pΓ0 q, and so A p pΓ0 /Nq is also Arens regular. By (i), Γ0 /N is discrete, and so N is open in Γ0 , and hence mG0 pNq > 0, a contradiction. Thus mΓ0 p{eΓ0 }q > 0, and so Γ0 is discrete. It follows that Γ is discrete, as required. Corollary 6.4.3 Let Γ be a locally compact group such that ApΓq is Arens regular. Then Γ is discrete, and every amenable subgroup of Γ is finite. Further, ApΓq l ApΓq Ă ApΓq and ApΓq is also Arens regular and an ideal in its bidual. Proof By Theorem 6.4.2(iii), Γ is discrete. Let Γ0 be an amenable subgroup of Γ. Then Γ0 is a closed subgroup, and so, by Theorem 4.4.6(i), ApΓ0 q is a quotient of ApΓq, and hence ApΓ0 q is Arens regular. By Theorem 6.4.1, Γ0 is finite. By Proposition 3.2.3, ApΓq is a compact algebra, and so the further results follow from Theorem 6.1.25. No example of an infinite locally compact group Γ such that ApΓq is Arens regular is known; such a group Γ must be discrete, non-amenable, and not contain F2 , the free group on two generators, as a subgroup. Such groups are known; an attractive, short example is given by Monod in [243]. In the case where Γ is an infinite locally compact abelian group, ApΓq is not Arens regular, but it is shown by Graham in [144] that there may be restriction algebras that are Arens regular. The following result is an immediate application of Corollary 6.1.26; for this we note that, for each locally compact group Γ, the Banach function algebra ApΓq is weakly sequentially complete, that ApΓq has a BSE norm by Theorem 5.3.18(i), and that, in the case where Γ is discrete, so that ApΓq is a Tauberian Banach sequence algebra, ApΓq is a compact algebra by Proposition 3.2.3. Finally, we note that C BS E pApΓqq “ Bρ pΓq by Theorem 5.3.18(ii). However, it is not known whether there is any infinite, discrete group Γ that satisfies the equivalent conditions given in the theorem. Theorem 6.4.4 Let Γ be a discrete group. Then the following are equivalent: (a) the algebra ApΓq is Arens regular; (b) the algebra Bρ pΓq is Arens regular; (c) the algebra Bρ pΓq is an ideal in its bidual; (d) Bρ pΓq l Bρ pΓq Ă Bρ pΓq; (e) Bρ pΓq l Bρ pΓq Ă ApΓq.

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409

Example 6.4.5 Let Γ be a locally compact group, and then take p and r such that 1 < p < 8 and 1 ď r < 8. Consider the example Arp pΓq “ A p pΓq ∩ Lr pΓq of Example 4.4.13, so that Arp pΓq is a weak Segal algebra with respect to A p pΓq. Suppose that Γ is metrizable and that Arp pΓq is Arens regular. Then, by Proposition 6.4.2(ii), Γ is discrete. See also [151, §3, Corollary 4]. In particular, Γ is discrete whenever Γ is metrizable and the Lebesgue–Fourier algebra LApΓq on Γ is Arens regular. Conversely, when Γ is discrete, we have LApΓq “  1 pΓq, and so LApΓq is Arens regular. See also [131, Theorem 5.1]. Similarly, Arp pΓq “  r pΓq whenever 1 ď r ď p ∨ q, and so Arp pΓq is also Arens regular in this case.

The following theorem of Forrest is [122, Theorem 2]; it is the analogue of Corollary 6.4.3, but is not quite as strong. For generalizations of this theorem, see [110, §5.4]. Theorem 6.4.6 Let Γ be a locally compact group, and take p with 1 < p < 8. Suppose that A p pΓq is Arens regular. Then Γ is discrete and every abelian subgroup of Γ is finite. Again, there is no known infinite group Γ for which A p pΓq is Arens regular for any p with 1 < p < 8. We now consider when the Banach function algebras that we are discussing are strongly Arens irregular. The following result was first proved by Lau and Losert [214, Theorem 6.5] by a different argument from ours. Losert [234] has recently shown that ApΓq is not strongly Arens irregular whenever Γ is discrete and contains F2 , and so the condition in the following theorem that Γ be amenable cannot be dispensed with. Theorem 6.4.7 Let Γ be an amenable, discrete group. Then ApΓq is strongly Arens irregular. Proof We have noted that ApΓq is always weakly sequentially complete, and, for each discrete group Γ, the Banach sequence algebra ApΓq is an ideal in ApΓq . In the case where Γ is also amenable, ApΓq has a CAI, and so Theorem 6.1.20 applies to show that ApΓq is strongly Arens irregular. It seems that nothing non-trivial is known about when A p pΓq is strongly Arens irregular, save when p “ 2. p be a LCA group, so that the Banach function algebra ApΓq is strongly Let Γ “ G Arens irregular, and take E P Rc pΓq. Then, by Theorem 4.3.32(iii), IpEq has a bounded approximate identity, and so IpEq is also strongly Arens irregular by Proposition 6.1.6. Suppose that IpEq is Arens regular (and non-zero) and Γ is metrizable. Then, by Theorem 6.4.2(i), Γ is discrete. Also, IpEq is weakly sequentially complete and it has an identity, and so Γ \ E is compact, and hence finite. Thus IpEq is finite

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dimensional. This shows that IpEq is Arens regular if and only if L1 pGq is Arens regular. Let G be an infinite, compact abelian group, with dual group Γ. For a subset S of Γ, we define MS pGq “ {μ P MpGq : p μ | pΓ \ S q “ 0} ,

LS1 pGq “ L1 pGq ∩ MS pGq .

Then MS pGq is a weak-∗-closed ideal in MpGq and LS1 pGq is a closed ideal in MS pGq that is weakly sequentially complete as a Banach space. The algebra LS1 pGq is a Tauberian Banach sequence algebra on S , and MS pGq is a dual Banach sequence algebra on S . Suppose that the set S is a Riesz set, which means that LS1 pGq “ MS pGq. Then it is proved by Ülger in [317, Corollary 2.5] that LS1 pGq is Arens regular. Some special cases of the fact that L1 pGq is strongly Arens irregular whenever G is a locally compact group are very easy; a stronger form of the next proposition will be given in Theorem 6.4.13. Proposition 6.4.8 Let G be a compact group. Then pL1 pGq,  q is strongly Arens irregular. Proof Since G is compact, L1 pGq is an ideal in its bidual by Theorem 4.1.37. Since L1 pGq has a CAI and L1 pGq is weakly sequentially complete as a Banach space, the result follows from Theorem 6.1.20. Example 6.4.9 Consider the case where G “ T, so that pL1 pTq,  q is strongly Arens irregular. The subset N of Z is a Riesz set (this is the famous M. and F. Riesz theorem [201, §3.13]), and so the closed ideal L1N pTq is Arens regular. Thus we have a Banach sequence algebra ApZq that is SAI and a closed ideal that is Arens regular, but not SAI. Let G be a locally compact group. As in Theorem 4.1.8, the measure algebra pMpGq,  q on G is a dual Banach algebra with isometric Banach-algebra predual C 0 pGq. The hyper-Stonean envelope (see Example 3.6.5) of the space G is denoted by r As before, the canonical projection is πG : G r Ñ G8 , the dual space of MpGq is G. r and the bidual space is MpGq r “ MpGq . Here MpGq is an unital, C 0 pGq “ CpGq, commutative C ∗ -algebra, and its identity, the constant function 1, when regarded as a functional on MpGq, is just the augmentation character of Definition 4.1.19. We have noted that the dual space L8 pGq of the group algebra L1 pGq is a C ∗ algebra with respect to the pointwise product, and again the constant function 1, when regarded as a functional on L1 pGq, is just the augmentation character restricted to L1 pGq. Definition 6.4.10 Let G be a locally compact group. Then the two unital Banach r l q and pMpGq, r  q are the algebras formed by identifying MpGq r algebras pMpGq, with pMpGq , l q and pMpGq ,  q, respectively.

6.4 Arens regularity for algebras on locally compact groups

411

It follows from Theorems 2.4.10 and 4.1.28(ii) that C 0 pGqK is a closed ideal in r that C 0 pGq Ă WAPpMpGqq “ WAPpGq, and that MpGq, r “ MpGq  C 0 pGqK . MpGq r Ñ MpGq is a continuous algebra In particular, the projection map πG : MpGq r Ñ G when we identify points epimorphism that extends the previous map πG : G r r{eG } “ π´1 p{eG }q for the of G and G with their corresponding point masses. We set G G fibre above the identity eG of G. In the case where the group G is compact, we have identified the algebras pMpΦX q, l q and pMpGq,  q, where X “ CpGq, and so the map πG coincides with the quotient map qX that was used earlier. Let G be a locally compact group, and denote left Haar measure on G by m, and r of so we can regard Φ :“ Φm as a clopen subset of the hyper-Stonean envelope G r G, following the notation of page 211; we set Φ{eG } “ Φ ∩ G{eG } . We recall from page 212 the definition of the canonical net   μ| B :BÑϕ μB “ μpBq associated to a positive measure μ on G and a character ϕ P Φμ . The following theorem extends [182, Theorem 3.2]. Theorem 6.4.11 Let G be a locally compact group. Then: r for each ϕ P Φ and ψ P G r{eG } ; (i) ϕ l ψ “ ψ  ϕ “ ϕ P Φ Ă G (ii) in the case where the group G is compact, M l N “ M l πG pNq ,

N  M “ πG pNq  M

r . pM P MpΦq, N P MpGqq

r{eG } and a set B P BG such that 0 < mpBq < 8. Proof (i) First, we fix ψ P G For each ε > 0 and each A P BG with mpAq < 8, there exists V P NeG such that mpAt´1 \ Aq < ε mpBq

pt P Vq .

For each μ P MpGq+ and C P BG with μpCq > 0, we have     χA pstq dmB psq dμC ptq “ χAt´1 psq dmB psq dμC ptq B C B C   mpAt´1 \ Aq dμC ptq . ď χA psq dmB psq + mpBq B C Thus, in the case where C Ă V, it follows from equation (4.1.9) that xχA , mB  μC y ď xχA , mB y + ε . By Proposition 3.6.7, we can take the limits lim CÑψ to see that

412

6 Arens regularity

xχA , mB l δψ y ď xχA , mB y + ε . This holds for each ε > 0, and so xχA , mB l δψ y ď xχA , mB y. However, this inequality also holds if A be replaced by G \ A, and so xχA , mB l δψ y “ xχA , mB y . It follows that xλ, mB l δψ y “ xλ, mB y pλ P L8 pGqq. Since mB P MpΦq and L8 pGq “ CpΦq, we have mB l δψ “ mB . Finally, we take the limits limBÑϕ to r see that ϕ l ψ “ ϕ P Φ Ă G. r Similarly, ψ  ϕ “ ϕ P Φ Ă G. (ii) We return to the above formula mB l δψ “ mB , which holds for each ψ P r G{eG } and B P BG with mpBq > 0. r Since G is compact, there exists s P G with πG pψq “ s. Now suppose that ψ P G. r{eG } , and so mB l δψ l s´1 “ mB , whence Then ψ l s´1 P G mB

l

ψ “ mB  s “ mB  πG pψq .

This formula extends to give mB l N “ mB l πG pNq for each N which is a linear r and then, by taking weak-∗ limits, for each combination of point masses in MpGq, r N P MpGq. We now take limits limBÑϕ to establish that ϕ l N “ ϕ l πG pNq for each ϕ P Φ r and then take linear combinations of point masses in Φ and further and N P MpGq, weak-∗ limits to see that M

l

N “ M l πG pNq

r ; pM P MpΦq, N P MpGqq

r l q. this last step is valid because the map R N is weak-∗ continuous on pMpGq, r Similarly, N  M “ πG pNq  M pM P MpΦq, N P MpGqq. Clause (i), above, shows that pΦ{eG } , l q is a left-zero semigroup for every locally compact group G. Corollary 6.4.12 Let G be a locally compact group, let M P MpΦq, and take ψ P r{eG } . Then M l δψ “ δψ  M “ M. G The above result, in the case where ψ P Φ{eG } , says that the element δψ is a mixed identity for MpΦq “ L1 pGq . r is identified with β G, in Let G be a discrete group. Then MpGq “  1 pGq and G the sense that δu l δv “ δu l v pu, v P β Gq . For a general locally compact group G, we have β Gd “ G and pβ Gd , l q is a compact, right topological semigroup which is a subsemigroup of pMpβ Gd q, l q. r that In the case where G is not discrete, the space Φ “ ΦmG is a clopen subset of G is disjoint from G. In general, there is some semigroup structure on subsets S and T

6.4 Arens regularity for algebras on locally compact groups

413

r (in the sense that δu l δv is a point mass whenever u P S and v P T ), but this is of G r see the memoir of Dales, Lau, and Strauss [59, Chapter not true of the whole of G; r 8]. Here we shall describe the semigroup structure of Φ Ă G. We have already proved in Proposition 6.4.8 that L1 pGq is strongly Arens irregular whenever G is a compact group; we shall now show that, further, there is a specific DLTC set. r is a DLTC set Theorem 6.4.13 Let G be a compact group. Then the subset Φ{e} of G for L1 pGq. Proof Set A “ L1 pGq. Take an element M P MpΦq such that M l δϕ “ M  δϕ pϕ P Φ{eG } q, so that πG pMq P CpGq “ MpGq, and then take λ P A “ L8 pGq. For each g P A, we have xπG pMq · λ , gy “ xλ, g  πG pMqy “ xλ · g, πG pMqy “ xλ · g, My because λ · g P CpGq. However xλ · g, My “ xM · λ, gy by definition, and so πG pMq · λ “ M · λ in A . Let ϕ P Φ{eG } . By Corollary 6.4.12, δϕ is a mixed identity for MpΦq, and so M l δϕ “ δϕ  M “ M. Since M l δϕ “ M  δϕ , we have M  δϕ “ M. Thus, for each λ P A , we have xλ · πG pMq, δϕ y “ xλ · M, δϕ y “ xλ, M  δϕ y “ xλ, My . This shows that the function λ · πG pMq is constant on the fibre Φ{eG } . By Proposition 3.6.8 (taking μ “ m), the function λ · πG pMq is continuous at the point e, and so λ · πG pMq P CpGq. By Proposition 4.1.18(ii), πG pMq P L1 pGq, say πG pMq “ f . It follows that M · λ “ f · λ, and hence xλ, My “ xλ, δϕ l My “ xM · λ, δϕ y “ x f · λ, δϕ y “ xλ, f y . This holds for each λ P A , and so M “ f P L1 pGq. r that is a DLTC set We do not know whether there is always a finite subset of G 1 for L pGq whenever G is compact. Let G be a locally compact group, set A “ L1 pGq, and let X be a faithful, leftintroverted A-submodule of L8 pGq such that X “ XA. We recall from Corollary 6.1.15 that we regard MpGq as a subset of Zt pX  q. Our next aim is to show that MpGq “ Zt pX  q in the special case where X “ LUCpGq. This was first shown by Grosser and Losert [156] in the abelian case, and by Lau in the general case [212]; see also the theorem of Lau and Pym in [218]. We shall establish a stronger result (in the case where G is σ-compact), based on Theorem 12.24 of the memoir [58]; an independent proof of the same result was given by Budak, I¸sik, and Pym in [29, Theorem 1.2(ii)], and a similar result was proved by Filali and Salmi in [115, Theorem 14]. The theorem of Lau is generalized to cover the case of weighted versions of LUCpGq in [57, Theorem 11.9], [116, Theorem 5], and [248, Theorem

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6 Arens regularity

1.2]. For generalizations of the main result to cover topological groups that may not be locally compact, see the paper of Ferri, Neufang, and Pachl [108]. Let G be a locally compact group, set A “ L1 pGq, and let X be unital C ∗ subalgebra of C b pGq with C 0 pGq Ă X such that X is a faithful and left-introverted A-submodule of L8 pGq and X “ XA. We recall from Theorem 4.1.33 that we identify pX  , l q with pMpΦX q, l q and then we identify pMpΦX q | G, l q with pMpGq,  q and pMpΦX \ Gq, l q with pC 0 pGqK , l q, so that pMpΦX q, l q “ pMpGq,  q  pC 0 pGqK , l q . Definition 6.4.14 Let G be a locally compact group, set A “ L1 pGq, and let X be unital C ∗ -subalgebra of C b pGq with C 0 pGq Ă X such that X is a faithful and leftintroverted A-submodule of L8 pGq and X “ XA. A function F : ΦX Ñ MpΦX q is G-continuous at ϕ P ΦX if limtÑϕ Fptq “ Fpϕq, where we require that t P G in the limiting process. Theorem 12.24 of [58] is the following. Theorem 6.4.15 Let G be a locally compact, non-compact group. Then there is a subset {a, b} of ΦLUCpGq such that μ P MpGq whenever μ P MpΦLUCpGq q and Lμ | ΦLUCpGq is G-continuous at both a and b, and so Zt pLUCpGq q “ MpGq. In fact, we shall give the proof only in the special case that G is also σ-compact; the full proof is given in [29] and [58]; the proof is a modest extension of the one given in Theorem 6.3.7. Let U be a fixed compact, symmetric neighbourhood of eG in G, and let pKn q be a sequence of compact subsets of G such that U Ă Kn pn P Nq and the sets Kn cover G. By modifying this sequence, we may suppose that each set Kn is symmetric, that  UKn2  int Kn+1 pn P Nq, and that {int Kn : n P N} “ G. We choose a sequence ptn q in G such that (6.4.1) tn+1 P int Kn+1 \ UKn2 pn P Nq . For each n P N, we have UKn “ UKn Ă Kn+1 . By Proposition 4.1.34(iii), UKn is a neighbourhood of Kn in ΦX ; it follows that Kn Ă int Kn+1 . Set T “ {tn : n P N}. Let C and D be disjoint subsets of T . Then C ∩ UD “ ∅, and so, by Proposition 4.1.34(iii), C ∩ D “ ∅. Thus the natural map from β Gd onto ΦX has a restriction to β T that is a homeomorphism from β T onto T . We now claim that each t P T ∗ is right cancellable in the semigroup ΦX . To see this, take ϕ, ψ P ΦX with ϕ  ψ, and assume that ϕ l t “ ψ l t. Take K and L to be disjoint, compact neighbourhoods of ϕ and ψ in ΦX , respectively. By Proposition 4.1.34(i), there is a symmetric neighbourhood W of eG with K ∩ WL “ ∅, and we may suppose that W Ă U. Since ϕ l t “ ψ l t, there exist m1 , n1 , m2 , n2 P N with m1 < n1 and m2 < n2 such that pKm1 ∩ Kqtn1 ∩ WpKm2 ∩ Lqtn2  ∅ .

6.4 Arens regularity for algebras on locally compact groups

415

By equation (6.4.1), this implies that n1 “ n2 . However, in this case, it follows that K ∩ WL  ∅, a contradiction. Thus u is right cancellable in ΦX , giving the claim. (For further results on right cancellable elements in ΦX , see the paper by Filali and Pym [114].) It follows from Proposition 1.2.30(ii) that μ “ μ l u for each μ P M R pΦX q and each u P T ∗ . For each s P G, take Npsq “ min{m P N : s P Km }, essentially as in the proof of Theorem 6.3.7. Take u P U and s P G with Npsq “ n. Then us P Kn+1 . Also s P Km+1 when us P Km , and so Npusq P {Npsq ´ 1, Npsq, Npsq + 1}. Now take s P G with Npsq “ m and take n > m. Then stn P Kn+1 , but stn  Kn´1 , and so Npstn q P {n, n + 1}. Similarly, Npstn q P {Npsq ´ 1, Npsq, Npsq + 1} whenever Npsq > n + 1. Next set γpsq ≡ Npsq pmod 12q for s P S , so that we have a map γ : S Ñ Z 12 . Let A “ {tn : γptn q “ 2} and B “ {tn : γptn q “ 8}, and choose a P A∗ and b P B∗ so that μ “ μ l a “ μ l b pμ P M R pΦX qq. Set X1 “ {s P G : γpsq P {2, 3}} , X2 “ {s P G : γpsq P {8, 9}} , Y1 “ {s P G : γpsq P {0, 1, 2, 3, 4, 5}} , Y2 “ {s P G : γpsq P {6, 7, 8, 9, 10, 11}} , Z “ {s P G : γpsq P {5, 6, 7, 8, 9, 10, 11, 0}} . so that A Ă X1 Ă Y1 , B Ă X2 Ă Y2 , Y1 ∪ Y2 “ G, and Y1 ∩ Y2 “ ∅. We see that X1 ∩ UZ “ ∅, and so X1 ∩ Z “ ∅ . (6.4.2) We also have ϕ l a P X1 and ϕ l b P X2 when ϕ P ΦX , and so supp pμ l aq Ă X1 ,

supp pμ l bq Ă X2

pμ P MpΦX qq .

Finally, we have ϕ l t P Z for each t P T and each ϕ P Y2 \ G, and so supp pν l tq Ă Z

pt P T, ν P Y2 \ Gq .

(6.4.3)

Now fix μ P M R pΦX q such that Lμ | ΦX : ΦX Ñ MpΦX q is G-continuous at both a and b. We set μ1 “ μ | G ,

μ2 “ μ | pΦX \ Gq ,

ν1 “ μ2 | Y1 ,

ν2 “ μ2 | pY2 \ Y1 q ,

so that μ “ μ1 + ν1 + ν2 , and hence μ l a “ limpμ1 l t + ν1 l t + ν2 l tq , tÑa

where t P X1 in the limit. Since supp pμ l aq Ă X1 , it follows from (6.4.2) and (6.4.3) that μ l a “ limpμ1 l t + ν1 l tq | X1 , tÑa

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and so μ “ μ l a ď μ + ν1 . Similarly, μ ď μ + ν2 , and so 2 μ ď 2 μ1  + ν1  + ν2  . However, μ “ μ1  + ν1  + ν2 , and so ν1 “ ν2 “ 0, and μ “ μ1 P MpGq. It follows that μ P MpGq for each μ P MpΦX q such that Lμ | ΦX : ΦX Ñ MpΦX q is G-continuous at both a and b, and this completes the proof of Theorem 6.4.15 in the case where G is also σ-compact. Take H to be a subset of G that is not contained in any compact subspace of G. We remark that we could choose the sequence ptn q to be contained in H, and so we could suppose that a, b P H. Let G be a locally compact group, and again set A “ L1 pGq

and

X “ LUCpGq “ A A .

As before, X  “ A /X K with quotient (restriction) map qX : A “ MpΦq Ñ X  “ MpΦX q , so that, as in Theorem 4.1.33, pX  , l q is an unital Banach algebra that contains MpGq as an unital, closed subalgebra. As in equation (2.3.12), we have qX pM l Nq “ M l qX pNq pM, N P A q , and so there is a natural product A × X  Ñ X  that identifies X  as a Banach left A -module. In particular, we can define μ · ϕ for each μ P MpΦq and ϕ P ΦX . Definition 6.4.16 A measure μ P MpΦq is G-continuous at a point ϕ P ΦX if limtÑϕ μ · t “ μ · ϕ, where we require that t P G in the limiting process. We shall now prove that every group algebra L1 pGq is strongly Arens irregular, and seek to find ‘small’ DLTC sets for this algebra. Historically, Lau and Losert first proved [213, Theorem 1] that L1 pGq is strongly Arens irregular for each locally compact group G, following the earlier influential paper [182] of I¸sik, Pym, and Ülger. A significant paper is that of Lau and Ülger [220], which gave a new proof that each L1 pGq is strongly Arens irregular; this paper contains an historical summary of related results up to 1993. In [247], Neufang gave a shorter proof of a stronger version of the result. A further paper of Filali and Salmi [115] establishes this result in an attractive way, and unifies this result with several related results. Finally, the paper [29] of Budak, I¸sik, and Pym proves a stronger result in the non-compact case, namely that, for a locally compact, non-compact group G, there are just two elements in L1 pGq that form a DLTC set for our algebra; this result does not apply to compact groups, such as T. The following is [29, Theorem 1.2(iii)]. Theorem 6.4.17 Let G be a locally compact, non-compact group. Then L1 pGq is strongly Arens irregular and there is a two-point DLTC set for L1 pGq.

6.4 Arens regularity for algebras on locally compact groups

417

Theorem 6.4.18 Let G be a locally compact group. Then pL1 pGq,  q is strongly Arens irregular. Proof This follows from Theorem 6.4.17 when G is not compact and from Theorem 6.4.13 when G is compact. Example 6.4.19 Let G be a locally compact group, and take A “ L1 pGq and X “ LUCpGq. By Theorem 6.4.15, Zt pX  q “ MpGq and, by Theorem 6.4.17, p q p q Zt pA q “ A, and so Zt pA q ∩ X   Zt pX  q when G is not discrete.

Let G be a locally compact group. The fact that LUCpGq Ă WAPpGq implies that G is compact was first proved by Granirer in [146]; a different proof is given by Lau in [212, Corollary 4]; our proof is based on [220]. Proposition 6.4.20 Let G be a locally compact group such that LUCpGq is a subset of WAPpGq. Then L1 pGq is an ideal in its bidual, and G is compact. Proof Set A “ L1 pGq. By Theorem 4.1.24, A A “ LUCpGq for every locally compact group G. By Theorem 4.1.28(ii), WAPpAq “ WAPpGq. Thus our hypothesis is that A A Ă WAPpAq. Since WAPpGq “ WAPpG op q, also AA Ă WAPpAq. By Theorem 6.4.18, A is SAI. Thus it follows from Corollary 6.1.9 that A is an ideal in A . By Theorem 4.1.37, G is compact. The semigroup algebra L1 pS q of the following theorem was defined in Definition 4.1.13. The result is proved by a small variation of the proof of Theorem 6.4.17. Theorem 6.4.21 Let G be a locally compact group, and let S be a closed subsemigroup of G such that eG P S and such that mG pV ∩ S q > 0 for each open neighbourhood V of eG . Suppose that S is not compact. Then L1 pS q is strongly Arens irregular and there is a two-point DLTC set for L1 pS q. Corollary 6.4.22 The Banach function algebra pL1 pR+ , +q,  q is strongly Arens irregular and there is a two-point DTC set. We introduced the notion of a Banach algebra being extremely non-Arens regular in Definition 2.3.39. The following extension of the fact that group algebras are not Arens regular is proved by Filali and Galindo in [109]. Theorem 6.4.23 Let G be an infinite locally compact group. Then the group algebra L1 pGq is extremely non-Arens regular. The class of Beurling algebras was defined in Definition 4.1.39. The following theorem, which is an analogue of Theorem 6.3.31, is given by Neufang in [248, Theorem 1.2] as a consequence of an impressive general factorization theorem. An upper bound for the size of a DLTC, in the case of certain groups G, is given by Filali and Salmi in [116, Theorem 10].

418

6 Arens regularity

Theorem 6.4.24 Let G be a non-compact, locally compact group, and let ω be a continuous weight on G. Suppose that there is a scattered set on which the weight ω is diagonally bounded. Then the Beurling algebra L1 pG, ωq is strongly Arens irregular. For related results on when Beurling algebras are extremely non-Arens regular, see [112]. We conclude with a major theorem of Losert, Neufang, Pachl, and Stepr¯ans [236] that resolved a long-standing question. Further extensions, when the group G is replaced by certain ‘hypergroups’, are given by Losert in [235]. Theorem 6.4.25 Let G be a locally compact group. Then the measure algebra pMpGq,  q is strongly Arens irregular.

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© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 H. G. Dales and A. Ülger, Banach Function Algebras, Arens Regularity, and BSE Norms, CMS/CAIMS Books in Mathematics 12, https://doi.org/10.1007/978-3-031-44532-3_0

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Index of Terms

A-invariant, 142 C ∗ -algebra, 94 C ∗ -algebra, commutative, 94 C ∗ -algebra, group, 270 C ∗ -algebra, postliminal, 102 C ∗ -algebra, reduced group, 276 C ∗ -algebra, representation on Hilbert space, 98 C ∗ -algebra, scattered, 101, 103 C ∗ -isomorphic, 95 C ∗ -isomorphism, 95 Gδ -set, 10 W ∗ -algebra, 99 p-convolution operator, 290 p-pseudo-functions, 290 p-pseudo-measures, 290 A absolutely continuous, 167 absolutely convergent Taylor series, 213 absolutely convex hull, 6 accumulation point, 10 affine map, 7 AL-space, 29, 30, 60, 62 algebra, 72 algebra, factors, 72 algebra, factors weakly, 72 algebra, faithful, 76 algebra, group, 79 algebra, opposite, 72 algebra, radical, 73 algebra, semigroup, 79, 82, 86, 148, 155, 180, 199, 237, 357, 368, 383, 386, 388 algebra, semisimple, 73 algebra, weighted semigroup, 80, 146, 383 almost periodic, 113, 244

AM-space, 29 AM-space, with a unit, 29 annihilator, 26 approximate identity, 84 approximate identity, bounded, 84 approximate identity, bounded pointwise, 198 approximate identity, contractive, 84 approximate identity, left, right, 84 approximate identity, multiplier-bounded, 87, 282 approximate identity, pointwise, 198 approximate identity, sequential, 84 approximate identity, weak, 86 approximate units, 84 approximation property, 58, 69, 371 approximation property, bounded, 58, 69 Archimedean, 7 Arens products, 104 Arens regular, 110, 114, 121, 385 augmentation ideal, 240 B Banach ∗-algebra, 77 Banach A-bimodule, 91 Banach A-bimodule, bidual, 91 Banach A-bimodule, dual, 91 Banach A-bimodule, essential, 93 Banach A-bimodule, neo-unital, 93 Banach A-bimodules, isomorphic, 92 Banach A-module, 91 Banach algebra, 77 Banach algebra representation, 80 Banach algebra, compact, 106 Banach algebra, dual, 138, 147 Banach algebra, dual, isometric, 138

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 H. G. Dales and A. Ülger, Banach Function Algebras, Arens Regularity, and BSE Norms, CMS/CAIMS Books in Mathematics 12, https://doi.org/10.1007/978-3-031-44532-3_0

431

432 Banach algebra, of power series, 266 Banach algebra, reflexive, 77 Banach algebras, isometrically isomorphic, 77 Banach algebras, isomorphic, 77 Banach function algebra, 153 Banach function algebra, contractive, viii, 154, 347 Banach function algebra, dual, 153 Banach function algebra, equivalent to a uniform algebra, 153 Banach function algebra, natural, 153 Banach function algebra, normal, 156 Banach function algebra, pointwise contractive, viii, 198 Banach function algebra, reflexive, 153, 268 Banach function algebra, regular, 156 Banach function algebra, self-adjoint, 153 Banach function algebra, strongly regular, 156 Banach function algebra, Tauberian, 156 Banach function algebra, uniform, 153 Banach function algebra, unital, 153 Banach lattice, 19 Banach lattice algebra, 78, 80, 234, 235 Banach lattice algebra, homomorphism, 78, 236 Banach lattice algebra, representation, 80 Banach lattice, Dedekind complete, 19 Banach lattice, dual, 28, 29 Banach lattice, real, 18 Banach left A-module, 91 Banach left A-modules, isomorphic, 92 Banach operator algebra, 79, 122 Banach sequence algebra, 170 Banach sequence algebra, dual, 170 Banach space, 16 Banach space, c-injective, 53 Banach space, dual, 21, 46 Banach space, injective, 53 Banach space, reflexive, 22 Banach-algebra embedding, 77 Banach-algebra embedding, isometric, 77 Banach-algebra homomorphism, 77 Banach-lattice isomorphic, 60 Banach-lattice isomorphism, 60 Banach-lattice, homomorphism, 60 Banach-lattice, isometric, 60 Banach-lattice, isometry, 60 band, 8, 19, 235 band, projection, 8 basis, subbasis, 10 belongs locally, 160

Index of Terms Beurling algebra, 80, 147, 253, 265, 327, 417 bidual, 22 bidual algebra, 104 big disc algebra, 225 biregular, 375, 376 Boolean algebra, 12, 13 Boolean algebra, complete, 12 Borel function, 13 Borel function, bounded, 13 Borel measure, 13 Borel measure, regular, 13, 230 Borel sets, 13 boundary, 209 boundary, Choquet, 209, 343 boundary, closed, 209 boundary, Šilov, 209 bounded variation, 167, 178 BSE algebra, v, 310, 313 BSE function, v, 311 BSE norm, vi, 310 C cancellable, 5 cancellable, left, right, 5 canonical embedding, 135 canonical net, 212, 411 Cantor group, 254 Cantor set, 14, 209 cardinality, 2 centre, 72 character, 253 character, augmentation, 6, 82, 240, 282, 386, 410 character, evaluation, 153 clusters, 31 clusters, 0-, 31 Cole algebra, viii, 218, 226, 331, 347 commutant, 72 commutes, 72 compact operator, 51 compactification, 11 compactification, Bohr, 244, 304 compactification, one-point, 11 ˇ compactification, Stone–Cech, 15, 383 complemented, 17, 48, 64 complemented, c-, 48 complemented, in bidual, 49 completely continuous operator, 51 component, 11 cone, 7 cone, positive, 7 conjugate index, 2 continuous functional calculus, 95

Index of Terms continuous map, 11 continuously differentiable, 169 Continuum Hypothesis, 2 contraction, 40 contractive, 208 converges in order, 4 convex, 6 convex hull, 6 convolution product, 79, 232 copy, 40 coset ring, 231 coset ring, closed, 231, 280 cozero set, 13 D Dales–Davie algebra, 169, 329 Dedekind complete, 4, 19 Dedekind complete, CpKq, 20 Dedekind complete, MpKq, 20 Dedekind complete, B r pE, Fq, 61 Dedekind complete, dual, 29 dense-in-itself, 10 dentable, 38 determining for the left topological centre, 363, 385 determining set, 155 disc algebra, 213, 219, 224, 302, 330, 342 disjoint, 7 disjoint complement, 8 distance, 11 Dixmier projection, 49 Douglas algebra, 226 dual Banach algebra, 138, 234 dual group, 253 Dunford–Pettis property, 58, 59, 101, 102 E eigenspace, 78 embedding, 11 embedding, C ∗ -, 95 epimorphism, 74 equicontinuous, 14 equivalent to the  1 -norm, 344 extreme point, 6 extremely non-Arens regular, 116, 361, 417 F Feinstein algebra, 175, 355 fibre, 108, 210 Figà-Talamanca–Herz algebra, 286, 339, 406 fixes, 40 Fourier algebra, 254, 273, 406 Fourier transform, 83, 254

433 Fourier transform, inverse, 254 Fourier–Stieltjes algebra, 255, 270 Fourier–Stieltjes algebra, reduced, 277 Fourier–Stieltjes transform, 255 function algebra, 152 function, left uniformly continuous, 242 function, positive-definite, 272 function, right uniformly continuous, 242 functional, normal, 99 G Gel’fand topology, 81 Gel’fand transform, 82 Gleason metric, 218 Gleason part, viii, 219, 345 Grothendieck inequality, 100 Grothendieck space, 38, 56, 106, 232 Grothendieck’s constant, 57 group, 4 group C ∗ -algebra, 270 group algebra, 79, 122, 233 group morphism, 5 group von Neumann algebra, 275 group, amenable, 241, 279 group, free, 5 group, maximally almost periodic, 244, 253 group, topological, 12 group, unimodular, 231 growth, 15, 398 H Haagerup tensor product, 381 Haar measure, 230 homeomorphic embedding, 40 homeomorphism, homeomorphic, 11 homomorphism, 74 homomorphism, ∗-, 75 homomorphism, C ∗ -, 95 hull, 157 hyper-Stonean, 21, 100, 145 hyper-Stonean envelope, 210, 410 I ideal, 5 ideal in bidual, 105 ideal, ∗-, 75 ideal, left, maximal modular, 73 ideal, left, modular, 72 ideal, left, right, 5, 72 ideal, maximal, 72 idempotent, 5, 72 identity, left, right, 72 interpolation sequence, 209, 226 interpolation set, 209

434 introverted, 123, 124, 295 introverted, left, 123, 124, 134, 242, 247, 363 introverted, right, 123 invariant subspace, 98 involution, 75 involution, linear, 75, 112 isometric embedding, 40 isometric lattice embedding, 60 isomorphic, 74 isomorphism, 74 iterated-limit condition, 113, 168 iterated-limit criterion, 33 J Jacobson radical, 73 James algebra, 174, 353 James space, 47, 174 K KB-space, 61, 236 L Laplace transform, 255 lattice, 3, 391 lattice homomorphism, 4 lattice isomorphism, 4 lattice norm, 18 lattice, complete, 4 lattice, Dedekind complete, 4 Lebesgue–Fourier algebra, 283, 289, 337, 409 left-regular representation, 80, 81, 236, 276 left-regular representation, of Γ, 275 limit point, 10 linear homeomorphism, 40 linear isomorphism, 7 linear space, 6 linear space, ordered, 7 linearly homeomorphic, 40 linearly isometric, 40 linearly isomorphic, 7 Lipschitz algebras, 168, 303, 328 Lipschitz functions, 168, 348 locally compact group, 230 locally compact space, 10 locally convex space, 21 M maximal, minimal element, 3 mean, 241 mean, invariant, 241 mean, left-invariant, 241 mean, right-invariant, 241

Index of Terms measure, 12 measure algebra, 234 measure space, 12 measure space, σ-finite, 12 measure space, finite, 12 measure space, probability, 12 measure, continuous, 21 measure, discrete, 21 measure, normal, 21, 99 measure, positive, 12 measure, probability, 20 measure, regular, 13 measure, representing, 215 measure, support, 20 measure, total variation, 12, 20 metric space, 11 metric space, complete, 11 metric space, totally bounded, 11 metrizable, 11 metrizable, completely, 11 mixed identity, 126–129, 133, 134, 412 modular function, 231 module homomorphism, 75 module homomorphism, Banach, 92 modulus, 7, 19 monoid, 4 monomorphism, 74 multiplier, 76 multiplier algebra, 76, 162 multiplier, left, right, 76 N natural Banach function algebra, 153 negative part, 7 nil, 74 nilpotent, 74 norm-one characters, 190 normal, 94 normal family, 16 normalized point mass, 398 normed algebra, 76 normed algebra, unital, 77 normed lattice, 18 norming, 25, 381 norming subset, 25 nuclear algebra, 80, 92, 107, 146 null sequence, 17 O open map, 11 operator, 39 operator space, 381 operator, pq, pq´summing, 57 operator, p -summing, 57

Index of Terms operator, approximable, 52 operator, bidual, 42 operator, compact, 51 operator, completely continuous, 51 operator, dual, 42 operator, finite-rank, 51 operator, Hilbert–Schmidt, 97 operator, nuclear, 65 operator, order-bounded, 8, 61 operator, positive, 8, 61 operator, regular, 8, 61 operator, trace-class, 97 operator, unconditionally converging, 54 operator, weak-∗ continuous, 42 operator, weakly compact, 51 operator, weakly precompact, 51 orbit, left, right, 244 order-bounded, 3, 19 order-closed, 4 order-continuous, 4, 20 order-dual, 9 order-ideal, 8, 19 order-interval, 3 P pairing, 23 partition of unity, 159 peak interpolation set, 209 peak set, peak point, 157 peak-point conjecture, 218 perfect, 10 pointwise approximate identity, 198 pointwise approximate identity, bounded, 198 pointwise approximate identity, contractive, 198 pointwise bounded, 14 pointwise contractive, 198 pointwise convergence, 153 positive, 95 positive part, 7 positive square root, 95 power set, 2 power-bounded, 81 pre-annihilator, 26 predual, 46 predual, Banach algebra, 138 predual, concrete, 46, 138 predual, isometric, 46, 138 predual, unique, 138 product topology, 11 projection, 7, 48 projection, band, 8, 19 property pVq, 37, 66, 98, 121, 232, 377, 383

435 Q quasi-nilpotent, 74 quotient map, 7 quotient norm, 17 quotient operator, 43 R Radon–Nikodým property, 38, 65, 69, 101, 371 Rajchman algebra, 170, 258, 284, 338 regular norm, 61 relatively compact, 10 representation, 98, 270 representation, ∗-, faithful, 98 representation, ∗-, universal, 98 representation, irreducible, 98 resolvent set, 74 restriction algebra, 158 Riesz set, 410 S Schatten p-class, 97 Schur property, 35, 55, 59, 85, 102, 121, 162, 283 Segal algebra, 166, 258 Segal algebra, weak, 164 semi-character, 6 semi-direct product, 73 semigroup, 4, 79, 80 semigroup algebra, 79 semigroup monomorphism, epimorphism, isomorphism, 5 semigroup morphism, 5 semigroup, abelian, 4 semigroup, cancellative, 5 semigroup, centre, 4 semigroup, dual, 12 semigroup, free, 5 semigroup, idempotent, 5 semigroup, left topological, 12 semigroup, left-zero, 5 semigroup, opposite, 5 semigroup, product, 5, 89 semigroup, right topological, 12 semigroup, right topological, compact, 12, 248, 384 semigroup, right-zero, 5, 149, 387 semigroup, semi-topological, 12 semigroup, separating, 83 semigroup, topological, 12, 384 semigroup, weakly cancellative, 5, 147 semilattice, 5 separating ball property, 189 separating ball property, strong, 190

436 separating ball property, weak, 189 set of synthesis, 157, 264 spectral radius formula, 81 spectrum, 74, 78, 81 state space, 81, 98, 215 state, pure, 81, 215 Stone space, 12 Stonean space, 11, 13 strong boundary point, 157 strong unit, 8 strongly Arens irregular, 361, 385 strongly Arens irregular, left, right, 361, 385 strongly pointwise contractive, 200, 205, 208, 348 sublattice, 8 submodule, 75 submodule, faithful, 123 subspace, 11 support, 13 Swiss cheese, 224 symmetric, 230 T tensor diagonal, 64 tensor norm, injective, 68 tensor norm, projective, 63 tensor product, 63 tensor products, Banach algebras, 89 tomato can algebra, 224 topological centre, 363, 385 topological centre, left, right, 360, 385 topological group, 12 topological semigroup, 12 topological semigroup, left, right, 12 topological space, σ-compact, 10 topological space, compact, 10 topological space, completely regular, 15 topological space, connected, 11 topological space, extremely disconnected, 11 topological space, first countable, 10 topological space, locally compact, 10 topological space, locally connected, 11 topological space, scattered, 10, 14, 38, 51, 59, 101, 380 topological space, second countable, 10 topological space, separable, 10 topological space, sequentially compact, 10 topological space, Stonean, 11 topological space, totally disconnected, 11 topologically invariant mean, 193, 282 topology, compact-open, 14 topology, strong operator, 96

Index of Terms topology, ultra-weak, 96 topology, weak, 24 topology, weak operator, 96 topology, weak-∗, 24 trace duality, 93 trace map, 93 translation invariant, 230 translation invariant, left-, right-, 230 truncated convolution multiplication, 256 U ultrafilter, 12, 15, 384 uniform algebra, 153, 208 uniform algebra, not complemented, 342 uniform norm, 9 unitary representation, 273 unitary, unitary group, 94 unitization, 156 upper, lower bound, 3 V Varopoulos algebra, 183, 269, 340, 342, 374, 380 vector lattice, 7 Volterra algebra, 256, 367 von Neumann algebra, 99 von Neumann algebra, enveloping, 121 W weak p -summing norm, 56 weak closure, 24 weak Segal algebra, 164 weakly almost periodic, 113, 244 weakly bounded, 24 weakly Cauchy, 34 weakly compact operator, 51 weakly sequentially complete, 34, 65, 276 weakly unconditionally Cauchy, 37 weight, 80 weight, diagonally bounded, 402 weight, radical, 83 weight, regulated, 266 weight, semisimple, 83 weight, symmetric, 80 weight, weakly diagonally bounded, 402 weighted semigroup algebra, 80 weights, equivalent, 80 Wiener–Pitt phenomenon, 257 Z zero product, 80 zero set, 13

Index of Theorems and Examples

Π2 -extension, 57 Ülger, 86, 114, 117–119, 189, 191, 192, 245, 375, 376, 410 Šilov’s idempotent, 82 A Akemann, 100 Arens, 104, 110, 121, 213 Arikan, 116 Arsac, 278 Ascoli–Arzelà, 14, 251 B Baire category, 11 Baker and Rejali, 400 Banach’s isomorphism, 41 Banach–Alaoglu, 26 Banach–Dieudonné, 28 Banach–Mazur, 50 Banach–Steinhaus, 41 Basener, 218, 331 Berglund, Junghenn, and Milnes, 12 Bermúdez and Kalton, 42 Blecher and Read, 173 Bochner–Schoenberg–Eberlein, vi, 293, 313, 333, 335 Bourgain and Pisier, 34, 65 Budak, I¸sik, and Pym, 413, 416 Bédos, 303 C Cannière and Haagerup, 282 Carleson’s corona theorem, 225 Chou and Xu, 288 Chu, Iochum, and Watanabe, 103 Cigler, 259

Civin and Yood, 110, 112 classification, 347 closed graph, 41 Cohen’s factorization, 93 Cole, 218 Cole, Ghosh, and Izzo, 218, 225 Cowling, 290 Craw and Young, 399 D Dabhi and Upadhyay, 358 Dales and Dedania, 402 Dales and Lau, 398, 404, 406, 413 Dales and Loy, 177 Dales, Lau, and Strauss, 211, 413 Dales–Davie, 169, 329 Daws, 123, 138, 146, 285, 371 Daws and Spronk, 290 Daws, Pham, and White, 235 Day, 110, 123 Derighetti, 286 disc algebra, 37 Dixmier, 256 Dorofaeff, 282 Duncan and Hosseiniun, 115 E Eberlein–Šmulian, 28 Emmanuel and Hensgen, 66 Eymard, 270, 271, 275, 279, 333, 335 F Fatou’s lemma, 226 Feinstein, 177, 226, 227 Fernández-Polo and Peralta, 98 Ferri, Neufang, and Pachl, 414

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 H. G. Dales and A. Ülger, Banach Function Algebras, Arens Regularity, and BSE Norms, CMS/CAIMS Books in Mathematics 12, https://doi.org/10.1007/978-3-031-44532-3_0

437

438 Filali and Galindo, 116, 417 Filali and Pym, 415 Filali and Salmi, 402, 413, 416, 417 Forrest, 231, 287, 407, 409 G Gardella and Thiel, 146 Gardner, 95 Garnett, 219 Gel’fand–Naimark, commutative, 95 Gel’fand–Naimark, non-commutative, 98 Ghahramani and McClure, 367 Ghasemi and Koszmider, 101 Ghosh and Izzo, 227 Gilbert, 286 Goldstein, 26 González and Kania, 39 Grabiner, 267 Graham, 342, 408 Granirer, 116, 276, 287, 289, 417 Granirer and Leinert, 277 Grosser, 110 Grosser and Losert, 413 Grothendieck, 31, 57 H Haagerup, 100, 282 Hahn–Banach, 25 Hausdorff–Young, 262 Hennefeld, 113 Herz, 285–288, 290 Hewitt and Zuckerman, 258 Hindman and Strauss, 12, 384, 385 Host’s idempotent, 280 Hu, 282 Hu and Neufang, 361 Hölder’s inequality, 23 I Inoue and Takahasi, 260, 348 Izzo, 218, 225 Izzo and Papathanasiou, 225 I¸sik, Pym, and Ülger, 416 J Jensen, 101, 102 Johnson’s uniqueness-of-norm, 78 Jones and Lahr, 198, 199 K Kakutani and Kodaira, 231 Kalton, 53 Kamali and Bami, 336 Kamowitz, 171

Index of Theorems and Examples Kaniuth and Ülger, 338 Kaniuth, Lau, and Ülger, 338 Kantorovich–Banach, 61 Kanuith and Ülger, 309 Kaplansky’s density, 98, 99, 277 Katznelson–Tzafriri, 81 Khalil, 282 Krein–Milman, 34 Krein–Šmulian, 28 Kumar and Rajpal, 383 Kusuda, 101 Körner, 225 L Larsen, 84 Lau, 128, 387, 413, 417 Lau and Losert, 409, 416 Lau and Loy, 123 Lau and Pym, 413 Lau and Wong, 407 Lau and Ülger, 102, 283, 416 Leptin, 279 Ljeskovac, 380 localization lemma, 160 Losert, 276, 288, 409, 418 Losert, Neufang, Pachl, and Stepr¯ans, 418 Loy, 77 M Markov–Kakutani fixed point, 60, 191 Mazur, 28 McKissick, 224 Miao, 289 Milne, 181 Mirkil, 263 Monod, 408 Montel, 16, 330 N Nakano, 22 Neufang, 380, 381, 383, 414, 416, 417 O O’Farrell, 169 open mapping, 41 P Pachl and Stepr¯ans, 390 Pełczy´nski, 54, 209 Pfitzner, 37, 98 Pham, 272 Phillips, 49, 55 Pier, 286 Pisier, 100

Index of Theorems and Examples Pitt, 183 Plancherel, 255 Pontryagin duality, 253 Pym, 113 Pym and Saghafi, 367 R Radon–Nikodým, 235 Renaud, 195 Riesz, 410 Riesz representation, 25, 47, 233 Riesz–Kantorovich formulae, 8, 20, 28 Rosenthal, 55 Rosenthal’s  1 -, 50 Rossi’s local maximum modulus, 218 Rudin, 209 Runde, 138, 290 Ryll-Nardzewski fixed point, 60, 246 S Saccone, 37 Saeki, 250 Saghafi, 368 Sakai, 99 Schauder–Tychonoff fixed point, 59, 202 Schwarz–Pick, 219 Sidney, 227 Sobczyk, 49

439 T Takahasi and Hatori, 293, 315, 330, 336 Tanbay, 35 Taylor, 280 Tietze extension, 81 Titchmarsh’s convolution theorem, 256 Tychonoff product theorem, 11 U uniform boundedness, 41 V van der Walt, 21 Varopoulos, 268, 342 Vidav–Palmer, 110 von Neumann, double commutant, 98 W Wang, 225 Watanabe, 105 Wendel, 238 White, 404 Wilken, 218 Willis, 241 Y Young, 123, 238, 386

Index of Symbols

p F, |·|π q, 63 pE ⊗ q F, |·|ε q, 68 pE ⊗ p E  , |·|π q, 80, 92, 107, 146 pE ⊗ pF  , l q, 124 pL p pGq, q, 377 r l q, pMpGq, r  q, 410 pMpGq, pMpβ S q, l q, pMpβ S q,  q, 385 pS , q, 388 p 1 pS q, q, 79, 82, 386, 388 p 1 , · q, 370 pBpE, Fq, |·|op q, 39 pBpE × F, Gq, q |·|op q, 40 pNpE, Fq, |·|ν q, 65 pB r pE, Fq, |·|r ), 61 pLpE, Fq, ďq, 8 0G , 230 1G , 232 1S , 2, 6 A ⊗ B, 88 A “ pA , l q, 104 A · A, A A, 91 ApKq, 213, 330 ApΔq, 254 ApΓq, 254, 273, 333, 337, 338, 407, 408 ApΦS q, 83, 155 ApΦQ+• q, 199 ApZq, 144 ApDq, 37, 213, 219, 224, 302, 330, 342, 343 ACpIq, 167, 328 APpAq, 113, 243 APpGq, 244 p B, 89, 181 A⊗ A+ , 76 A+ pDq, 213, 224, 327 A , 72, 156

Ar2s , A2 , 72 Aop , 72 A0 , 170 A0 pωn q, 350 AH , 161, 196 Aω , 148, 178, 397 Ac , 330 A p pΓq, 285, 286, 339, 340, 353, 409 A p pΓq, 407 Arp pΓq, 289, 340, 353, 409 B  I, 73 BpΓq, 255, 270, 333 BVCpIq, 167, 328 B b pXq, B bR pXq, 13 Bω , 177, 308, 327, 356 Bρ pΓq, 277, 333, 408 B p pΓq, 290, 339 Br pxq, 11 rr1s pλq, 65 Br1s pxq, B B 0 pΓq, 258, 284, 338 CpA q, 309 CpGq, 250 CpK, Aq, 117 CpX, Yq, CpXq, 11 Cpβ Nq, 96 CV p pΓq, 290 C ∗ pΓq, 270 Cδ∗ pΓq, 275, 295, 303 C p1q pIq, 328 C p8q pIq, 169 C pnq pIq, 169, 328 C b pXq, 13, 18, 19, 78 Cρ∗ pΓq, 276 C pIq, 256 C BS E pAq, v, 310

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 H. G. Dales and A. Ülger, Banach Function Algebras, Arens Regularity, and BSE Norms, CMS/CAIMS Books in Mathematics 12, https://doi.org/10.1007/978-3-031-44532-3_0

441

442 C 0,R pKq, C 0 pKq+ , 13 C 00 pKq, 13 C 0 pKq, 13, 78, 210, 301, 326 C 0 pK, Aq, 78 C 0 pK, Iq, 14 C Rb pXq, C b pXq+ , 13 DpI; pMk qq, 329 DpI; pMn qq, 169 E  , 21 E  , 22 EpS q, 5 E – F, E ∼ F, 40 E ‘8 F, E ‘ p F, 17 E + , 7, 19, 78 E •, 6 E ∼ , 9, 61 Eω , 397 Errs , Er1s , 16 E R , 19 Gd , 230 H 8 , 225 H 8 pUq, 330 H 8 + C, 226, 330 HT , 88, 378 IpS q, 157 IE , 7 J, J  , 174 JpS q, 157 J8 , J x , 156 K ⊗ a L, 66 KA , ex KA , 81, 215 KG , 57 K8 , 11 K{x} , 210 L p pΩ, μq, 19 LpAq, v, 295 LpA, Ωq, 295 LOpλq, ROpλq, 244 LUCpGq, 414, 417 LUCpGq, RUCpGq, UCpGq, 242 L1 pΩ, μq, 30 L1 pGq, 233, 416, 417 L1 pG, ωq, 252, 253 L1 pS q, 237, 417 L1 pIq, 256 L1 pRq, 255 L1 pR+ q, 336 L1 pR+ , ωq, 266 L1 pR+ , ωq, 367 LS1 pGq, 410 L10 pGq, 240 L8 pGq, 244, 262, 353 L8 pΩ, μq, 18 L8 pTq, 226

Index of Symbols L p pGq, 231, 250, 262, 353, 368 p Lq pGq, 264 L p pGq ⊗ L p pΩ, μq, 18, 19, 23, 62 L8 pGq, 244 La , Ra , 76 La , Ra , La , Ra , 91 L s , R s , 4, 384 L pR pΩ, μq, 18 MpGq, 144, 234, 418 MpKq, 20, 210 MpKq+ , 20 MpRq, 255 MpR+ q, 256 Mpβ S q, 62, 385 Mpβ Nq, 302 MS pGq, 410 M8 , 153 Mω , 178, 356, 368 p Mω , 368 Mω ⊗ Mϕ , 82, 153 Md pGq, Mc pGq, Mac pGq, M s pGq, 235 Md pKq, Mc pKq, 21 M x , 153 M R pKq, 20 NpKq, 21 OpUq, 16, 330 PpKq, 213 PDpΓq, 272 PF p pΓq, PM p pΓq, 290 Pn , 9 PrpKq, 20 RpKq, 213, 218, 224, 330, 331 R ⊗ S , 67 R∗ , 99 Rλ , 113 R s μ, 31 S + T, 6 S Lp2, Rq, 241 S · T , S T , 72 S ď T, 8 S d , S dd , 8 S r2s , S 2 , S rns , S n , 72 S op , 5 S c , S cc , 72 S 2 , 281 S p , 260, 261, 337 S p pGq, 260 S ∧, S ∨, 5 S tpBq, 12 T  , T  , 42 TS , 2 UV, U 2 , 4 UΔV, 2 U ´1 , 4

Index of Symbols Vpc 0 q, 380 VpKq, 183 VpK, Lq, 183, 340, 342, 380, 381 VpΓq, 374 V NpΓq, 275 V K , 26, 142 V ´1 U, UV ´1 , 4 WAPpAq, 113, 243 WAPpGq, 244, 249 WAPpΓq, 271 X ∗ , 15 Xω , 265 Z, 2 Z1 , . . . , Zn , 2 rMs, 296 C, C• , 2 CrrXss, CrXs, 265 D, 2 Δ, 14, 254 ΔG , 231 Δn , 9, 170 F2 , 241, 282, 406, 407 Γ0 pAq, ΓpAq, 209 I, 2 I∨ , 367 Λpβ S q, 385 N, Nn , Z, Z+n , Z+ , Z´ , Z• , 1 N∗ , 15 N∨ , 5, 83, 393 N∧ , 5, 83, 148, 394 Ωps, tq, 398 Ω l , Ω , 398 ΦA , 74 ΦS , 6, 82, 199 Φμ , 211 ΦB0 pΓq , 284 Φ MpGq , 257 Φ{ϕ} , 108 Φm , 411 ΦQpAq , 296 Π, Π, 2 Q, Q+ , Q+• , Q• , 2 R, R+ , R+• , 2 T, 2 r 410 G, Gr{eG } , 411 r 154, 210, 302 K, rc , 210, 302 K Z∨ , 5 ℵ0 , ℵ1 , 2 αpμq, 255 β Kd , 301 β G, 244, 248

443 β S , 15, 384 β T ω , T ω∗ , 398 β X, 15, 96 β N, 15 χT , 2 χs , 2 δn , 9 δ s , 2, 9, 21 δu , 385 δ s,t , 2  1 , 172, 302, 350  1 pGq, 79  1 pKq, 21  1 pS q, 23, 79, 155, 385, 387 p  1 pT q, 64, 89  1 pS q ⊗ 1  pS , ωq, 146  1 pS , ωq, 22, 80, 83, 87, 155, 397, 400, 403  1 pF2 , ωq, 406  1 pN, ωq, 406  1 pN∨ , ωq, 83, 401  1 pN∧ , ωq, 83, 85, 148, 178, 356, 400  1 pΦA q, 343  1 pQ, ωq, 404  1 pQ+• q, 199  1 pR+• , 400  1 pZ, ωq, 265, 327, 402, 405  1 pZ+ , ωq, 265, 327  1 pZ∨ q, 86, 148, 180, 357, 368 p c 0 , 355 1 ⊗ p c 0 , 183 1 ⊗ p  2 , 64 2 ⊗ 8  , 9, 172, 193  8 pEα q, 17  8 pS q, 170  8 pS q,  R8 pS q, 9  8 pS , 1/ωq, 22 p  8 , 380 8 ⊗ p  , 9, 173, 327, 377  p pAn q, 79, 117  p pEq,  8 pEq, 18  p pEα q, 17  p pS q, 9, 79  p pZ+ , ωq, 267  p pZ+ , ωr q, 268 p  8 , 183, 327 p⊗ p  q , 183, 351, 377 p⊗ np pEq, 18 t f , rt f , 230 ex S , 6 κE , 22 λt , 274 xx, yyE,F , 23 lin S , 6 |S |, 2

444 |μ|, 12 |a| p , 95 | f |S , 9 ZpA q, 360 pq prq Zt pA q, Zt pA q, 360 μ ∨ ν, μ ∧ ν, 20 μ l ν, μ  ν, 385 μ ν, 233 μ∗ pBq, 234 μ+ , μ´ , |μ|, 20 μ p,n pxq, 56 · 1 · 2 , · 1 ∼ · 2 , 41 · 8 , 18 · BS E , 309 · op , 39, 84 ω, ω1 , 2 uw B , 96 p B , 154 ΦA , 297 πG , 411 πq,p pT q, Πq,p pE, Fq, 57 rad A, 73 ρpaq, 74 σpE, E  q, σpE  , Eq, 24 σpE, Fq, 24 σpT q, 78 σpaq, 74 supp μ, 20 supp f , 13 τE , 152 τ f , 297 τ p , 153, 298, 313 θ E , 133 ε x , 153 ϕS , 82 ϕS , 386 fp, 83 fr, fq, 230 aS , 72 bpΓq, 283 b G, 254, 304 bv, bv 0 , 178 c0 pEn q, 17 c 00 pS q, 170 c 0 pAn q, 117 c 0 , 9, 172, 326 c 0 pS q, 9, 170 c 0 pS , 1/ωq, 23, 146 c0 pEq, 17 dA , 218 eA , 72 eG , 230 eS , 4

Index of Symbols f ⊗ g, 182 hpIq, 157 mG , 230 mΔ , 254 qF , 124 s · λ, λ · s, 31 s ∧ t, s ∨ t, 2, 3 s´1 U, U s´1 , 4 o → s, 4 sα ´ sα ↑ s, sα ↓ s, 4 u l v, u  v, 384 u l v, u  v, 384 x K y, 7 x ∼ y, 218 x+ ,x´ , |x|, 7 y0 ⊗ λ0 , 51 ApEq, 52, 78, 122 BpEq, 40, 78 BpE, Fq, 39 KpEq, 52, 78, 122, 371, 373 KpE, Fq, 52 NpEq, 65, 80, 122 UpHq, 96 WpEq, 52, 78 WpE, Fq, 52 ZX p f q, 13 F2 , 5 S2 , 5 ApE, Fq, 52 BpE, Fq+ , B r pE, Fq, B b pE, Fq, 61 B r pEq, B b pEq, 61 EA , 126, 365 F pEq, 52, 78 F pE, Fq, 51 F pS q, 258 F μ, 255 F p pΓq, 262 F f , 254 G, 82 Gμ , 211 KK , 10 LpE, Fq, 7 LApΓq, 283, 337 Lμ, 255 L r pE, Fq, L b pE, Fq, 8 L f , 266 MpAq, 76, 127, 162 MpL1 pGqq, 238 M 00 pAq, M 0 pAq, 163, 321 M  pAq, Mr pAq, 76 N x , 10 PpS q, 2 QpApΓqq, 303

Index of Symbols QpAq, 294, 295, 298, 349 S1 pHq, 122 S p pHq, 97 UpAq, 95 U p , 15 V, 256, 367 pq prq Zt pβ S q, Zt pβ S q, 385 a A , 79 Aα , 225 BX , 13 IpAq, 72 QpAq, 74 Rc pΓq, 231, 280, 338 ZpAq, 72 ZpS q, 4 Zt pF  q, 363 c, 2 M l N, M  N, 103 TIMpϕq, 193, 282 TIMpxq, TIMpϕqrms , 193 aco S , 6

445 co S , 6 lipα I, Lipα I, 168, 303, 328 lipα I, Lipα I, 348 (CH), 2 Xd , 10 BpL p pIqq, 382 AP, 58 BAP, 58 BLAI, BRAI, BAI, CAI, 84 BPAI, CPAI, 198 Derighetti, 290 DLTC set, DTC set, 363, 385 Larsen, 262 Losert, 279 LSAI, RSAI, SAI, 361 MBAI, 87 PAI, 198 SBP, 190 So, 96 SSBP, 190 Wo, 96 WSBP, 190