http://dx.doi.org/10.1090/surv/060
Selected Titles in This Series
60 Iain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace C*-algebras, 1998 59 Paul Howard and Jean E. Rubin, Consequences of the axiom of choice, 1998 58 Pavel I. Etingof, Igor B. Frenkel, and Alexander A. Kirillov, Jr., Lectures on representation theory and Knizhnik-Zamolodchikov equations, 1998 57 Marc Levine, Mixed motives, 1998 56 Leonid I. Korogodski and Yan S. Soibelman, Algebras of functions on quantum groups: Part I, 1998 55 J. Scott Carter and Masahico Saito, Knotted surfaces and their diagrams, 1998 54 Casper Goffman, Togo Nishiura, and Daniel Waterman, Homeomorphisms in analysis, 1997 53 Andreas Kriegl and Peter W. Michor, The convenient setting of global analysis, 1997 52 V. A. Kozlov, V. G. Maz'ya> and J. Rossmann, Elliptic boundary value problems in domains with point singularities, 1997 51 Jan Maly and William P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, 1997 50 Jon Aaronson, An introduction to infinite ergodic theory, 1997 49 R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, 1997 48 Paul-Jean Cahen and Jean-Luc Chabert, Integer-valued polynomials, 1997 47 A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May (with an appendix by M. Cole), Rings, modules, and algebras in stable homotopy theory, 1997 46 Stephen Lipscomb, Symmetric inverse semigroups, 1996 45 George M. Bergman and Adam O. Hausknecht, Cogroups and co-rings in categories of associative rings, 1996 44 J. Amoros, M. Burger, K. Corlette, D. Kotschick, and D. Toledo, Fundamental groups of compact Kahler manifolds, 1996 43 James E. Humphreys, Conjugacy classes in semisimple algebraic groups, 1995 42 Ralph Freese, Jaroslav Jezek, and J. B. Nation, Free lattices, 1995 41 Hal L. Smith, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, 1995 40.3 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 3, 1998 40.2 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 2, 1995 40.1 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 1, 1994 39 Sigurdur Helgason, Geometric analysis on symmetric spaces, 1994 38 Guy David and Stephen Semmes, Analysis of and on uniformly rectifiable sets, 1993 37 Leonard Lewin, Editor, Structural properties of polylogarithms, 1991 36 John B. Conway, The theory of subnormal operators, 1991 35 Shreeram S. Abhyankar, Algebraic geometry for scientists and engineers, 1990 34 Victor Isakov, Inverse source problems, 1990 33 Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, 1990 32 Howard Jacobowitz, An introduction to CR structures, 1990 31 Paul J. Sally, Jr. and David A. Vogan, Jr., Editors, Representation theory and harmonic analysis on semisimple Lie groups, 1989 30 Thomas W. Cusick and Mary E. Flahive, The Markoff and Lagrange spectra, 1989 (Continued in the back of this publication) Mathematical Surveys and Monographs
Volume 60
Morita Equivalence and Continuous-Trace C*-Algebras
lain Raeburn Dana P. Williams
American Mathematical Society
^DEO Editorial Board Georgia M. Benkart Tudor Stefan Ratiu, Chair Peter Landweber Michael Renardy
1991 Mathematics Subject Classification. Primary 46L05; Secondary 46L35, 46L40, 43A65.
ABSTRACT. We give a modern treatment of the classification of continuous-trace C*-algebras up to Morita equivalence. This includes a detailed discussion of Morita equivalence of C*-algebras, a review of the necessary sheaf cohomology, and an overview of recent developments in the area. The book should be accessible to anyone familiar with the basics of C*-algebras up to the GNS- construction.
The authors were supported by grants from the Australian Research Council, the Uni versity of Newcastle, and the Edward Shapiro Fund at Dartmouth College.
Library of Congress Cataloging-in-Publication Data Raeburn, Iain, 1949- Morita equivalence and continuous-trace C*-algebras / Iain Raeburn, Dana P. Williams, p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 60) Includes bibliographical references and index. ISBN 0-8218-0860-5 (alk. paper) 1. C*-algebras. I. Williams, Dana P., 1952- . II. Title. III. Series: Mathematical surveys and monographs ; no. 60. QA326.R34 1998 512/.55-dc21 98-25838 CIP
Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to [email protected]. © 1998 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 03 02 01 00 99 98 To our parents: Margaret and Fraser Raeburn on their golden wedding anniversary, and in memory of Ruth and James Williams Contents
Introduction ix
1 The Algebra of Compact Operators 1
2 Hilbert C*-Modules 7 2.1 Hilbert Modules 8 2.2 Bounded Maps on Hilbert Modules 16 2.3 Multiplier Algebras 23 2.4 Induced Representations 30
3 Morita Equivalence 41 3.1 Imprimitivity Bimodules 42 3.2 Morita Equivalence 47 3.3 The Rieffel Correspondence 52 3.4 The External Tensor Product 62
4 Sheaves, Cohomology, and Bundles 67 4.1 Sheaf Cohomology 68 4.2 Fibre Bundles 92 4.3 The Dixmier-Douady Classification of Locally Trivial Bundles .... 102
5 Continuous-Trace C*-Algebras 115 5.1 C*-Algebras with Hausdorff Spectrum 116 5.2 Continuous-Trace C*-Algebras 121 5.3 The Dixmier-Douady Classification of Continuous-Trace C*-Algebras 126 5.4 Automorphisms of Continuous-Trace C*-Algebras 140 5.5 Classification up to Stable Isomorphism 143
6 Applications 155 6.1 The Brauer Group 155 6.2 Pull-back C*-Algebras 160 6.3 Induced C*-Algebras 163
vii viii Contents
7 Epilogue: The Brauer Group and Group Actions 173 7.1 Dynamical Systems and Crossed Products 174 7.2 The Equivariant Brauer Group 177 7.3 The Brauer Group of a Point 181 7.4 Group Cohomology and Moore Cohomology 184 7.5 The Brauer Group for Trivial Actions 192 7.6 The Brauer Group for Free and Proper Actions 193 7.7 The Structure of the Brauer Group 196
A The Spectrum 201 A.l States and Representations 201 A.2 The Spectrum of a C*-Algebra 210 A.3 The Dauns-Hofmann Theorem 223 A.4 The State Space of a C*-Algebra 226
B Tensor Products of C*-Algebras 235 B.l The Spatial Tensor Product 236 B.2 Fundamental Examples 244 B.3 Other C*-Norms 246 B.4 C*-Algebras with Hausdorff Spectrum 254 B.5 Tensor Products of General C*-Algebras 257
C The Imprimitivity Theorem 269 C.l Haar Measure and Measures on Homogeneous Spaces 269 C.2 Vector-Valued Integration on Groups 274 C.3 The Group C*-Algebra 280 C.4 The Imprimitivity Theorem 285 C.5 Induced Representations of Groups 293 C.6 The Stone-von Neumann Theorem 298
D Miscellany 303 D.l Direct Limits 303 D.2 The Inductive Limit Topology 304
Index 309
Bibliography 317 Introduction
The Gelfand-Naimark theorem says that a commutative C*-algebra A with identity is determined up to isomorphism by its spectrum T, in the very strong sense that A is isomorphic to the algebra C(T) of continuous functions on T. There are also non- commutative C*-algebras with spectrum T: for example, the algebra C(T, Mn(C)) of continuous functions from T into the matrix algebra Mn(C). In this book we study theorems which classify algebras with spectrum T and automorphisms of these algebras in terms of topological invariants of the space T. One of the goals of algebraic topology is to associate to a space T algebraic objects whose properties reflect the topological structure of T. The ones of interest to us are cohomology groups Hn(T;7j): very roughly speaking, iJn(T;Z) counts n-dimensional holes in the space T by chalking up a copy of Z for each such hole. Our C*-algebras with spectrum T are classified by classes in the 3-dimensional cohomology group H3(T; Z), and their automorphisms by classes in H2(T; Z); these theorems were first proved, respectively, by Dixmier and Douady in 1963 and by J. Phillips and Raeburn in 1980. They have been used extensively in recent years: in the analysis of C*-dynamical systems and their crossed products, in the K-theory of C*-algebras [150], in differential geometry [14], and in mathematical physics [16]. We shall discuss all the background material needed to prove these classification theorems and apply them to C*-dynamical systems, including the definitions and properties of the cohomology groups themselves. There are several versions of the Dixmier-Douady Classification Theorem, in volving different families of C*-algebras and classifying up to different equivalence relations. The simplest concerns locally trivial bundles over T with fibres isomor phic to the algebra of compact operators on an infinite-dimensional Hilbert space, and identifies them up to bundle isomorphism; we shall prove this one first. For applications to operator algebra, it is more satisfactory to work with continuous- trace C*-algebras, as Dixmier and Douady did. They used continuous fields of C*-algebras, which had been developed by Fell to prove structure theorems for important classes of C*-algebras. However, Fell's theory never became standard C*-algebraic equipment, so anybody wanting to use the Dixmier-Douady classifi cation faced a heavy start-up cost. We have used instead Rieffel's theory of Morita equivalence for C*-algebras, which is now recognized to be a fundamental tool for C*-algebraists. It has been folklore for the past 20 years that the Dixmier-Douady classification can be couched in terms of Morita equivalence (see, for example, [5] or [140, 20]), but this version has remained logically dependent on the special structure
IX X Introduction theories developed by Fell and Dixmier-Douady. We aim to give a direct treatment, starting from the basic theory of C*-algebras. Morita equivalence was adapted to C*-algebras by Rieffel in the 1970's in con nection with a C*-algebraic version of the Mackey machine [144, 147], and it has since become a standard tool for analyzing group C*-algebras and crossed prod ucts [148, 62]. Although Rieffel's papers were carefully motivated and expounded, their primary concern was representation theory, and the general theory of Morita equivalence they contained only gradually emerged. We have therefore decided to give a detailed introduction to this general theory. To do this in modern language, we have to discuss Hilbert modules, and we have preferred to be self-contained: the recent expositions of Hilbert modules by Lance [94] and Wegge-Olsen [167] are tilted towards quantum groups and if-theory rather than Morita equivalence, and hence their motivating examples are different. The classification theorems necessarily involve sheaf cohomology. It is slightly nonstandard sheaf cohomology: one has to deal with sheaves of nonabelian groups, and hence the techniques of homological algebra do not apply directly. (And many functional analysts would probably prefer to avoid these techniques anyway.) So we have chosen to give a concrete introduction to sheaf cohomology and fibre bundles based on cocycles. Thus our proof of the bundle-theoretic Dixmier-Douady clas sification is entirely self-contained, and should be accessible to readers from other areas as well as operator algebraists. Our treatment of continuous-trace C*-algebras reflects our emphasis on Morita equivalence: for us, a continuous-trace C*-algebra A with spectrum T is almost by definition a C*-algebra which is locally Morita equivalent to the commutative C*- algebra C(T), and the Dixmier-Douady class 6(A) £ H3(T;Z) is the obstruction to the existence of a global Morita equivalence. To make all this precise, we need background material on the spectrum and primitive ideal space of a C*-algebra, and we have included an introduction to the subject in Appendix A. We originally intended to phrase all our results and calculations in terms of Morita equivalence. However, as we progressed we found this attitude restrictive, and noticed in particular that it is often easier to compute 6(A) using local rank- one projections and partial isometries, as in [127, §2]. We have come to believe that the power and enduring interest of the Dixmier-Douady theory lies in its many formulations, and we hope that our treatment will prove useful from several viewpoints. We have therefore been careful to tie up our main theorem with the bundle-theoretic version by including a discussion of the Stabilization Theorems of Kasparov and Brown-Green-Rieffel. One punchline of our book is a version of the Dixmier-Douady Theorem in which the map A »—> 6(A) induces an isomorphism of a Brauer group of continuous-trace algebras with fixed spectrum T onto the cohomology group H3(T:Z). This lovely theorem is part of the folklore we are aiming to tidy up: it was mentioned almost in passing by Taylor in 1975 [161], and discussed in seminar notes by Green around 1978 [61]. However, even to formulate it, we need nontrivial facts about the spectra of tensor product C*-algebras; again we have developed the necessary material in an Appendix. When we started this book, one goal was to give a readable and complete intro- Introduction xi duction to the research area which has been described by Rosenberg as the study of the "fine structure of the Mackey machine" [151]. Loosely speaking, this concerns actions of groups on continuous-trace algebras and the structure of the associated crossed products. We are still several books short of this goal, so we have com promised by providing an overview of the area and of the remaining background. This is the content of Chapter 7. We have used as a unifying theme the equivari- ant Brauer group introduced in [20]. The structure theory of this group involves a variety of cohomological invariants and principal bundles, and describes many fascinating relations between them; there are many more, we suspect, waiting to be found.
We have made a serious effort to keep required background to a minimum. There is by now a fairly standard first course in C*-algebras, covering the Gelfand- Naimark Theorem, the continuous functional calculus, positivity, and the GNS- construction. We assume that readers have seen this; everything we need is in the first 3 chapters of Murphy's book [111], for example. We make the usual conventions of the subject: for example, homomorphisms of C*-algebras are always *-preserving, and ideals are closed and two-sided unless otherwise stated. We have had to use some general topology and functional analysis, but we have tried not to overdo this, and to give detailed references. (Our preferred authority here is Pedersen's book [125], which covers both topology and functional analysis, is elegantly written, and has a good and practical selection of topics.) On the other hand, we have been careful to develop all the algebraic topology we need, and have arranged our material so that parts involving locally compact groups and integration can be avoided without loss of continuity. We hope that students who have seen the basics of C*-algebras at the Honours level in the UK or Australia, the Diploma level in Europe, or the beginning graduate level in North America can read this book and enjoy doing so. We have been careful to make the different sections of the book as self-contained as possible, so that those who want to know about Hilbert modules and Morita equivalence, the spectrum of a C*-algebra, tensor products of C*-algebras or basic sheaf cohomology should be able to turn to the relevant section and learn something useful without absurd difficulty.
Reader's Guide
Chapter 1 is about the algebra of compact operators and its properties. This algebra is fundamental both in the structure theory of nice C*-algebras and as motivation for the theory of Morita equivalence. More experienced readers could probably skip or skim this chapter, but others may find it helpful to see our notation and viewpoint in a relatively elementary setting. Chapter 2 is a self-contained introduction to Hilbert modules, multiplier alge bras, and Rieffel's general theory of induced representations. Induced representa tions are defined using tensor products of Hilbert modules, but these are technically much easier than the tensor products of C*-algebras discussed in Appendix B, and xii Introduction we do not assume previous familiarity with tensor products of any sort. Rief- fel's motivation came from the unitary representation theory of locally compact groups, and we have discussed this application in outline at the end of each section. However, this material can be safely and logically skipped by those who are not interested or not familiar with integration on locally compact groups. For those who are, we have included a detailed discussion in Appendix C. Chapter 3 concerns imprimitivity bimodules and Morita equivalence of C*- algebras, and builds on our previous treatment of Hilbert modules. We discuss in detail the Rieffel correspondence between ideals and quotients of Morita-equivalent C*-algebras. From this and the definition of the topology on the primitive ideal space, we obtain the existence of the Rieffel homeomorphism between the primitive ideal spaces of Morita-equivalent C*-algebras. Section 3.4, on the external tensor product of imprimitivity bimodules, uses the basic properties of the spatial tensor product discussed in Appendix B.l. Since this material is not used until Chapter 6, Section 3.4 can be skipped at this stage. Chapter 4 is a self-contained introduction to sheaf theory covering the ma terial necessary for basic applications to C*-algebras, which should be accessible even to homologically-challenged functional analysts. The only slightly nonstan dard algebraic background required is the direct-limit construction discussed in Appendix D.l. Those seeking instant gratification in the form of applications to C*-algebras should see Proposition 4.27, which will be much more informative to those who have read Chapter 1. All the sheaf cohomology needed for Chapter 5 is contained in Section 4.1. In Section 4.2 we discuss fibre bundles and their clas sification in terms of sheaf cohomology: these ideas are used freely in the recent literature on crossed products of continuous-trace C*-algebras, as outlined in Chap ter 7. We can then give the Dixmier-Douady classification for locally trivial fibre bundles with fibre the algebra K,{H) of compact operators. This version of the Dixmier-Douady classification theorem uses only the material in Chapters 1 and 4. In the last few pages of Chapter 4, we explain why this version of the theorem is not ideal for those working in C*-algebras. Chapter 5 is the heart of the book: the classification of continuous-trace C*- algebras. To define this class, we need to consider C*-algebras with Hausdorff spectrum. To appreciate the material in Section 5.1, one needs a working knowledge of the spectrum and primitive ideal space of a C*-algebra; all the necessary details are in Appendix A. In Section 5.2, we define continuous-trace C*-algebras, discuss the alternative characterizations, and give many nontrivial examples. Our treatment of the Dixmier-Douady classification theorem and the analogous classification of automorphisms depends on all the earlier chapters. We give two versions of the Dixmier-Douady Theorem: the classification up to Morita equiv alence, and the classification of separable stable continuous-trace C*-algebras up to isomorphism. The latter depends on the Brown-Green-Rieffel Theorem, which says that Morita equivalence is the same as stable isomorphism, and which we prove in Section 5.5. This is a deep theorem, and the discussion in Section 5.5 is necessarily at a higher level than most of the main text. We conclude the chap ter with a brief discussion of the pros and cons of the different versions of the Dixmier-Douady classification. At this stage we do not complete the classification Introduction xiii of automorphisms, since the most efficient route to this involves more sophisticated use of tensor products of C*-algebras. Chapter 6 begins with our punchline: the identification of the Brauer group Br(T) of continuous-trace C*-algebras with H3(T;Z). The multiplication in Br(T) is a balanced C*-algebraic tensor product; to understand it, we need the description of the spectrum of a tensor product from Appendices B.1-B.4. Once we have this balanced tensor product, we can quickly finish the classification of automorphisms of continuous-trace algebras. The other sections of Chapter 6 concern two con structions which yield interesting examples of continuous-trace (7*-algebras. The pull-back construction of Section 6.2, which is defined using the balanced tensor product, shows how to make the Brauer group into a functor. The inducing con struction of Section 6.3 is a C*-algebraic version of Mackey's original construction of the Hilbert space of an induced representation; it yields a class of nontrivial continuous-trace C*-algebras whose Dixmier-Douady class we can explicitly com pute. The first two Appendices discuss aspects of the general theory of C*-algebras which would not normally be covered in a first course on the subject, but which are required in the book. The first is a relatively gentle introduction to the spectrum of a C*-algebra which highlights the examples of interest to us. The second is a self-contained treatment of tensor products of C*-algebras whose main goal is to prove that the spectrum of the tensor product of two continuous-trace C*-algebras is the product of their spectra. This result is certainly known, but it is hard to point to a detailed proof in the existing literature. Only the first four sections of Appendix B are used in the book proper; the fifth has been included to provide a detailed source for other known facts. The third Appendix contains a proof of Rieffel's formulation of Mackey's Imprimitivity Theorem; this material is logically independent of the rest of the book, and is included because we would be giving a lop-sided view of Morita equivalence if some applications to representation theory were not mentioned. Hooptedoodle. This book occasionally breaks for a Hooptedoodle. These contain comments which go beyond the prevailing scope and level, but which might help put the material in context. The term Hooptedoodle comes from the prologue of John Steinbeck's Sweet Thursday, in which one of the main characters criticizes the previous book Cannery Row:
Sometimes I want a book to break loose with a bunch of hooptedoodle. ... But I wish it was set aside so I don't have to read it. ... Then I can skip it if I want to, or maybe go back to it after I know how the story comes out. Steinbeck then dutifully labels his digressions as hooptedoodles. Material in our hooptedoodles is not used in the main text, and can easily be skipped. Indeed, they should be skipped if they don't seem to be helping!
Acknowledgments. We have tried to present a detailed account of the theory of continuous-trace C*-algebras, along with the background necessary to read the current literature. Our account is logical rather than historical: we have not made any attempt to give detailed attributions, and we apologize to those we have not xiv Introduction mentioned*. We do acknowledge obvious debts to Rieffel and to Dixmier-Douady; where our subject matter overlaps, we have been unable to improve on Lance's treatment of Hilbert modules. We thank those who helped us with this book. In addition to those who taught and introduced us to the subject, we thank the participants in the various seminars we gave at Newcastle based on parts of the book. We specifically thank Neal Fowler, Astrid an Huef, Wojciech Szymahski, David Webb, and especially Paul Muhly for reading large parts of drafts and for their helpful suggestions. The remaining mistakes are the other author's fault.
*We did start out with better intentions, but soon realized we could cause even more offense this way. Our memories are not what we think they once were. Appendix A The Spectrum
The Gelfand theory identifies the maximal ideal space of a commutative Banach algebra as a crucial topological invariant — indeed, it determines a commutative C*-algebra up to isomorphism (by the Gelfand-Naimark Theorem). Attempts to produce an analogue of the Gelfand-Naimark Theorem for noncommutative alge bras have not been completely successful, but they have given us two important generalizations of the maximal ideal space: the spectrum and the primitive ideal space. Both sets carry natural topologies, and it is important in applications that we can describe the topology on the spectrum in several different ways. In this ap pendix we shall discuss two of these descriptions and some of their basic properties. Although the GNS-construction of representations of C*-algebras from posi tive functionals is really part of our stated prerequisites, its extension to algebras without identities raises a few subtleties which are not always fully discussed in elementary treatments. We therefore begin by reviewing this construction. In Sec tion A.2, we discuss an ideal-theoretic approach to the topology on the spectrum, in which the topology is lifted from the hull-kernel topology on the set of primitive ideals. We use examples to illustrate the behavior of this topology and its implica tions for the structure theory of C*-algebras, and then prove some basic properties of the spectrum, including a fundamental result called the Dauns-Hofmann Theo rem. We can also define a topology by viewing the spectrum as the image under the GNS-construction of a collection of positive functionals, and pushing the weak-* topology forward. In Section A.4 we explain how this works, and prove that it gives the same topology, so that one can switch freely between the two descriptions.
A.l States and Representations
A representation of a C*-algebra -A is a homomorphism of A into the C*-algebra B(H) of bounded linear operators on a Hilbert space H\ just as we always assume that homomorphisms of C*-algebras are *-preserving, so representations n of C*- algebras automatically satisfy 7r(a*) = 7r(a)*. We often write Hn for the underlying Hilbert space H of a representation n : A —> B(H). We are interested in sets
201 202 The Spectrum of representations which are large enough to describe A but small enough to be manageable, so we need to decide when two representations are essentially the same and to be able to decompose representations into smaller ones. Two representations n : A —> B(Hn)^ p : A —> B{TLP) are unitarily equivalent (or just equivalent) if there is a unitary isomorphism U of Hp onto TL^ such that 7r(a) = Up(a)U* for all a e A; we then think of TT and p as being the same representation, and write TT ~ p. It is easy to check that unitary equivalence is an equivalence relation in the usual sense. If TT : A —• B(Hn), p : A —• B(HP) are representations, the formula 7r 0p(a)(ft, fc) := (rr(a)h, p(a)k) defines a representation 7r 0 p of A on the Hilbert space direct sum W^ 0 Wp, called the direct s-ura of 7r and p. Conversely, we think of an equivalence of a representation TT with a direct sum pi 0 p2 as being a decomposition of TT in terms of the smaller representations p\ and p2. A closed subspace /C of H^ is invariant under TT if 7r(a)k G /C for every aGA and k G /C. If /C is invariant and y belongs to the orthogonal complement /C"1, then (k | 7r(a)2/) = (7r(a)*fc | ?/) = (7r(a*)fc | y) = 0 for all fe G /C, so K1- is also invariant. For example, H^ = {(ft, 0) } and W^ = { (0, k) } are an( invariant for TT 0 p. Conversely, if /C is invariant, both 7r|/c • a •-» ^{O)\K i ^IAC-1 are representations of A, and we can recover n as the direct sum 7r|^ 0 7r|^x; more precisely, the canonical unitary isomorphism U : (fc, y) i—> /c + ^/ of /C 0 /Cx onto W^ implements an equivalence between 7r|/c 0 7r\^± and 7r. A representation TT is irreducible if there is no closed invariant subspace apart from { 0 } and Ti^, and hence no nontrivial direct sum decomposition. Equivalently: Lemma A.l. A representation TT of a C*-algebra A is irreducible if and only if the only operators commuting with n(A) are multiples of the identity operator 1^. Proof. If 7r has V as a nontrivial invariant subspace, then the orthogonal projection P of H onto V commutes with every 7r(a), and is not scalar. Conversely, if there is a non-scalar operator T commuting with 7r(.A), either the real or imaginary part of T is a non-scalar self-adjoint operator S commuting with TT(A). Some spectral projection P for S is not 0 or 1, and it too commutes with TT(A).* Then P(H) is a nontrivial subspace of H which is invariant for TT. •
Hooptedoodle A.2. For representations on infinite-dimensional Hilbert spaces, the notion of irreducibility we have defined above is sometimes called topological irreducibility to distinguish it from algebraic irreducibility: a representation is algebraically irreducible if it has no invariant subspaces, closed or not. Remarkably, these two notions coincide for representations of a C*-algebra [28, Corollary 2.8.4].
* Alternatively, for those who have seen the continuous functional calculus rather than the classical spectral theorem: S non-scalar and selfadjoint implies that cr(S) consists of more than one point. Thus there are nonzero / and g in C(cr(S)) such that fg = 0. Since f(S),g(S) G C*(S) C n(A)'', f(S)H and g(S)H are nonzero mutually orthogonal invariant subspaces, and TT is reducible. A. 1 States and Representations 203
The essential subspace of a representation IT of A is the closed subspace
/C := span{ 7r(a)h : h G Hn, a G A } of Tin. The representation n is said to be nondegenerate if its essential subspace /C is all of Tin; this is equivalent to 7r(l) = 1 if A has an identity, and in general to the assertion that 7r(ei) —> 1 strongly (i.e., ir(ei)h —> h for all ft G Hn) whenever { ei } is an approximate identity for A. The essential subspace /C of a representation is always invariant, and n is equivalent to TT\JC ® TT\JC± = 7r|^ 0 0. More generally, if / is an ideal in A, then 1 K := span{7r(a)ft : ft G Hn,a G /} is invariant, but TT is not zero on /C* unless J is an essential ideal. Conversely, any nondegenerate representation of an ideal. J extends canonically to a nondegenerate representation n of A on the same space (this a special case of of Proposition 2.50).
Remark A.3. That invariance of /C implies invariance of K,1- is a feature of He- represent at ions of *-algebras; in fancy language, it says invariant subspaces are reducing. In other parts of mathematics, this is not automatic: for example, in dealing with representations of groups by invertible operators (rather than unitary ones), one has to distinguish between invariant and reducing subspaces.
The fundamental theorems about representations of C*-algebras say that there are lots of them. Thus the Gelfand-Naimark Theorem^ says that every C*-algebra has a faithful representation — that is, a representation n with ker7r = {0}. We shall be particularly interested in irreducible representations, and here the first main result says that there are enough to separate points of A (Theorem A. 14). All these representations are built using the GNS-construction, which starts from a positive functional on A. Here "GNS" stands for Gelfand-Naimark-Segal. Recall that an element a of a C*-algebra A is positive (written a > 0) if there is another element b of A with a = 6*6. Basic but highly nontrivial theorems assert that a is positive if and only if a (a) C [0, oc) [111, 2.1.8 and 2.2.4], and that ||a||26*6 — b*a*ab > 0 for every a, b G A. (To see the latter when A does not have an identity, note that because a (a) is by definition the spectrum in A1, positivity is determined in A1; since ||a||2l — a*a > 0 in A1, it follows from [111, 2.2.5] that 6*(||a||2l — a*a)b > 0 in A1, and hence in A.) A linear functional p on A is positive if a > 0 =5* p(a) > 0; equivalently, p is positive if p(b*b) > 0 for all b G A. A positive linear functional of norm 1 is called a state. We need a basic lemma:
Lemma A.4. Let f be a positive functional on a C*-algebra A. Then for all a,b e A we have (a) f(b*a)=J(a7b); (b) (The Cauchy-Schwarz inequality) |/(6*a)|2 < f (b*b)f (a*a).
"I"There are two Gelfand-Naimark Theorems, and both are important: the first says that a commutative C*-algebra is canonically isomorphic to an algebra of continuous functions, and the second that every C*-algebra can be realized as a closed *-subalgebra of some B{7i). It is this second theorem which we shall prove in this section. 204 The Spectrum
Proof. Fix a, b G A. Then, for any A G C,
0 < f((Xa + b)*(\a + 6)) = |A|2/(a*a) + A/(a*6) + \f(b*a) + /(6*6). (A.l) First of all, since |A|2/(a*a) -h f{b*b) is always real, we have Im(A/(a*6) + A/(6*a)) =0 for all A. Taking A = 1 and then A = i gives the equality of the imaginary and real parts of f(a*b) and f(b*a). Next, we put A = xf(b*a) for x G R in (A.l) to obtain 0 < x2|/(6*a)|2/(a*a) + x\f(a*b)\2 + x\f(b*a)\2 + f(b*b) = x2\f{b*a)\2f{a*a) + 2x\f(b*a)\2 + f(b*b). Since the right-hand-side is a quadratic in x which is always > 0, this implies 4|/(6*a)|4 - 4\f(b*a)\2f(a*a)f(b*b) < 0, which gives (b). • For the GNS-construction, we start with a state r on a C*-algebra A. We let
NT := {aeA:r(a*a) = 0}, and deduce from Lemma A.4 that r(b*a) = 0 if either a or b lies in A^T. Thus there is a well-defined inner product on A/Nr such that
(a + NT |6 + A^T) = r(6*a).
We let HT be the Hilbert space completion of A/NT. Another application of Lemma A.4 shows that iVT is a left ideal in A, so A acts by left multiplication 2 on A/Nr\ since \\a\\ b*b — b*a*ab has the form c*c, we have
2 2 \\a • (b + 7Vr)|| = r(b*a*ab) = ||a|| r(6*6) - T(C*C) 2 2 2 <||a|| r(6*6) = ||a|| ||6 + iVr|| , so the elements of A act as bounded operators on A/NT, and extend to bounded operators 7rr(a) on the completion. We have now proved, at least in outline, the following proposition: Proposition A.5. For every state r on a C*-algebra A, the GNS-construction gives a nondegenerate representation irT of A on a Hilbert space HT.
These GNS-representations 7rr are characterized by the existence of a cyclic vector, a vector h G H^ is cyclic for TT if the set { n(a)h : a G A } spans a dense subspace of H^. More precisely: Proposition A.6. If p is a state on a C*-algebra A, then there is a unit vector hp in Hp which is cyclic for TTP and satisfies p(a) = (7rp(a)hp \ hp) for all a £ A. Conversely, if h is a unit cyclic vector for a representation TT : A —• B(Hn), then r : a i-» (7r(a)h | h) is a state on A, and the map a i-» 7r(a)h induces a unitary isomorphism U ofHT onto H^ such that 7r(a) = U7rT(a)U* for all a G A. A.l States and Representations 205
This Proposition is quite easy if A has an identity: the vector hp := 1 + Np is cyclic for np. For the general case, we need a lemma. To see where the first part comes from, note that if A has an identity and p is a positive functional on A, then the Cauchy-Schwarz inequality and the inequality a*a < \\a\\2l in A imply that
\p(a)\2 = b(l*a)|2 < p(l)p(a*a) < p(l)||a||2p(l), and hence that ||p|| = p(l).
Lemma A.7. Suppose that A is a C* -algebra which does not have an identity, and pe S{A). Then (a) if { ei } is an approximate identity for A, then p{ei) —> 1, and (b) the formula r(a + Al) = p(a) + A defines a state r on the C*-algebra A1 obtained by adjoining 1 (Example 2.J±1). Proof Since approximate identities in C*-algebras are increasing, and p is positive, {p{ei) } is an increasing net of positive real numbers bounded above by 1. Thus it converges to a real number L with L < 1. Note that we have
2 e = (ej)*eiej<(ejrief=ei.
Thus if a G A, then the Cauchy-Schwarz inequality gives
2 2 2 \p(eia)\ < p(e )p(a*a) < p^af < L\\a\\ , (A.2) which is only compatible with ||p|| = 1 if L > 1. Thus p(ei) —> 1. Equation (A.2) also implies that |p(a)|2 < p(a*a). Together with the identity p(a*) — p{a) from Lemma A.4, this gives
r((Al + a)* (XI + a)) = T(\X\21 + Xa + Xa* + a* a) = \X\2 + 2 Re(Ap(a)) + p(a*a) >\X\2-2\X\\p(a)\+p(a*a) >\X\2-2\X\\p(a)\ + \p(a)\2 >(\X\-p(a))2>0.
Thus r is positive and has norm T(1) = 1 . •
Proof of Proposition A.6. As we mentioned above, 1 + iV^ is cyclic for TTP if A has an identity. If A does not have an identity, we can use the previous lemma to extend p to a state r on A1. The inclusion of A in A1 induces an isometry V of Hp into Hr which takes a + Np to a + Nr and intertwines TTP and 7rr, in the sense that V7Tp(a) = 7rT(a)V for all a G A. Thus we can identify Hp with the subspace VHP of Hr- Indeed, Hp is then the essential subspace span{ 7rT(a)HT } of 7rr |A, and 1 nvU = 7T/9 © 0 on Hp 0 Hp . The projection hp of the vector 1 + 7VV on Hp satisfies
7rp(a)hp = 7rr(a)(l + Nr) = a + NT, 206 The Spectrum
so hp is cyclic for 7rp, and satisfies
{*p{a)hp | ftp) = (7rT(o)(l + NT) | 1 + NT) = r{a) = p(a). (A.3)
(Note that since p is a state, (A.3) implies that
l=||p||<|IM2
NT = { a e A : (?r(a*a)ft \ h) = 0} = {a e A : 7r(a)ft = 0 }, so there is a well-defined linear map Uo : A/NT —> Hn such that t/o(a+A^r) = ir(a)h. It is isometric:
(U0(a + ATT) | ^o(& + NT)) = (7r(6*a)ft | ft) - r(b*a) = (a + NT\b + NT).
Thus Uo extends to an isometric linear map U of the completion 7iT onto span{ 7r(a)ft : a G A }, which is all of 7in because ft is cyclic. So U is unitary. The computation
U7Tr(a)(b + JVr) - U(ab + JVr) = 7r(a6)ft - 7r(a)(ir(b)h) = 7r(a)U(b + ATr) implies that UnT(a)U* = 7r(a) because U is unitary. •
Corollary A.8. If IT is an irreducible representation of a C*-algebra A and ft G H^ is a unit vector, then TT is equivalent to the GNS-representation n^ associated to the vector state (j:an (7r(a)ft | ft).
Proof Since /C = span{7r(a)ft : a £ ^4} is invariant under 7r, the irreducibihty of 7r implies that K is either {0} or Hn. Because n is nondegenerate (otherwise its essential subspace is a nonzero invariant subspace), we have 7r(e^)ft —> ft; thus ft is a non-zero vector in /C, and K must be all of Hn. •
To produce representations of a C*-algebra A with given properties, it is enough to produce appropriate states and apply the GNS-construction. In particular, to get a representation which is nonzero on a given element a, we look for a state p satisfying p(a) ^ 0 . Since this problem only becomes harder if we make the algebra larger, we can suppose that A has an identity, and use the following characterization of states.
Lemma A.9. Suppose that f is a linear functional of norm 1 on a C*-algebra A with identity 1. Then f is a state if and only if /(l) = 1. A.l States and Representations 207
Proof. We saw in the discussion preceding Proposition A.7 that every state / sat isfies /(l) = 1. So suppose that /(l) = 1, and that there exists anaGi such that f(a*a) £ [0, oo). Since cr(a*a) C [0, oo) it is geometrically clear (draw a picture!) that there exist A G C and R > 0 such that
a(a*a) CB(X;R) but f(a*a) £ B(\;R). (A.4)
The first of these assertions implies a(a*a — Al) C B(0; R), by the spectral mapping theorem. Then
r(a*a - Al) < R => \\a*a - Al|| < R ^\f(a*a)-\f(l)\ Lemma A.10. Suppose A is a C*-algebra and a is a self-adjoint element of A. Then there is a state p of A such that \p(a)\ = \\a\\. In particular, for any a £ A, there is a state p such that p(a*a) = ||a||2. Proof. As above, we may as well assume that A has an identity. Consider the C*- subalgebra C*(a) of A generated by a and 1. This is a commutative C*-algebra with identity, and hence the Gelfand transform is an isometric isomorphism of C*(a) onto the algebra C(A) of continuous functions on the maximal ideal space A. Since a is a continuous map on a compact space, there exists 0 G A such that |0(a)| = |a(0)| = Halloo = ||a||. an By the Hahn-Banach theorem, there exists p e A* such that p\c*(a) — Theorem A.11 (The Gelfand-Naimark Theorem). Every C*-algebra A has a faithful nondegenerate representation. Proof. For each non-zero a e A, Lemma A. 10 gives a state pa such that pa(a*a) = 2 ||a|| . Let 7rPa : A —> B(HPa) be the corresponding GNS-representation, with cyclic vector hPa. Then 7rPa (a) ^ 0 because 2 2 0 < ||a|| = pa(a*a) = (nPa(a*a)hPa | hPa) = ||7rPa(a)/ipJ| , s a and hence TT := ©aG^ a_^0 ^Pa ^ faithful representation of A, which is non- degenerate because it is a direct sum of such representations. • The spectrum will consist of equivalence classes of irreducible representations, and for this to be a useful invariant, we need to know there are enough irreducible 208 The Spectrum representations to distinguish the points of A. To find them, we need to know when a GNS-representation is irreducible. The standard characterization uses the geometric structure of the space S(A) of states on A. The key point is that S(A) is convex: for every 0, ip G S(A) and A G [0,1], the convex combination \(j)+(l-\) Proof. Suppose that n := irp is not irreducible: there is an invariant subspace /C of Tip such that the orthogonal projection P of Hp onto /C satisfies P ^ 0 ^ 1 — P. We aim to write p as a non-trivial convex combination of states. Since /C is invariant, 7r(a)P : Hp —• /C and n(a)P — P7r(a)P; taking adjoints shows that PIT (a) = Pn(a)P = ir(a)P for all a G A. It follows from this and the cyclicity of hp that \\Php\\ ^0: Php = 0- 7r(A)Php = 0=> P7r(A)hp = 0 : PHp = 0 P = 0. 2 Similarly ||(1 - P)hp\\ ± 0. Thus A := \\Php\\ belongs to (0,1), and P/ p/i (1 - P)hp (1 ~ P)h 0(a) := ( ^ V fl ^(a) := 7r(a) p IP/in [l-P)hp (1-P)hp define states p(a) = (ir(a)hp \ hp) = (n(a)hp | Php) + (7r(a)hp \ (1 - P)hp) = (ir(a)Php | Php) + (rr(a)(l - P)hp \ (1 - P)hp) x P)hp (l-p)ftp ^wtii m)^M<^l(i-^) M (i - P)/IP = A(a) + (l-AMa). If we had p = 0, then the equation Pn(a)P = 7r(a)P would imply 2 (7r(a)hp | Zip) = ||P/ip||~ (7r(a)/ip | Php) for all a G ^1. 2 2 But this is only possible if \\Php\\ hp = Php, which in turn implies \\Php\\ Php = 2 2 P hp = Php, and ||P/ip|| = 1, which contradicts A G (0,1). Similarly we cannot have p — ip, so we have a nontrivial convex decomposition of p, and p is not an extreme point of S(A). A.l States and Representations 209 Conversely, suppose that n := TTP is irreducible and that p{a) = (n(a)h \ h) for some unit vector h £ Hn. We need to show that p is an extreme point; thus we suppose that p = A 4- (1 — \)ip for states 0 and ip and A £ (0,1), and aim to prove that 0 = p. Since ip and (j) are positive Np = {a e A : p(a*a) = 0} C N^ = {a e A : (j)(a*a) = 0 }. Since 7r(a)h = 7r(b)h exactly when a — b e Np, the Cauchy-Schwarz inequality (Lemma A.4(b)) implies that [ir{a)h^{b)h\ :=A0(6*a) is a well-defined sesquilinear form on the dense subspace n(A)h := { 7r(a)h : a 6 A } of Tin. Furthermore, [7r(a)/i,7r(a)ft] = A0(a*a) = p(a*a) — (1 — \)ip(a*a) < p{a*a) = ||7r(a)/i||2. It follows from the polarization identity (2.16) that [•, •] is bounded on ir(A)h and can be extended to a bounded sesquilinear form q on all of H^. Therefore there is a bounded operator T £ B{H^) such that q{h, k) = {h\ Tk) for all h.keH^ ([111, Theorem 2.3.6]); in particular, (n(a)h | Tir(b)h) = [7r(a)ft, 7r(6)h] = A0(6*a) for all a, 6 G A Since 0 is positive and 7r(i4)/i is dense, it follows that T is a positive operator of norm at most 1. Moreover we claim that T commutes with TT(A). For if a,b,c£ A, then (n(a)h | Tir(c)ir(b)h) = A0((cfc)*a) = A0(6*(c*a)) = (ir(c*a)h | Tir(b)h) = (7r(a)/i|7r(c)T7r(6)/i); since 7r(A)h is dense, we have 7r(c)T = Tn{c) for all c 6 i, as claimed. Since 7r is irreducible and T is positive, there is a nonnegative scalar z such that T = zl (Lemma A.l). If { e* }^/ is an approximate identity for A, then for all a £ A, A0(a) = limA(e;a) = lim(ir(a)h \ Tir(ei)h) = z(n(a)h \ h) = zp(a). Then Lemma A.7(a) implies that A = Irnii \(f>(ei) = limiZpfe) = z. Therefore (f) = /?, which is what we set out to show. • Lemma A. 13. For every element A of a C*-algebra A, there is a pure state p on A such that p(a*a) = \\a\\2. 210 The Spectrum For the proof of this lemma, we need a special case of the the Krein-Milman Theorem: every nonempty weak-* compact convex set K in the dual X* of a Banach space X is the closed convex hull of its extreme points — in other words, K is the intersection of all the weak-* closed convex sets containing the extreme points of K. Here, we only need to know that every such K has extreme points, but this is a significant part of what the theorem says. For the details, see either [125, 7.5.4] or the Appendix in [111]. Proof. Let E denote the set of states p of A satisfying p(a*a) = ||a||2, which is nonempty by the Lemma A. 10 and is easily seen to be a weak-* compact convex subset of A*. The Krein-Milman theorem implies that E has an extreme point p. We claim that p is necessarily a pure state, i.e. that p is also an extreme point of the larger set S(A). So suppose (/),ip G S(A), A e (0,1) and p = \(j) + (1 - A)^. Then the string of inequalities p(a*a) = A0(a*a) + (1 - A)^(a) < A||a||2 + (1 - A)||a||2 - ||a||2 = p(a*a) forces equality in the middle, which is only possible if (a*a) = ||a||2 = ip(a*a). But then Proof. Given a, we choose p as in the previous Lemma. Then np is irreducible by Lemma A. 12. The inequalities 2 \\af = p(a*a) = (7rp(a*a)hp | hp) = \\np(a)hp\\ < \\irp(a)\\\ combine with the standard property ||7r(a)|| < ||a|| of all C*-algebra homomor- phisms to force H^^a)||2 = ||a||2. • A.2 The Spectrum of a C*-Algebra The spectrum A of a C*-algebra A is the set* of unitary equivalence classes of irreducible representations of A. We begin our study of the spectrum with a couple of key examples. Example A.15. We claim that every irreducible representation of IC(H) is equiv alent to the identity representation id : K,(Ti) —» B(H), so that K{7i) = {id}. To establish the claim, we need to recall the description of JC(Tl) as span{ h<8> k ; ft, fc G W }, *For each irreducible representation TT and each h G Hn, the set {n(a)h} is dense in Tin, so the dimension of Ti^ is bounded by the cardinality n of A. Since all Hilbert spaces of a given dimension are isomorphic, each irreducible representation of A is equivalent to a representation in one fixed Hilbert space of dimension n. Thus the equivalence classes of irreducible representations do form a set. A.2 The Spectrum of a C*-Algebra 211 where ft 0 k(l) := (I \ k)h (Proposition 1.1). Now suppose that n : K,(H) —> B(H7T) is an irreducible representation, and fix a unit vector e G H. Then e 0 e is the projection of H onto the span of e, and hence P = n(e 0 e) satisfies P2 = P = P*, and is the projection onto the closed subspace PJi^. Fix a unit vector £ G P(W7r), and consider the subspace 7r(/C)£ = span{ 7r(T)£ : T G /C(W) }. This is invariant, is non-zero because £ = 7r(e 0 e)£ is in it, and hence by the irreducibihty of n is all of HIT. Define U :H —> Hn by [7ft- := 7r(ft 0 e)£. Then U is isometric: ([7 = (^|ft)(£|^(e0e)O = (^|ft)||£||2 = (^|ft). It is surjective because every element in the spanning set {7r(ft 0 fc)£ } is in the range of [7: 7r(ft 0 k)£ = ?r(ft 0 P)7r(e 0 e)f = (e | k)7r(h 0 e)f = (e | k)Uh. Thus [7 is a unitary transformation of H onto W^, and since 7r(h®k)(Ug) = 7r((ft 0 *•)(# 0 e))£ = (# | /c)7r(ft0e)£ = ( f(t). Since the equivalence relation on 1-dimensional representations is trivial^, we can therefore write A C(T) = {et:teT}. To see the claim, note that ir(C(T)) is commutative, and hence every operator 7r(/) is in the commutant 7r(C(T))'. Thus if TT is irreducible, then n : C(T) —> Cl-^, and every subspace of Hn is invariant; this is only possible if dim 7^ = 1, and then 7r is essentially a nonzero homomorphism of C(T) into C. But every such homomorphism is an evaluation map et (see, for example, [125, 4.2.5]). A powerful tool in the analysis of commutative C*-algebras is the identification of the sets of nonzero homomorphisms and maximal ideals, via the map (j) H-» ker HfTi is 1-dimensional, it is isomorphic to the Hilbert space C. Since B(C) = C is commutative, conjugating by a unitary has no effect. Thus the number in C = B(C) we get from an operator in B(C) by identifying 7i with C is independent of the identification. In particular, this means that each evaluation map gives rise to a representation in a canonical way. 212 The Spectrum and denoted Prim A. (But we caution that this will soon stand for the topological space obtained by endowing the set of primitive ideals with a natural topology: see Definition A.21 on page 214.) Before discussing the topology, we list some basic properties of primitive ideals. Recall that for us, all ideals in C*-algebras are closed and two-sided unless other wise stated; primitive ideals are closed because representations of C*-algebras are automatically continuous. Proposition A. 17. Let A be a C*-algebra A. Then (a) every closed ideal I in A is the intersection of the primitive ideals containing it; (b) if I is a primitive ideal in A, and J, K are two ideals such that J D K C I, then either J C I or K C I. Proof, (a) We have to show that if a G A and a £ /, then there is a primitive ideal P of A with I C P and a £ P. Consider the element a + / of the quotient C*-algebra A/I, which is nonzero because a £ I. There is an irreducible representation TT of A/1 such that ||7r(a + I)\\ = \\a -f I\\ ^ 0. But if q : A —> A/1 is the quotient map, then the composition TT O q is an irreducible representation of A with a £ ker TT o q. (b) Let TT : A —> B(H) be an irreducible representation with ker7r = I. If J<£ I, then 7r(J) ^ 0, and hence V := TT(J)H is nonzero. Because J is an ideal, V is invariant: for any j £ J, h G W, ir{a)(ir(j)h) = 7r(aj)h G TT(J)W, and the linearity and continuity of 7r(a) force n(a)V C V. Thus the irreducibility of TT implies 7r( J)H = H. But now TT(K) (TT(J)H) C TT(K n J)H C TT(I)W = 0 implies K C ker TT = I. • Remark A. 18. (a) Part (b) of Proposition A. 17 says that every primitive ideal is prime. The concept of prime ideals comes from ring theory, and especially the ring Z. The ideals in Z have the form riL for some n G N, and raZ C nZ <=> n | m. The ideal nL is prime exactly when n is a prime number. (b) In any separable C*-algebra, the converse of (b) is also true: any closed prime ideal is also primitive (Theorem A.49). Whether the same is true in an arbitrary C*-algebra is not known. Example A. 16 says that the spectrum of C(T) is naturally in one-to-one corre spondence with T via the map t i—• et. Since the compact space T is normal, two representations et and es have distinct kernels, and the map 11-» ker et is a bijection of T onto Prim C(T). It is important to remember that the topology we are about to define on the primitive ideal space Prim A is intended to be the usual topology on T when A = C(T), in the sense that 11-» keret is a homeomorphism. A.2 The Spectrum of a C*-Algebra 213 Definition A.19. If A is a G*-algebra, and F is a subset of Prim A, we define the closure F of F to be F={PePrimA: f] I c P}. ieF For motivation, consider the algebra C(T), where F = { ker et : t G S } for some uniquely defined subset S of T. Then p| / = f| ker6, = {/ = { / G C(T) : /|s = 0 } = { / G C(T) : /|s = 0 }. So we trivially have ClieF ^ ^ ker e* f°r all £ G S. Conversely, if s £ 5, Urysohn's Lemma gives / G C{T) such that /|^- = 0 and f(s) = 1, and hence an element of n OieF I which is °t in ker es. Thus F = { ker et : t G 5 } corresponds to the usual closure of S in T. Lemma A.20. Let A be a C* -algebra. The subsets F of Prim A such that F — F are the closed sets in a topology for Prim A. Proof. We verify that F \—> F satisfies the Kuratowski closure axioms: (a) 0 = 0^ (b) FcF for all F C Prim A; (c) T = F for all F C Prim A; (d) FUG = F U G for all F,G C Prim A It is then quite easy to check from first principles that { F : F = F } are the closed sets in a topology. Parts (a) and (b) are trivial, provided one remembers that p|/G0/ is all of A. For (c), just observe that f]IeF I C P for all P G F implies C\ieF I c flpeF ^5 which is only compatible with F C F if OieF ^ ~ OpeF P- ^ F C H then F_C F, and it follows that FuGc FUG. So it remains to prove that FUG C F U G. But n /cp<=*(rv)n(rv)cp, ieFuG ieF JeG which because P is prime (Proposition A. 17(b)) implies that P G F or P G G. • This topology on Prim A is known variously as the Jacobson topology (after its inventor, in the context of ring theory), or the hull-kernel topology (after the constructions in the first part of Proposition A.27 below). In Corollary A.28, we will describe the open sets explicitly. Since the kernels of equivalent representations are the same, there is a canonical surjection n i—> ker n of A onto Prim A. This map is not always an injection (see Hooptedoodle A.33) but is for many G*-algebras of interest to us, such as C(T). Using the map n \—> ker7r, we can pull the hull-kernel topology back to a topology on A. When one talks about the spectrum of a G*-algebra, one usually means the set A endowed with this topology. Thus we formally define: 214 The Spectrum Definition A.21. The primitive ideal space Prim A of a C*-algebra A is the set of primitive ideals of A endowed with the hull-kernel topology. The spectrum of A is the set of unitary equivalence classes of irreducible representations of A with the topology inherited from Prim .A: thus Sciis open if and only if { ker7r : n G S } is open in Prim A Remark A.22. Notice that we have already slipped into a standard sloppiness: we confuse an irreducible representation ix with its equivalence class [IT] in A. This will only cause problems if we forget that we are really making a choice of representative for the class [7r], or forget to check that some mapping is well-defined on A. Thus, for example, we were careful above to note that equivalent representations have the same kernel, so that [n] i—• ker-zr is well-defined. We are now ready to discuss the examples most relevant to this book. We have already seen that Prim C(T) is just T, in the sense that 1i—» kere^ is a homeomor- phism. (Before Lemma A.20, we saw that this map converts the usual closure oper ation to the one we have defined on Prim C(T)\ it follows that it swaps closed sets with closed sets and open sets with open sets, and is therefore a homeomorphism.) Since et i-+ kere^ is a also a bijection, we also have C{T) = T. Example A.15 says that K{H) — {id}, and hence Prim/C(H) = {0}. The topology here is not interesting, but Proposition A. 17(a) is: it implies that { 0 } is the only proper ideal in /C(W), or in other words that K,(H) is simple. Our next examples are combinations of these two, and the techniques we use to determine their spectra are used throughout the book. The simplest is the algebra C(T, Mn(C)) of continuous functions from a compact Hausdorff space T into the C*-algebra Mn(C) of complex n x n matrices, which is a C*-algebra under the pointwise operations inherited from Mn(C) (so that, for example, (fg)(t) is the usual product of the matrices f(t) and g(t)), and the sup norm H/Hoo := sup{ ||/(*)||^(C) : * e T}. A function / e C(T,Mn(C)) can be naturally viewed as a matrix of continu ous functions, and hence as an element of the C*-algebra Mn{C(T)) discussed on page 34; indeed, this gives a isomorphism of C(T, Mn(C)) onto Mn(C(T)), which has to be isometric because both algebras carry a unique C*-norm. We shall use this isomorphism freely in the examples to follow. Example A.23. Suppose A is the C*-algebra C(T, Mn(C)) described above. Then each irreducible representation is equivalent to an evaluation map et : a i—> a(t) of n A into Mn(C) = £(C ), and there is exactly one such t. Thus we can write i = {ef : t G T}, provided as usual that we remember that we have only written down a representative for each class: we get different representations if we change n the identification of Mn(C) with J3(C ), but they are unitarily equivalent*. The description of Prim A as {kere^ : t £ T}, on the other hand, is canonical. The maps 11—> [et] and t \—> keret are both homeomorphisms. n x *If S,T are two isomorphisms of Mn(C) onto B(C ), then S o T~ is an automorphism of B(Cn), and hence has the form Ad U for some unitary operator U on Cn (Proposition 1.6). The unitary operator U implements an equivalence between the representations T o et : a H-> T(a(t)) and S o et : a i—• S(a(t)). A.2 The Spectrum of a C*-Algebra 215 Proof. The arguments we use to see all this will come up again and again. First suppose that n : A —» B(H) is an irreducible representation. There is a copy of C(T) sitting inside A, as multiples { fln } of the identity matrix ln G Mn(C): we can view fln either as the function t H-» f(t)ln in C(T, Mn(C)), or as the diagonal matrix // o ... o\ 0 / ... 0 \o o ... // of continuous functions in Mn(C(T)). The map / i—> f\n is a unital homomorphism, and hence the representation / i—> 7r(/ln) is nondegenerate. Further, since (fln)a = a(/ln) for all a G A, 7r(/ln) commutes with the range of 7r; since IT is irreducible, this implies that 7r(/ln) G Cl?^. for all / G C(T). Since 7r is certainly nonzero, Example A. 16 implies that there exists t G T such that 7r(/ln) = f(t)ln for all / € C(T). We next claim that /t:={oG C(T, Mn(C)) : a(t) = 0} C ker7r. For suppose a(t) = 0, and e > 0. We can choose a neighbourhood N of t such that ||o(s)|| < £ for s G iV, and / G C(T) such that 0 < / < 1, /(£) = 0 and f(s) = lfovs^N. Then \\a(s) — f(s)a(s)\\ < £ for all s G T, and hence ||7r(a) — 7r(/ln)7r(a)|| < £. Since we know that 7r(/ln) = fif)!^ = 0, we deduce that ||7r(o)|| < e; because £ was arbitrary, we must have ||7r(a)|| = 0. Since It C ker7r, there is a representation n of A/It on the same space Hn such that 7r(a + It) = 7r(a); 7r is irreducible because it has the same range as n. Since 1 a i—> a(£) induces an isomorphism {ker et : t G TV } = {keret : t GiV} for every subset TV of T. We know that the left-hand side has the form { ker et : t G M } for M := { s : Hteiv ^er e* ^ ker e« }> an<^ we nave to prove M = N. Since the elements of A are continuous functions on T, a = 0 on N implies a = 0 on A7", so certainly N C M. On the other hand, if 5 ^ AT, there exists / G C(T) such that = ano s = ano /ITV 0 - /( ) ^' ~ then /In is in kere^ for all t € N but not in keres, which implies s £ M. Thus we must have M = N, and the claim is proved. • Example A.24. Now suppose that A = Co(T, JC(H)) for some locally compact space T and infinite-dimensional Hilbert space H. As for the previous example, 11—> [e^] is a homeomorphism of T onto A, but there are technical complications in 216 The Spectrum following the same line of argument. First, CQ[T) does not sit inside A as multiples of the identity: rather we have a homomorphism t of Cb(T) into the centre ZM(A) of the multiplier algebra of A, given by (i(f)a)(t) := f(t)a(t). Then, since T is locally compact rather than compact, i is nondegenerate in the sense of Section 2.2 rather than unital: we can approximate any a G A by t(f)a for suitable / in Co(T). (Choose / to be 1 on {t : ||a(t)|| > e}.) By Proposition 2.50, we can extend 7r to a unital representation TX of the multiplier algebra M(A), and consider the representation / i-> 7t(t(f)), which is nondegenerate because i is, and in particular is nonzero. Because t(f) is in ZM(A), 7t(i(f)) is in 7r(A)' = Cl^w, and as before there exists t G T such that 7t(i(f)) = f(t)lu^- As in the previous example, we can see that It := {a G A : a(t) = 0} is contained in ker7r, and the result of Example A. 15 implies that n is equivalent to the evaluation map et : a h-» a(£). One can also modify the argument of the previous example to see that in [et] is a homeomorphism, using elements of the form f(e ® e) rather than /ln. Example A.25. Consider three C*-subalgebras of C([0,1],M2(C)): Ai = {/€C([0,1],M2(C)):/(0)€C12} ^2 = {/ e C([0,1],M2(C)) : /(0) = (J g) for some A G C} 4s = {/ G C([0,1],M2(C)) : /(0) = ($ °) for some A,/i G C} Using the techniques of the previous examples, one can see that t H [et] is a homeomorphism of [0,1] onto both A\ and .A2. On the other hand, while every irreducible representation of As factors through an evaluation map et, there are two which factor through eo: the representations 7Ti : / i-» /(0)n and 7r2 : / i-» /(0)22 are inequivalent. As a topological space, As = (0,1] U {7Ti, 7r2 }, where (0,1] has its usual topology, and sets of the form { TT\ } U (0,6) and {7r2 } U (0,6) are also open. Thus sequences converging to 0 in (0,1] converge to both TTI and 7r2 in A3, and A3 is definitely not Hausdorff. However, it is Ti, because both the unusual points are closed. (Curiously, one is more likely to come across algebras like A<± and As than the unital algebra A\.) We shall now discuss the main properties of these topologies on Prim A and A. First, we consider their behavior with respect to ideals and quotients; this is of fundamental importance for our localization arguments in Chapter 5. We begin with set-theoretic descriptions in the next Proposition, and then prove that these descriptions respect the topologies in Proposition A.27. Proposition A.26. Let I be an ideal in a C*-algebra A, and let q : A —• A/1 be the quotient map. Then (a) the map TT 1—> 7r| / is a bisection of {IT G A : n\i =/=• 0 } onto I, and P \—> P H I is a bisection of {P G Prim A: I <£ P} onto Prim I; (b) the map TT I-> n o q is a bisection of (A/1) onto {p G A : p\j = 0}, and Q 1—> q~x(Q) is a bisection of Prim A/1 onto {P G Prim A : I C P} with inverse P 1—> P/I. A.2 The Spectrum of a C*-Algebra 217 Proof, (a) Suppose n is irreducible and n\i ^ 0. Since Tr(I)Hn := span{ n(a)h : a G I, h G W^ } is a closed invariant subspace of H^ we must have n{I)H^ = H^. Thus 7r| j is nondegenerate. If V is invariant for 7r|j, then the equation 7r(a)(7r(z)fc) = 7r(ai)k (A.5) implies that V is invariant for 7r on all of A Hence V must be { 0 } or W^, and TT\I is irreducible. If 7r ~ p, then we trivially have TT\J ~ p|j, so the map ir i—> 7r|/ is a well-defined map of{7rG.A:7r|j^0} into 7. The equation (A. 5) also implies that if 7T\I ~ p|/, then 7r ~ p on all of A, so the map 7r I—• 7r| / is injective. If we start with a representation p £ 7, then because 7 is an ideal, p extends to a representation p of .A (apply Proposition 2.50 with X = H). The irreducibility of p trivially implies that of 7r, SO this proves that the map 7r I—> 7r|/ is onto. Since (ker7r) fl I = ker(7r|j), it follows immediately that P \-> P D I is a well- defined map of Prim A onto Prim/. To see that this map is injective, suppose Pi, P2 € Prim A have I (jLP^ and Pi fl J = P2 fl J. Then the primeness of Pi implies that P2 C Pi, and by symmetry we must have Pi = P2. (b) For any representation n of A/1, the representation n o q of A has exactly the same range as 7r; hence TV O g is irreducible if and only if 7r is, and ir o q ~ po q if and only if n ~ p. Thus 7r 1—• 7r O # is a well-defined injective map. If p € A has 7 C ker p, then there is a representation p of A/1 on Wp such that p{a +1) = p(a), and hence such that p o q = p; then p is irreducible because p is. Thus n »-> 7r O # is surjective from (A//) onto {pG^4:p|/=0}. The corresponding facts for Prim A/1 follow easily. • Proposition A.27. Let A be a C*-algebra. (a) F 1—> fc(F) := p|{ J : J e F} is a bijection between the closed subsets F of Prim A and the ideals in A, with inverse given by h(I) := {P eVvimA : I C P}. One calls k(F) the kernel of F and h(I) the hull of I. (b) If I is an ideal in A, then P 1—» P fl I is a homeomorphism of the open set {P € Prim A : I 7 = p|{PePrimA:IcP}. 218 The Spectrum This implies, first, that the set F :— { P G Prim A : I C P } is closed: PeF^ p| JcP^=>lcP<=>PeF; JeF and, second, that / = k(F), so that k o h is the identity on the set of ideals. On the other hand, k(F) C P «£=>- P G P, so that we can recover a closed set F as { P : fc(F) C P }, and ft, o k is the identity on the set of closed subsets of Prim A. (b) The canonical maps 6 from A to Prim A are open and continuous, and we have a commutative diagram 7rH->7r|j {TT G A : TT|/ ^0} • J 0 r {Pe Prim A : / £ P} ^^ Prim/ in which the horizontal arrows are bijections by Proposition A.26 and 1 {ireA:n\I^0} = 0- {Pe Prim A : I <£ P }. Since S := {P G Prim A : / ^ P} is open by (a), it is enough to show that P i—> P D J is bicontinuous. But a subset N is open in Prim A if and only if there is an ideal J in A such that N = {P e Prim A : J (£_ P}, and hence M is open in S if and only if there is an ideal J in A such that M = {PG Prim A: I (jL P and J£ P}. Since primitive ideals are prime, I£P and J£P^=>/nJ£P, so we can deduce that the open sets in S have the form M = {PePrimA:lnJ <£P} for some ideal J of A. But PGM^/nJ?:p^/nJ(z:/nP ^JflPG {Q G Prim/: in J ^ Q}, and hence P i—> P fl / is bicontinuous. (c) The closed sets of T = { P G Prim A : / C P } have the form T fl p|{ P G Prim A :JcP} = {PG Prim A : / U J C P } for some ideal J of A. By writing if for the ideal generated by / and J, we see that the closed sets of T have the form TK = {PePhmA:K CP} A.2 The Spectrum of a C*-Algebra 219 for some ideal K of A containing I. Since the closed subsets Prim A/1 are precisely those of the form SL = { Q £ Prim A/1 : L C Q } for some ideal L of A/1, we see l that q~ maps the closed set SL to Tq-i^L)-> and its inverse maps TK to 5#yz- ^ Corollary A.28. The open sets in Prim A are precisely those of the form Oj = {P:J£P} for some ideal J of A. Remark A.29. A considerable effort has gone into describing the topology on the primitive ideal spaces of interesting C*-algebras (for example, see [48, 50, 51, 10, 170, 135, 114, 74, 33, 35, 38, 151]). The first part of the previous proposition shows why: when we know which subsets of Prim A are closed, we know all the ideals in A. Indeed, since the correspondence F H-> k(F) is order reversing, we even know the lattice of ideals in A. Theorem A.14 implies that the functions n i—> ||7r(a)|| : A —> R determine the norm of a via ||a|| = sup^ ||7r(a)||. These functions also have some interesting continuity properties which we shall need; the following general results can be substantially sharpened when the spectrum is Hausdorff (see Lemma 5.2). Lemma A.30. Suppose that A is a C*-algebra and that a G A. (a) The function n i—> ||7r(a)|| is lower semicontinuous on A; that is, { TT G A : ||7r(a)|| < k } is closed for all k>0. (b) For each k > 0, { n G A : ||7r(a)|| > k } is compact. Proof (a) Let C = {TT G A : ||7r(o)|| < k}. Since ||7r(a)||2 = ||7r(a*a)||, it suffices to prove the assertion for a > 0. Let a (a) be the spectrum of a. Because a > 0, the norm of a is its spectral radius, so C = {7reA:a(7r(a))c [0,*]}. If C is not closed, then there are p G C and a £ [0, k] such that a G a(p(a)). However, there is a continuous function / : R —> R such that / = 0 on [0, k] and f(a) = 1, and then f(p(a)) = p(f(a)) ^ 0. On the other hand, 7r(/(a)) = 0 for all 7r G C\ since p G C means kerp D 0neC ker7r, this implies p(f(a)) = 0, which is a contradiction. This proves (a), (b) To prove that C = {neA: \\n(a)\\ > k } (A.6) is compact, let { Ci }iei be a family of closed sets in C with the finite intersection property. Let Ji = HTTGC ker7r, and let J be the ideal generated by \Ji^I Ji-> so that J = f]7TenC. ker7r. Fix n G N. By definition of the quotient norm there exists b in the linear span oi[JieI Ji such that ||a + j||>||a + 6|| * (A.7) 220 The Spectrum Note that b G span ([jieF Jj) for some finite subset F of /. If n G f]ieF Ci, which is nonempty by the finite intersection property of { d }, then 7r(6) = 0; since 7TGC, this implies that ||a + fc|| >||7r(a + 6)|| = ||7r(a)||>fc. (A.8) Since n was arbitrary, (A.7) and (A.8) imply \\a + J\\ > k. It now follows from Theorem A. 14 that there is an irreducible representation p of A/J such that \\p(a + J) || > k; thus there exists r] € A such that kerrj D J and ||ry(a)|| > k (see Proposition A.26(b)). In other words, rj G d for all i (because J D Ji for all i) and rj £ C. Since each C; is closed in C, we deduce that 77 G Cf for all z. Thus C is compact, as claimed. • At this point, we have the basic facts about the spectrum which we need for our localization arguments. However, we would be leaving a false impression of the possible complications if we only mention algebras of continuous functions with values in algebras of compact operators. So we shall briefly survey other important results in the area, without any claims to completeness. Among other things, we shall explain the acronyms CCR and GCR which occasionally crop up in the text. First we want to discuss the effect of the previous Proposition on the exten sion problem for C*-algebras: to what extent do an ideal i" and the quotient A/1 determine a C*-algebra A? Even for commutative algebras, this is a complicated question. An ideal in Cb(T) has the form Ip := { / : /|F = 0 } = Co(T\F) for some closed subset F of T, and the map / •—• /|F induces an isomorphism of Co(T)/Ip onto Co(F). Thus the extension problem asks: given two locally compact Hausdorff spaces S and F, how can we embed S as an open subset of a locally compact space T so that the complement T \ S is homeomorphic to Fl There is a trivial solution (take T to be the disjoint union of S and F), but there could be many more. Given two C*-algebras J5, C, the extension problem seeks algebras A containing a copy of B as an ideal with the quotient isomorphic to C. Again there is a trivial solution: the C*-algebra direct sum B®C, with ||(6, c)|| := max{ ||6||, [|c|| }. Proposition A.27 says that this problem is at least as complicated as the topological problem for Prim B and Prim C, and in fact it is even more so: primitive ideal spaces need not be Hausdorff, so there are potentially even more ways to fit the two bits together. The fundamental example here is the Toeplitz algebra T; we shall have no need of its realization as the (7*-algebra generated by a family of Toeplitz operators, so we merely define T to be the C*-algebra generated by the unilateral shift, and mention that a famous theorem of Coburn justifies the name (see, for example, [111, 3.5.18]). 2 Example A.31. Let H = £ have the usual basis { en }^=11 so that the unilateral shift S is the isometry characterized by Sen = en+i. We claim that the C*-algebra T generated by S is an irreducible subalgebra of B(H) containing the algebra JC(H) of compact operators, with corresponding quotient T/JC(H) isomorphic to C(T). Since T is irreducible, { 0 } is a dense subset of PrimT, so T cannot be isomorphic to/C(«)eC(T). A.2 The Spectrum of a C*-Algebra 221 Proof of the claim. We first prove that T acts irreducibly on Ji. So suppose V is a non-zero closed subspace of 7i which is invariant under T. Since S*e\ = 0 and 5*en = en_i for n > 1, the operator 5*5 — SS* is the rank-one projection n e\ ® ei onto the span of e\. Since 5 ei = en, the space Tei := {Tei :TGT} is dense in H. Write e\ = h -h fc with ft, G V and fc G V±. If (fc | ei) 7^ 0, then (ei 0 ei)(fc) = (fc | ei)ei and hence ei also belongs to V-1; this implies that x 1 Tei C V and V- = H, which contradicts V / {0}. Thus (fc | ei) = 0, ex = ft belongs to V, and Tei C V, so V is all of H. To see that T contains the compacts, it is enough to show that each rank-one operator ft ® fci s in T. Because Tei is dense in W, we can find R,T eT such that Rei ~ ft and Tei ~ &• But now ft 0 fc ~ i?(ei 0 e~i)T* G T, as required. It remains to identify the quotient T/K{H) with C(T). Since T is generated by 5, the quotient is generated by the image q(S) under the quotient map q. Because 5*5 = 1 and SS* — 5*5 is compact, u = q(S) is unitary, and hence has spectrum Me := spaH{ f(T)h :heH, f G C(cr(T)), and /(A) = 0 for |A| < e }. We first claim that ||Tfe|| > e\\h\\ for all h G Me. To see this, fix h G Me and let Jc = {/ G C(a(T)) : /(A) - 0 for |A| < c}. If / G J€ and #(A) = A for all A G cr(T), then ^2|/|2 > e2|/|2; thus T2/(T)*/(T) > 62/(T)*/(T), and ||T/(7>||2 = (T2f(Tyf(T)h | ft) > e2{f(TYf(T)h \ h) = 62||/(T)ft||2. Let { /i } be an approximate identity in Je. Then fi(T)h —± /i, so 2 2 2 ||T/if = lim Hr/iCTJftll > e lim ||/t(T)/i|| = e \\hf, as claimed. Thus T is bounded below on Me, and since T is self-adjoint, it follows that T is invertible on Me. Now let Pn be the projection onto Mi/n, and define f0 ifO /n(A) = {2(A-i) ifi Then fn G Ji/n and fn —> 9 uniformly on cr(T), where ^(A) = A for all A G o~(T). Thus /n(T) -^ T and Pnfn(T) = /n(T). If each M1/n were finite-dimensional, then Pn and hence fn(T) would be finite-rank operators, and T would be compact. So there exists n such that M\jn is infinite-dimensional, and this is the required subspace M. This proves that / is all of B(H), as required. Suppose that n is an irreducible representation of C(H) whose range contains a compact operator. The range must then be contained in K{H-K)'. otherwise the inverse image of ICi^H^) would be a proper nonzero ideal in C(H). For the same reason, n must be faithful. On the other hand, C(H) has an identity, so the space Hn must be finite-dimensional, which is impossible because one can find infinitely many mutually orthogonal projections in C(7i) (just take the images of infinitely many mutually orthogonal projections in B(H) of infinite rank). Thus no irreducible representation of C(7i) can contain a compact operator, as claimed. • Hooptedoodle A.33. We describe a family of C*-algebras A for which Prim A = {0} but A is uncountable. Let n be a sequence of positive integers { nn }^L± with each Kn > 2, A.3 The Dauns-Hofmann Theorem 223 and set K,\(n) := K1K2 • • • K>n. There is a natural unital monomorphism (j)n of MK?(n) into MKi(n+i) obtained by sending a £ MKj(n) to the block diagonal matrix in MK!(n+i) with a repeated «n+i times down the main diagonal; alternatively, identify MKt(n+1) with MKj(n) 0 MKn+1 (see Example B.19), and define n(a) := a ® 1. Now fix a separable infinite-dimensional Hilbert space TL and view each MK\(n) as a unital subalgebra of B(7i) by viewing a € MK?(n) as an infinite block diagonal matrix with a's on the diagonal; each (f>n is now given by inclusion. The Glimm algebra or UHF-algebra associated to K is the norm closure A(K) of (Jn AfK!(n) in 5(H). If we define eK:N^NU{oo}by e«(r) = sup{ m G N : rm divides some /c!(n) }, then some hard work shows that A(K) is isomorphic to A(K') if and only if eK = eK/ [111, Theorem 6.2.3 and Example 7.3.3]. It follows that there are uncountably many nonisomorphic Glimm algebras. It is easy to see that each Glimm algebra is simple [111, Theorem 6.1.3], so that PrimA(K,) — {0}. We shall now see that the spectrum A(K)A is uncountable. For each X Tnen n ra, fix a set A(n) = { A?,..., A«n } C [0,1] with j^rZi r = 1- there is a state p on MKn such that P (a) = 7^. , \aa', n n if each Af = l/«n, then p is the usual normalized trace on MKn. Let r be the product 1 n state p r2 ( , j = 7r1(a) + Sr1(d) = a^an + ^7022 + aSdu + f36d,22- n n+1 In general, we have r = r |MK!(TI) , so the rn combine to give a functional on |Jn MK\(n); A A n it turns out that there is a unique state r of the closure A(K) such that r |MK,(n) = T for all n. If each AJJ is either 0 or 1, then rA is a pure state [124, Lemma 6.5.5], and the associated irreducible representations are equivalent if and only if the sets A(n) are the same for all large n [124, Proposition 6.5.6]. A.3 The Dauns-Hofmann Theorem The Gelfand-Naimark Theorem identifies a commutative C*-algebra A with an algebra of continuous functions on its maximal ideal space. One might hope for a similar theorem identifying a noncommutative C*-algebra as an algebra of operator- valued functions on its spectrum or primitive ideal space, and indeed this was the original motivation for developing the theory of C*-bundles. This program was not completely successful — the resulting generalizations of the Gelfand-Naimark Theorem had nontrivial hypotheses and required the development of an entirely new kind of bundle — and this is one reason we have preferred a more algebraic approach in this book. However, the theory certainly had successes, and the Dauns- Hofmann theorem is a particularly striking one. It says that every C*-algebra is a module over the algebra of continuous functions on its primitive ideal space. While this formulation is purely algebraic, the result was motivated by the correspondence between bundles over a space T and C(T)-modules, and was originally proved by 224 The Spectrum bundle-theoretic methods [23]. The present direct proof was developed by Dixmier [27] and Elliott-Olesen [45]. Theorem A.34 (Dauns-Hofmann). Let A be a C*-algebra. For each P G Prim A, let np : A —> A/P be the quotient map. Then there is an iso morphism (j) of Cfe(PrimA) onto the center ZM(A) of M(A) such that for all f G Cb(PrimA) and a £ A 7Tp(0(/)a) = f(P)nP(a) for every P G Prim A. (A.9) We usually write f • a for (j)(f)a. For the proof we need some lemmas. For the first, recall that if / and J are ideals in a C*-algebra, then / + J is also a closed ideal: if TTJ : A —> A/ J is the quotient map, then I + J = 7rJ1(nj(I)). Lemma A.35. Suppose I and J are ideals in a C* -algebra A, and a is a positive element of I + J. Then there are positive elements b E I and c G J such that a = b + c. Proof. Since / and J are ideals in A1, we can assume that A has an identity. By definition, there are not-necessarily-positive elements b G / and c G J such that a = 6 + c. Since a = (a + a*)/2, we can assume 6 — 6* and c — c". Fix e > 0. Then h := \b\ + \c\ + el is positive and invertible. Let d = axl2h~xl2. Then |6| + \c\ < h, 1 2 fc-^fl&l + lcQft- / < l,andd(|6| + |c|)d* < a. Now let 61 =d|6|d*andci = d\c\d*; then 6i G i", c\ G J, both are positive, and 6i + c\ < a. Since 1 - d*d = /i-1/2(/i - a)/*"1/2 - /T1/2(|6| - 6 + |c| - c + el)^1/2 > 0, we have dd* < 1 and &i + ci = d(|6| + \c\)d* = d(h - el)d* = a - edd* > a - el. In other words, 0 < a — 6i — c\ < el. This implies that ||6i|| < ||a|| and ||ci|| < ||a||. We can repeat the above construction with a replaced by ai = a — 6i — c\ and e by e/2, and then continue inductively to obtain positive elements bk G / and Ck E J such that n 0 < a - (6i + 62 + • • • + bn) - (ci + c2 + • • • + cn) < (e/2 )l, fc 1 fc 1 \\h\\ < 2-( - )||a||, and ||cfc|| < 2-( - )||a||. The required elements are V = E^Li 6fc and c' = EfcLi c^- D Lemma A.36. Le£ / G C^(Prim A), a e A and e > 0. T/ien there exists be £ A such that \\np(be) ~ f(P)np{a)\\ < e for all P G Prim A. (A.10) Any other such b'e satisfies \\be — b'e\\ < 2e. A.3 The Dauns-Hofmann Theorem 225 Proof. It is enough to do this for / : Prim A —* [0,1]. Fix n, and define open sets O = {PePrimA: ^—^ < f(P) < *±i } for k = 0,1,... ,n. k n n Note that Prim A = Ujt=o ®k and that each P G Prim ^4 belongs to at most two Ofc's. Let Jk = f){P£?nmA:PtOk} be the ideal in A with primitive ideal space Okl so that P G Ok if and only if Jk Jo + Jl + ' • • + Jn = A. By Lemma A.35, there are positive elements a& G J^ such that a = ao + a\ + a/e ••• + an; notice that ||afc|| < ||a|| for all k. Let 6 = J]fc=o n ' Then for every P G Prim A, we have hP(b) - f(P)nP(a)\\ = \\*IIY—'fe=o" n -^p(ofc n ) " f(P)*p(ak) (A.H) •k P \El0^~f( ))Mak) But if P G Ofc, then |fc/n — /(P)| < 1/n. Since each P belongs to at most two Ofc's, (A. 11) is bounded by (2/n)||a||. Now the existence of be follows by taking n> 2||o||/e. If b'e also satisfies (A. 10), then ||6e-6'e||=sup||7rP(6e)-7rp(6'J|| P < sup(||7rP(6e) - /(P)7rp(a)|| + ||/(P)M<0 - 7rP(6'J||) P <2e, as required. • Proof of Theorem A.34- Given / and a, we consider the sequence {b2-n} generated by Lemma A.36. The uniqueness clause implies that this sequence is Cauchy, and hence converges to an element T(/, o). Equation (A. 10) implies that 7TP(T(/, a)) = f(P)irP(a) for all P G Prim A, and this equation characterizes T(f,a) because p)ker7Tp = {0}. The map b *->• T(/, 6) is an adjoint for T : a i—> T(/, a): to see this, we just need to verify that 7rp«T(/ , a),6)J = f(P)nP(a*b) = irP((a , T(f,b))J for P G Prim A. Thus we have a multiplier (j>(f) := T(/, •) of A. A similar calculation shows that If 7r G A and z G ZM(A), then TT(Z) commutes with 7r(A) and must be a scalar multiple of 1^ : say ft(z) := 1^(2)1?^. The map wn : ZM(A) —> C is a homo- morphism, and is nonzero because m^l) = 1; hence w^ belongs to the maximal ideal space A of ZM(A). Since ker wn = { z G ZM(A) : z • A C ker n }, u^ depends only on ker7r, and we have a well-defined map a : Prim A —> A such that a(ker7r) = wn. Recall that if J is an ideal in ZM(A), then Oj := {w e A: kerw ^ J} is a typical open set in A = Prim ZM(A) (Proposition A.27). On the other hand, J • A := span{ za : z G J, a G A } is an ideal in A, and the corresponding open set V-j-^ :={Pe Prim .A : P ^T^A) in Prim .A satisfies a~x{Oj) = {ker7r : TT G A and keru^ 7$ J } = { ker TT : TT e A and J - A (£ ker 7r } Thus a is continuous, and we can define ip : ZM(A) —> C&(Prim.A) by -0(z) = ioa, or equivalent ly '0(z)(ker7r)l^7r = z(w7T)l-]-c7v = TT(Z) for 7r G A Then for 7r G A and a e A we have 7r(za) = 7r(z)7r(a) = ^(z)(ker7r)7r(a) = n((f)(ip(z)) a), so2 = (f)(ip(z)). Thus 0 is surjective, as required. • Hooptedoodle A.37. If X is any topological space, then a compact Hausdorff space /3X together with a continuous map /3 : X —> /3X is called a Stone-Cech compactification of X if, for every continuous map k from X into a compact Hausdorff space Y, there is a unique continuous map k' : f3X —> Y such that k = k' o (3. Stone-Cech compactifications are essentially unique, and always exist — even for potentially pathological spaces such as Prim A — and Cb(X) = C(/3X). When A has an identity, Prim A is compact and /3(Prim A) = AZA is also known as a complete regularization of Prim A. Further references and details can be found in [112]. A.4 The State Space of a C*-Algebra For our motivating example A = C(T), the topology on Prim .A = A can be alter natively described by viewing A = {et : t £ T} as a subset of the Banach space dual A* and restricting the weak-* topology — indeed, both topologies reduce to A.4 The State Space of a C*-Algebra 227 the usual one on T (see, respectively, the motivating discussion before Lemma A.20, and [125, 4.2.5]). In this section we shall give a similar description of the topology on the spectrum A of a noncommutative C*-algebra A. For this, we realize A as the quotient of the space P(A) of pure states under the map p i—> TTP which sends a pure state p to its GNS-representation; the weak-* topology on P(A) then descends to a quotient topology on A Of course, there is a good deal of redundancy in this reaization of A: any vector state UJ : a H-> (n(a)h \ h) associated to an irreducible representation 7r gives a GNS-representation ix^ equivalent to n (by Corollary A.8). Theorem A.38. Let A be a C* -algebra, and give the set P(A) the weak-* topology. Then the GNS-map A : p i—> np is a continuous open surjection of P(A) onto A. After proving Theorem A.38, we discuss some results which we want to use in two of our peripheral sections. The first says, conversely to Proposition A. 17, that in a separable C*-algebra, prime ideals are primitive. This is used in the proof of Theorem B.45. The second is a technical result about states and ideals which is needed for the proof of the Brown-Green-Rieffel Theorem. For the proof of Theorem A.38 we need some more convex analysis. We have already met the Krein-Milman Theorem, which says that every weak-* compact set C in A* is the weak-* closed convex hull of its extreme points; now we need to know that if C is the weak-* closed convex hull of some other set .D, then every extreme point of C is in the weak-* closure of D. Detailed proofs of this result are given in [111, Theorem A. 14] and [81, Theorem 1.4.5]. We also need a version of the Hahn-Banach Separation Theorem obtained by applying the usual one to the dual of a Banach space X. Lemma A.39. Suppose that X is a Banach space, and (j) is a weak-* continuous linear functional on X*. Then there exists x G X such that f]{geX*:\g(Xi)\ Then g{xi) = 0 for all i implies (j>(g) = 0. (If 4>{g) ^ 0, then 4>(g)~lg belongs to the intersection but not to S.) Thus &(g) = (g(xi),. • • ,9(%n)) defines a linear map 3> : X* —> Cn such that ker $ C ker 0, and hence there is a linear functional n n T : C -> C such that (f) = T o$. Thus there is a vector (zu...,zn) G C such that (fi(g) = J2i zi9(xi) f°r an 9 £ -X"*- Then x := J2zixi wn^ do. • Corollary A.40. Suppose that X is a Banach space and that C and D are disjoint nonempty convex subsets of the dual space X*. If D is open in the weak-* topology, then there exist x£l and a G M such that Re/(x) < a < Reg(x) for all f G D and g G C. 228 The Spectrum Proof. This follows immediately from the Lemma and the usual Hahn-Banach Sep aration Theorem (see, for example, [125, Theorem 2.4.7]). • Lemma A.41. Let A be a C* -algebra with identity. Suppose D is a subset of S(A) such that a = a* and p(a) > 0 for all p G D => a > 0. (A.12) Then P{A) is contained in the weak-* closure of D. Proof. In view of the comments preceding Lemma A.39, it is enough to prove that the weak-* closed convex hull C of D is S(A). Since A has an identity, C C S(A). If C ^ S(A), then by Corollary A.40 there exist r G S(A), a G R, and a G A such that Rer(a) < a < Rep(a) for all p G C. Replacing a by ao = (a + a*)/2 and using that all positive functionals satisfy /(a*) = /(a), we have r(ao) < OL < p(ao) for all p G C But if p(ao) > a: for all /? G -D C C, then the assumption (A.12) implies ao — CL\A > 0, which in turn forces r(ao) > a, a contradiction. Therefore C = S(A), as required. D Let 7r : A —> B{T~L^) be a representation of a C*-algebra A. We say that a state r G 5(A) is associated to n if r(a) = {n{a)h \ h) for some /i G H^. Notice that if n and p are equivalent, then a state is associated to n if and only if it is associated to p. If 7r is irreducible, then a state r is associated to ix iff n is equivalent to 7rT (this follows from Proposition A.6 and Corollary A.8). More generally, if S is a subset of A, then the set of states associated to elements of S is precisely the set A~1(S) of states r such that nr G 5. The following result relating the hull-kernel closure of a subset F of A to the weak-* closure of A-1(F) is the key to the proof of Theorem A.38. Proposition A.42. Let n be an irreducible representation of a C*-algebra A, and let F be a subset of A. Then the following statements are equivalent: (a) 7r belongs to the closure F of F in A; (b) at least one state associated to n is in the weak-* closure of A~1(F); (c) every state associated to n is in the weak-* closure of A_1(F). Proof. Let \i — (Bp^Fp be the direct sum of a family of representatives for the classes in F. Let p(A)~ = p{A) + CIn be the C*-algebra generated by the image of p and the identity operator 1 (so that p(A)~ = p(A) if A has an identity). _1 Then we can view A (F) as the subset of S(/JL(A)~) consisting of the vector states associated to vectors in one of the direct summands Hp of H^. Thus if p(a) + Xlj-c is a self-adjoint element of p{A)~ and if r(p(a) + Al^) > 0 for all r G A-1(F), then p(a) + XlHp > 0 for all p G F, and p(a) + AlW/x = 0(p(a) + XlHp) > 0. Therefore we can apply Lemma A.41 with D = A_1(F), and deduce that every pure state on p>(A)~ is in the weak-* closure of A_1(F). A.4 The State Space of a C*-Algebra 229 Now suppose that n G F. Then ker-zr D (| ker p = ker /z, (A. 13) peF and any state r associated to 7r may be viewed as a pure state on 11(A). (It is pure on 11(A) because it is a vector state associated to an irreducible representation of A/ker /i = 11(A).) Since r extends to a state of fi(A)~ by Lemma A.7, and since the extension of a pure state is easily seen to be pure, it follows from the previous paragraph that (a) implies (c). Since (c) trivially implies (b), it will suffice to show that (b) implies (a). This amounts to proving that (A. 13) holds. By assumption there is a pure state r on A of the form r(a) = (n(a)h \ h) which belongs to the weak-* closure of A_1(F). Now suppose that a G A belongs to the right-hand side of (A. 13), and b G A is arbitrary. Then b*a*ab G p|kerp, so every 0 in A_1(F) vanishes on b*a*ab, and r must too. Since r(b*a*ab) = ||7r(a)7r(6)/i||2 and 7r(A)h = H^, we conclude that n(a) = 0. Thus (A. 13) is valid, and TTGF. • Proof of Theorem A.38. It follows from Lemma A. 12 that A is surjective. To see that A is continuous, suppose F is closed in A and r G A_1(F). Then by the (b) =4> (a) implication in the Proposition, we must have TTT G F = F, so r G A_1(F). Thus A_1(F) is closed. To see that A is open, suppose that U is open in P(A) and n G A(/7); say n = nT for r G U. Since r is not in the closure of P(A) \ U, it is not in the closure of the smaller set A-1 (^4 \ A(U)). But now the (a) => (c) implication in Proposition A.42 says that TTT is not in the closure of A \ A(C7), so A(U) is open. • Hooptedoodle A.43. With Theorem A.38 in hand, we have two rather different definitions of the topology on the spectrum of a C*-algebra. There is yet another. In essence, the idea is to view A as a quotient of the set Irr A of irreducible representations of A in a fixed Hilbert space of sufficiently large dimension; Irr A has a natural topology, in which it is a complete separable metric space when A is separable. Unfortunately, complications arise because C*-algebras can have irreducible representations of different dimensions. This "third definition of the topology on the spectrum" is discussed in [46]. A topological space P is separable* if it has a countable dense subset. A met ric d on P is compatible with the topology if d induces that topology on P, and a topological space which admits a compatible metric is called metrizable. The Urysohn Metrization Theorem says that every normal second countable space, and hence every second countable locally compact Hausdorff space, is metrizable [125, 1.6.14 and 1.7.9]. A separable topological space is called a Polish space if there exists a compatible complete metric. For example, (0,1) is a Polish space in its usual topology; the usual metric d(x,y) = \x — y\ is not complete, but there is a *Some authors call a locally compact group separable when its topology is second countable, i.e., has a countable base for the topology. This is potentially confusing: every second countable space is certainly separable, but there are locally compact groups which are separable as topological spaces but are not second countable (for example, the dual of the group M^ of additive reals with the discrete topology). We only use separable to mean the existence of a countable dense set. 230 The Spectrum homeomorphism of (0,1) onto R, so we can pull back the complete metric from R. Since closed subsets of complete spaces are complete, every closed subset of a Polish space is Polish. We can do rather better than this; recall that the countable intersection of open sets is called a Gs-set. Lemma A.44. Every Gs -subset of a Polish space P is Polish. Outline of the proof (For the details, see [2, §3].) We first claim that every open subset U of P is Polish. To see this, let d be a complete metric on P and define d(x, P\U):= inf { d(x, y) :y G P\U}. Then f(x) = l/d(x, P \ U) is a continuous function on U and U is homeomorphic to the graph r(/) of / in P x R. Since r(/) is closed in the Polish space P x R, it follows that U is Polish. Now suppose A is a G^-set in P. There are open sets Un C P such that A = Qn Un, and by the previous paragraph there are complete metric spaces Pn and homeomorphisms 4>n : Pn —> Un. Let B = { (Pn) G Y[Pn : MPi) = MPk) for k = 1,2,... }. n Again, it is not hard to see that B is a closed subset of the complete metric space f|nPn, and is therefore itself a complete metric space. It is not difficult to check : that the map (ft : B —> A defined by Pedersen [124] calls Q{A) the quasi-state space of A. Notice that Q(A) is a weak-* closed subset of A\, and is therefore Polish. Proposition A.46. If A is a separable C* -algebra, then P(A) is a Polish space. Proof. Let d be a complete metric on Q(A), and note that Q(A) is convex. Thus Fn := {0} U {(<£+V)/2 : Suppose that {pk} cFn\{0} and pk —> p. For each k there are 4>k,ipk £ Q{A) such that pk = |(0fc + ^k) and d( 1 i 2 = ||^(6+ >/£l)~ a II = c||(6+V^l)"1a(6+V^ir1||. (A.14) 2 On the other hand, 0 < a < ae 4- el = b + el, and since d < f implies that cdc* < cfc*, we have 0 < (b + >/il)-1a(6 + v^l)"1 < (6 + V^l)_1(^ + el)(6 + Ve'l)'1 = (b2 + el)(62 + 2y/~eb + el)"1 < 1. (A'15) Now (A.14) and (A.15) imply that ||6(&+ y/el)~la^ — a^ || < e. Since e is arbitrary and b G J, this implies that a1/2 G J, which in turn implies that a € J. • Lemma A.53. Suppose that J is a closed right ideal in a C*-algebra A. Then there is a net {e\} of positive elements in J such that a = lim^ e\a for all a G J. In particular, J is the closed right ideal generated by the positive elements in J. Proof. Since B — JD J* is a C*-subalgebra, there is an approximate identity { e\ } for B. If a G J, then aa* G B and 0 = lim^l — e\)aa* (in A1 if necessary). Since each e\ is a positive element of norm one, ||1 — e^H < 1, and 2 lim \\a - exa\\ = lim ||(1 - eA)aa*(l - eA)|| < lim ||aa*(l - eA)|| = 0. A.4 The State Space of a C*-Algebra 233 The last statement follows from the existence of the e\. • Proposition A.54. Suppose that A is a C*-algebra and that J and K are closed right ideals in A with J C K. If every state on A which vanishes on J also vanishes on K, then J = K. Proof. Since a right ideal in A is still a right ideal in A1, we may assume that A has an identity. We will show that every positive element of K is in J; this will suffice in view of Lemma A.53. Fix e > 0 and a e K such that a > 0. The set Se of states p such that p(a) > e is a (possibly empty) weak-* compact set. If p G *Se, then p is not identically zero on K, and by assumption cannot vanish identically on J. Therefore there exist ap G J and a weak-* neighbourhood Up of p in Se such that r(ap) ^ 0 for all r G Up. By compactness, there are elements ai,..., an G J such that the open sets Uai, • • •, Uan cover Se. Since by Cauchy-Schwarz we have 2 0 < \p(ai)\ < p(a*a,i) for all p€Uai, a r it follows that p{a\a\ H (- «n n) > 0 f° all p G 5e. Since 5e is compact, we can multiply the ai by a sufficiently large scalar and assume that p{a\a\ H ha£an) > p(a) for all p G S€. li p ^ Se, then p(el — a) > 0, so we can deduce that p{a\ai -\ h a*nan -f el - a) > 0 for all states p in 5(^4). It follows (see Remark 2.6) that a < a\a\ -{ h a^an + el, and Lemma A.52 implies that a G J. • Hooptedoodle A. 55. In general a C*-subalgebra B C Ais called hereditary if 0 < a < b in A and 6 G 5 imply that a € B. Hereditary subalgebras of A are in one-to-one correspondence with closed right ideals: the subalgebra corresponding to a right ideal J is J D J* [111, Theorem 3.2.1]. Appendix B Tensor Products of C*-Algebras In Chapters 5 and 6 we use tensor products of C*-algebras, and in this Appendix we provide a self-contained treatment of the facts we have used. Since this is a technically complicated aspect of the general theory of C*-algebras, we begin by reviewing the basic ideas. We discussed tensor products of vector spaces at the start of Section 2.4. One can view the tensor product V 0 W of two vector spaces either as the vector space spanned by certain elementary tensors v®w subject to rules which make the map (v,w) i—> v 0 w bilinear, or as an abstract vector space on which linear maps are given by bilinear maps on V x W. When 7i and K are Hilbert spaces, there is a natural inner product on the vector-space tensor product H0K, and hence a natural way to complete H 0 /C to get a Hilbert-space tensor product H 0 /C (see Lemma 2.59). In this case, though, it was definitely easier to use the construction of H 0 K rather than a universal characterization of H 0 /C in terms of a class of bounded bilinear maps on H x K (see Remark 2.63). If A and B are C*-algebras, there is an obvious way to make the vector-space tensor product A 0 B into a *-algebra (see Lemma B.l), but there are several different ways of norming AoB and completing it to give a C*-algebra. In practical terms, we have to choose whether we want a tensor product which is well-suited to concrete C*-algebras acting on Hilbert spaces, or one which has good universal properties and is better-suited to abstract C*-algebras. This is unfortunate: C*- algebras are a powerful tool largely because both concrete and abstract methods are available. On the other hand, there is a large class of nuclear C*-algebras for which all C*-algebra tensor products coincide, and one can often work entirely within this class. Thus, for example, the continuous-trace algebras studied in this book are nuclear, and almost anything we can do to them will give us other nuclear algebras (see Section B.5). To define the spatial tensor product, we represent A and B faithfully on Hilbert spaces H and /C, and show how to represent A 0 B faithfully on H 0 /C; we can 235 236 Tensor Products then pull the norm from B{TL 0 /C) back to AQ B, and complete to get a C*- algebra A®a B. The hard bit is to prove that the norm does not depend on the choice of representations (Theorem B.9). We discuss the spatial tensor product in Section B.l, and give some important examples in Section B.2. In many applications, such as those in Chapters 5 and 6, only the spatial tensor product appears. However, many of the arguments used to establish properties of A In Section B.4, we prove the main theorem about A®a B needed for our work on the Brauer group, which says that for C*-algebras A and B with Hausdorff spectrum, there is a natural homeomorphism of A x B onto the spectrum of A B.l The Spatial Tensor Product We begin by showing that A 0 B is a *-algebra. Lemma B.l. If A and B are *-algebras, then there is a unique *-algebra structure on the vector-space tensor product Ao B such that (a0 6)(c0 6) = ac0 6d and (a 0 6)*= a* 0 6* for all a,c £ A and 6, d £ B. Proof Fix a £ A and 6 £ B. Then (c, d) i—> ac®bd is bilinear from AxB into A05, and determines a linear operator La 0 Lb on A 0 B. The map (a, 6) \—> La 0 Lt> is bilinear from A 0 B into the vector space L(A 0 B) of linear operators on A 0 B, and hence induces a linear map M : A($ B —> L(AQ B). The required product of 5, t £ A 0 B is given by st := M(s)(t). Let (A0 B)~ be the conjugate of the vector space AQB. Then (a, 6) i-> a* 06* is a bilinear map from A x B to (A 0 B)~. Thus there is a linear map from A 0 B to (A 0 B)~ with the right property. The product and adjoint are uniquely determined by their values on elementary tensors because the elementary tensors span AO B. • B.l The Spatial Tensor Product 237 A natural strategy for endowing A 0 B with a C*-norm is to embed it as a *-subalgebra of some B(H): the norm of an element in A 0 B will then be the operator norm of the associated bounded operator. The key lemma shows how to construct an operator S 0 T on a tensor product H 0 /C from an operator S on H and an operator T on /C; we use the notation S 0 T to distinguish this operator from the element S0T of the tensor product B{H) 0B(/C), though we will shortly show that this distinction is unnecessary. Lemma B.2. Suppose that H and /C are Hilbert spaces, S G B(H), and T e B(IC). Then there is a unique bounded operator S 0 T onH^K such that S ®T(h®k) = Sh®Tk for heH and k e JC, (B.l) and\\S®T\\ = \\S\\\\T\\. Proof The map (ft, k) t-» Sh 0 Tk is bilinear, so there is a unique linear operator S 0 T on H 0 /C satisfying (B.l); the issue is to show that S 0 T is bounded with norm ||5|| ||T||, and hence extends to an operator on H 0 JC with the same properties. We first assume that S and T are unitary, and let £ G H O JC. We can write C = J^ILi ^ ® &i with the vectors fc; mutually orthogonal. Since T is unitary, the vectors Tki are also mutually orthogonal, and hence so are Shi 0 Tki. Thus 2 Shi Tkl 2 = Shi T 2 ||S 0 T(C)|| = \Z^-1 ® \\ XT_! W ® ^H n 2 T 2 n 2 2 = E Ji^ii ii ^ii = E 1ii^ii ii^ii = IICII2, so ||5 0 T|| = 1. Since B(H) and B(K) are spanned by unitaries*, it follows that S 0 T is a bounded operator for all 5 and T. The maps S i—» 5 0 1/c and T i—> 1^ 0 T are injective homomorphisms of B(H) and -B(/C) into £?(W0/C). Since injective homomorphisms between C*-algebras are isometric, we have ||S <8> lx:|| = ||<5|| and ||1^ 0 T|| = ||T||. Furthermore it is easily checked that (5 0 ljc)0-n ® T) = 5 0 T, and therefore ||S ®T|| = ||(5 0 l^ln 0T)||<||S||||r||. On the other hand, we can approximate ||S|| by ||5ft|| and ||T|| by ||Tfc|| for unit vectors ft and k; then ft 0 k is also a unit vector, and it follows from (B.l) that ||5®r||>||5||||T||. D Lemma B.3. Suppose 21 C B(H) and 03 C B(1C) are concrete C*-algebras (i.e., nondegenerate C*-subalgebras). Then there is an injective *-homomorphism i of $1 0 95 into B(H 0 /C) such that t(S 0 T) = S 0 T. *It is enough to show that if T 6 B(7i) and if 0 < T < 1, then T is a linear combination of unitaries. But U = T + iv7! - T2 is a unitary and 2T = U + U*. 238 Tensor Products Proof. Since (5, T) i—• S 0 T is a bilinear map into B(W0/C), there is a well-defined linear map *, of 21 © 95 into B(H<8> 1C), which takes 5 0 T to 5 0 T and is easily checked to be a *-homomorphism for the *-algebra structure of Lemma B.l. So it remains to prove that t is injective. To see this, suppose t G 21© 03 satisfies t(t) = 0, and write t = Y^j=i Sj ® Tj with { Ti,..., Tn } linearly independent. Fix h G H and choose an orthonormal basis { ei,..., em } for the span of { S\h,..., Snh } in 7Y. Then there are complex scalars \j such that Sjh = J^ Xijei for each 1 < j < n, and for any A; G /C we have En y—^n x—^ra Since the vectors e^ are independent, this implies ^2- \jTj = 0 for each 1 < i < m. Now the independence of the Tj implies that \j = 0 for all z, j, and we have Sjh = 0 for all j. Since h was arbitrary, this implies Sj = 0 for all j, and t = 23 7 ^ ® ^j — 0- Thus L is injective. D This Lemma allows us to identify 21 © 95 with a subalgebra of B(H 0 /C), and we will no longer distinguish between S 0 T in 210 95 and 5 0 T in J3(7Y 0 /C). We write 210C 95 for the closure of 21 0 95 in B (H 0 /C). Remark B.4. At first glance, it would seem that we have solved the problem of finding a tensor product norm for abstract C*-algebras A and B. There are nondegenerate faithful representations TT : A —* B(H7r) and 77 : J5 —> B{Hri)1 and then 7r(A) and r](B) are concrete C*-algebras isomorphic to A and 5, respectively. Since (a, 6) >—• 7r(a) 0 77(6) is bilinear, there is a unique linear map TT 0 77 : A 0 J3 —> B(H7r If 21 C B(H) is a concrete C*-algebra, a vector state on 21 is a state p G 5(21) of the form p : S 1—> (Sh | h) for some unit vector h G H. Although 21 will have lots of states which aren't vector states, for many purposes it suffices to consider vector states: Proposition B.5. 7/21 C B(H) is a nondegenerate C*-subalgebra, then every state p on$i is in the weak-* closed convex hull of the vector states of$l. The key lemma says that the norm of a self-adjoint element of 21 is determined by its values on vector states. This lemma will also be used in Section B.5. B.l The Spatial Tensor Product 239 Lemma B.6. Suppose that a is a self-adjoint element of a C*-algebra A. Then \\a\\=sup{\p(a)\:p€S(A)}. Moreover, ifCc S(A) and D is the weak-* closed convex hull of C, then S(A) C D if and only if ||a||=sup{|p(o)|:pGC} for every self-adjoint element a of A. Proof. The first assertion is part of Lemma A. 10. For the second, let s(a) = sup{ \p(a)\ : p G C}; we want to show that s(a) = \\a\\ if and only if S(A) C D. However, s(a) < sup{ \p{a)\ : p G D} = sup{ \p(a)\ : p G conv(C) } = sup | r>2,XiPi(a)\ : pi e C, Xi> 0, and y^ Xj = 1 j < sup < 22Xis(a) : Xi > 0 and V^_ A^ = 1 > = s(a). Therefore if S(A) C D, then s(a) > sup{ |p(a)| : p G 5(A) } = ||a||. To prove the other implication, suppose that po G S(A) \ D. Then there is a convex weak-* open neighbourhood U of po disjoint from D. By Corollary A.40, there are a G K. and a £ A such that Rep(a) < a < Repo(a) for all p € D. Now replace a by ao = (a + a*)/2. Since every positive linear functional satisfies p(a*) = p(a), we have p(ao) < a < po(ao) for all p £ D. Thus s(ao) < a < po(^o) < Ikoll- • Proof of Proposition B.5. Let C be the collection of vector states of 21. If T is in B(7i), define the numerical range of T to be nr(T) = sup{ \(Th | h)\ : h e H, \\h\\ = 1}. If T is positive, nr(T) = ||T|| (since T = S*S for some S € B{H)). If T = T*, then it follows from the functional calculus that either T + \\T\\l or — T + \\T\\1 is positive with norm 2||T||; suppose T + \\T\\1 > 0. Then 2||T||=sup{((T+117111^1^:^11 = 1} = sup{ {Th | h): \\h\\ = l} +\\T\\. It follows that for all self-adjoint operators T, ||T||=nr(T)=sup{p(T):TeC}. The result now follows from Lemma B.6. • Proposition B.7. Suppose that Qi C B(H) and *B C B(JC) are concrete C*- algebras, that p G 5(21), and that r G 5(03). Then there is a unique state p (g) r on 21 0C 05 such that p®r(S®T)= p(S)r(T) for all S G 21 and T G 03. (B.2) A state of this form is called a product state on 2C (g>c 03. 240 Tensor Products Proof. Since (B.2) determines /?0r on 210 03, uniqueness is clear. We claim that it will suffice to produce a positive functional / such that f(S 0 T) = p(S)r(T) and ll/H < 1. To see this, just choose 5 and T such that ||5|| = 1 = ||T||, p(S) ~ 1 and T(T) ~ 1. Then \\S 0 T\\ = 1 and f(S 0 T) = p(S)r(T) ~ 1, so / has norm 1 and is a state. Suppose first that both p and r are vector states, so there are unit vectors ft G H and k G K such that p(S) = (5ft | ft) and r(T) = (Tk | fc). The unit vector ft 0 fc determines a vector state u \T \-+ (T(ft0fc) \ ft0fc) on i?(W f(S 0 T) = \impx 0 rx(S 0 T) = Urnpx(S)rx(T) = p(S)r(T), this / will suffice. • Suppose 21 C B(7i) is a concrete C*-algebra and that ft is a unit vector in H. We will write [21ft] for the projection onto the subspace 2ift; we call [2lft] a cyclic projec tion and ft a cyclic vector for [2tft]. Note that since 2lft is invariant for 21, any cyclic projection P is in the commutant 21' := { T £ B(H) :TS = ST for all 5 G 21}. By a Zorn's lemma argument using the nondegeneracy of 21, there is a family{ Pi }iei of orthogonal cyclic projections such that ^/i Pi = lj-c (with convergence in the strong operator topology); we call it a maximal family of cyclic projections for 21. Since ||ft||2 = ^- ||Pjft||2 for all ft G H, and since each Pi belongs to the commutant 21/, we have ||T|| = sup \\TPi\\ for all T G 21. (B.3) Proposition B.8. Suppose that 21 C B(H) and 03 C B(K,) are concrete C*- algebras and that T G 210 03. Then as an operator onTC® JC, we have 2 \\T\\ = sup{ p 0 r(S*T*TS)/p 0 T(S*S) : p G 5(21), r G 5(03) and S G 21 0 03 satisfy p 0 T(S*S) ^ 0 }. (B.4) Proof If (j) G 5(21 0C 03) and 5 G 21 0C 03 satisfies 0(5*5) ^ 0, then the equation p^iT) := (j)(S*TS)/(l)(S*S) defines a positive linear functional. It is a state since liuix P(f)(U\) = 1 for any approximate identity { U\ } for 21 0C 03 [111, Theorems B.l The Spatial Tensor Product 241 3.3.1 and 3.3.3] + . Therefore the right-hand side of (B.8) is dominated by \\T*T\\ = imi2. Now let {Pi} be any maximal family of cyclic projections for 2l0c*B with (unit) cyclic vectors z{ for each Pim If T G 210 93 (or 21 0C OS), then ||T||2=sup{||TP,f :*G/} = sup{ ||T5^||2 : i e J, 5 G 210 03, ||5^|| < 1} = sup{ ||T5^||2/||5^||2 : i € J, 5 G 210 03, 5* + 0} = sup{^(5*T*T5)/u;,(5*5) : z G /, 5 e 2(0 03, 6^(5*5) ^ 0}, where u;^ is the vector state ^4 «—> [Azi \ Zj). Thus it is enough to show that we can choose the Pi and Zi so that each uji is a product state. It is certainly enough to arrange that each Zi is an elementary tensor. m an Choose unit vectors { h\ }AGA ^ d { ^ }MeM in /C such that 1^ = J2\l^h>] ut and 1^ = X^P^/J- P N = A x M, z^x^) = h\<^k^ and for each is e N, let P^ = [(21 0C 03)^]. Then { P^ }^eiv is an orthogonal family of cyclic projections, and the range of the projections ^2U Pv contains all elementary tensors h 0 k with = h G |JA 21/IA and /c G (J 03fcM. Since these tensors span a dense subspace, J2U ^ ^H®)C an(i we are done. D Now suppose that A and B are (abstract) C*-algebras. If p G S(A) and r G 5(5), then (a, 6) i—• p(a)r(b) is bilinear, and the universal property of tensor products gives a well-defined linear functional pQr on Ad)B satisfying pQr(a Theorem B.9. Suppose that A and B are C*-algebras. (a) IfteA®B,pe S(A), andr G S(B), then pQr(t*t) > 0. (b) For each t G A® B, define \\t\\l := sup{ p © r{s*t*ts)/p © T(S*S) : p G 5(A), r G 5(B) and 5 G A © P sate/?/ p © r(s*s) ^ 0 }. TTien || • ||a is a C*-norm on A® B satisfying \\a 0 b\\a = \\a\\ \\b\\. (c) If n : A -+ B(Hn) and r\ : B —> B{HT1) are nondegenerate representations, then there is a unique *-homomorphism n 0 rj : A © B —> B(7i^ 0 W^) «swc/i £/m£ 7r 0 7?(a 0 6)= 7r(a) 0 77(6). Moreover, for every t € AQ B we have Ik(8)77WII < 11*11*. t Alternatively, the Cauchy-Schwarz inequality for positive linear functionals says that if u G 1 2 1 5(A) and ifa.be A, then |u/(a*6)| < 6j(a*a) / ^(6*6) /2! Thus 0(5*T5) < (j)(S*S)1/2cj)(S*T*TS)1/2 < 11X110(5*5), where the last inequality follows from S*T*TS < ||T||25*5. Therefore \\ (d) If 7r and rj are faithful representations, then equality holds in (c). That is, it U || || h. I > ai®bi\\ = \\y n(ai) ® r](bi)\ for all di G A and bi G B. (e) If p G S(A) andr G S(B), then p Or is a-continuous; that is, \p(-)r(t)\ < \\t\\a for allt G AQ B. Proof. Suppose that ix and 77 are faithful nondegenerate representations. Since (a, b) 1—> 7r(a) 0 77(6) is bilinear, there is a linear map n 0 rj : A 0 B —> B{Hn 0 W^) characterized by 7r 0 ry(a 0 6)= 7r(a) 0 77(6), which is a *-homomorphism and is faithful (Remark B.4). The states of A are exactly those of the form p o IT for states p of 7r(A), and those of B have the form r o 77 for states r of rj(B). Notice that (p o 7r) 0 (r o 77) (£) = p 0 r(7r 0 ?7(t)). Since 7r 0 77 is a *-homomorphism, 7r 0 rj(t*t) is a positive element of n(A) 0C 77(B), and part (a) follows from Propo sition B.7. Proposition B.8 implies that \\t\\a is the operator norm of n 0 rj(t); (d) follows immediately, and the equality at the end of (b) follows from Lemma B.2. Part (e) follows because Proposition B.7 implies that product states are bounded with respect to the operator norm. On the other hand, even if n and 77 are not faithful, pon and r 077 are still states whenever p G S(TT(A)) and r G S(rj(B)), though not every state of A and B need arise this way. Thus (c) follows from (b). • Definition B.10. If A and B are C*-algebras, then the norm || • ||a defined in Theorem B.9 is called the spatial norm on AQB. The completion of AQB in || • \\a is denoted by A Corollary B.12. If A and B are C*-algebras, r G S(A), and p G S(B), then pQr extends to a state p 0^ r on A ®CT B. Proof This follows immediately from part (a) of the Theorem. • Proposition B.13. Suppose that : A —> C and ip : B —» D are homomorphisms between C*-algebras. Then there is a unique homomorphism Proof. As usual, the universal properties of the algebraic tensor product imply that there is a unique *-homomorphism 0 0 ip on A 0 B satisfying 0 0 ip(a 0 b) = 0(a) 0 ip(b). Suppose that TT : C —> B(J~t^) and 77 : D —> Biji,^) are faithful nondegenerate representations. Then ix 0 77 is isometric with respect to the spatial norm onC0D, and (TT077) o (0(g)7/;) = (7rO0) 0 (77o-0) is a representation of C 0 D which is bounded for the spatial norm (Corollary B.ll). It follows that 0 0 -0 is bounded with respect to the spatial norm oniQB and extends to a homomor- phism of A 0^ B, as asserted. If 0 and -0 are faithful, then 7r O 0 and 77 o ij; are faithful representations, so the final assertion follows also from Corollary B.ll. • Corollary B.14. Suppose B and C are C*-algebras, and A is a C* -subalgebra of B. Then the inclusion i of A in B extends to an embedding i 0 id of A 0CT C into B®aC. Proof We just apply the Proposition to i : A —+ B and the identity homomorphism id : C —> C; since both are injective, so is i 0 id. • In Section 6.1, we use the spatial tensor product to define an abelian group operation on a set of C*-algebras, and we will have to prove that this is commutative and associative. Proposition B.15. Suppose that A, B and C are C*-algebras. (a) There is an isomorphism 7 ofA®aB onto B (b) There is an isomorphism a of {A 0^ B) 0a C onto A 0^ (B 0a C) such that a((a 0 b) 0 c) = a 0 (b 0 c). The key is to realize that there are similar isomorphisms at the vector space and Hilbert space levels. So first let V, W, and Z be vector spaces. The map (v, w) 1—> w 0 v is bilinear, and hence there is a linear map /y:VoW^>WoV such that 7(7; 0 w) = w 0 v; doing the same thing to (w, v) \—> v 0 w gives an inverse for 7. In the same way, for fixed z, the map (v, w) ^—> v 0 (w 0 z) is bilinear, and thus there is a linear map Lz : V O W ^ V 0 (W 0 Z) such that Lz(v®w) = v0 (w0z). Now (£, z) »—» Lz(t) is bilinear on (V0 W) x Z, and there is a linear map a : (VQW)QZ —• F0(W0Z) such that a((v®w)®z)) = v®(w®z). The same process gives a linear map (3 :V Q (W 0 Z) —>> (V 0 W) 0 Z which is an inverse for a, so a is an isomorphism. If now W, /C and W are Hilbert spaces, it is easy to check that the isomorphisms of H 0 /C onto JCOH and (H0/C)0W onto W 0 (/C 0 W) preserve the inner products, and hence extend to unitary operators (7 : W 0 K -> /C 0 W and V : (W 0 K) 0 W -> H 0 (/C 0 W). To see that the vector-space isomorphisms 7 : AQB -+ BOA and a : (AoB)O C —> A©(#©C) extend to C*-algebra isomorphisms, choose faithful representations IT : A-> B{H1,), 77 : B -> £(7-^), and C : C -» B(WC). Then TT 0 77 and 77 0 TT are faithful representations of A0aB and #0^, by Corollary B.ll. If C7" : H^^H^ —> 7^ 0 Hn is the flip isomorphism on Hilbert spaces, then 77 0 TT(7(£)) = UTT 0 77(£)£T (B.5) 244 Tensor Products for all t G A 0 B\ to check this, write t = ^ ^ 0 ^ and apply both sides to an elementary tensor in H^ 0 H^. Since the right-hand side of (B.5) is an isomet ric *-homomorphism in t and rj 0 n is isometric, (B.5) implies both that 7 is a *-homomorphism and that 7 is isometric. Thus 7 extends to a C*-algebra iso morphism of A 0a B onto B 0a A, proving part (a) of Proposition B.15. We can similarly verify that 7T 0 (r? 0 C)(<*(t)) = V((TT 0 77) 0 C) (*)F* for alH G (i05)0C, and deduce that a extends to a C*-algebra isomorphism of the spatial tensor products. This completes the proof of Proposition B.15. B.2 Fundamental Examples There are some basic examples where the tensor product of two C*-algebras takes a familiar form, and these examples provide motivation and insight for the general theory. When one algebra is commutative, for example, we can identify the tensor product with an algebra of vector-valued functions. If T is a locally compact Haus- dorff space and A is a C*-algebra, then the set Co(T, A) of continuous functions / : T —> A such that t •—>• ||/(t)|| vanishes at infinity is a C*-algebra with pointwise operations and the sup-norm: (fg)(t):=f(t)g(t), f*(t):=f(ty, and ||/|| := sup ||/(t)|| (the completeness is established as in Example 1.7). Proposition B.16. Let T be a locally compact Hausdorff space and A a C*- algebra. Then there is an isomorphism 4> of Cb(T) 0a A onto Co(T,A) such that B(Hn), and let M : C0{T) -• B(£ (T)) be the usual (faithful) representation of Co(T) by multiplication operators: Mf(g)(t) = f(t)g(t) for / G Co(T) and g G £2{T). Corollary B.ll implies that M 0 n is a faithful representation of Co(T) 0^- A, and hence is isometric. There is also a 2 faithful representation R of C0(T, A) on £ (T, Hn) by multiplication: R(F)(£)(t) = Tr(F(t))£(t) for F G C0(T,A) and £ G ^(T,^). Let (7 be the isomomorphism of 2 2 £ (T)®Hn onto £ {T,n^) satisfying U(g®h)(t) = £(*)/i (see Example 2.61). Then a straightforward calculation shows that R{ Corollary B.17. If S and T are locally compact Hausdorff spaces, then there is an isomorphism ip ofCo(S)®aCo(T) onto Co(SxT) such that (j){f®g){s,t) = f{s)g(i) forfeC0(S),geC0(T). Proof In view of Proposition B.16, it suffices to prove that the map sending F e Co(S x T) to the function s i—• F(si •) is an isomorphism of Co(5 x T) onto Co (5, Co(T)). We like to think this is a routine exercise in general topology. • Our next examples involve the algebra Mn := Mn(C) of n x n-matrices with th entries in C. Let e^ G Mn be the matrix unit with 1 in the (i, j) slot and zeros elsewhere, so that { e^ }™.=i is a vector-space basis for Mn: for any (Xij) G Mn, we have (A^) = J^- A^e^. The multiplication on Mn is then determined by \eu ifj = k e^ = \0 if;/*, and the adjoint by e*- = e^. In the discussion preceding Lemma 2.65, we showed how the *-algebra Mn(A) of n x n-matrices with entries from a C*-algebra A can be made into a C*-algebra (in a necessarily unique way: there is only one complete C*-norm on any *-algebra). Every elementary tensor in Mn 0 A can be written as (A^) 0 a = J2 \j^%j ® a = J2eij ® \ja, so the map (a^) i-> X^e*j ® au *s a vector-space isomorphism of Mn(A) onto Mn 0 A; it is easy to check using the formula for multiplying matrix units that it is actually an isomorphism of *-algebras. Pulling the norm over from Mn(A) gives a C*-norm on Mn 0 A in which it is already complete. In fact this is the only C*-norm on Mn 0 A: Proposition B.18. If A is a C*-algebra, then Mn 0 A has a unique C*-norm. It is complete in this norm, so we can safely write Mn 0 A for Mn 0 A, and the map e (o>ij) •—• ^2 ij 0 aij is an isomorphism of the C*-algebra Mn(A) onto Mn 0 A. Proof. Let || • || denote the norm on Mn 0 A pulled over from Mn(A), and suppose || • ||7 is another C*-norm on A 0 Mn with corresponding completion Mn 07 A. Then the inclusion of (Mn 0 A, \\ • ||) in Mn 07 A is an injection of C*-algebras, and is therefore isometric — in other words, || • ||7 = || • || on Mn 0 A. The rest we proved above. • Example B.19. If m and n are positive integers, we obtain a *-isomorphism of Mm 0 Mn with Mmn by mapping an element of Mm 0 Mn to an m x m block matrix in M^Mn) and viewing this as an ran x ran-matrix. More formally, let { eij } be a set of matrix units for Mm, { fki } for Mn, and { grs } for Mmn. Then we define a linear isomorphism by sending the basis element e^ 0 fki for Mm 0 Mn to 9(i-i)m+k,(j-i)m+h an n m n mn Since Mn ^ B(C ), the previous example says that B(C )®B(C ) ^ B(C ). For a finite-dimensional Hilbert space we have B(TL) — ^C(H), so we can also view this as the finite-dimensional case of the next example. 246 Tensor Products Example B.20. If H\ and H2 are Hilbert spaces, then applying Corollary B.ll to the identity representations of K,(Hi) on Hi gives a faithful representation TT of ^C(Hi) 0a JC(H2) on Hi 0 H2 satisfying TT((/II 0*^)0 (h2 0 £^)) = (/ii 0 ft2) 0 (*i 0*2). Since the various rank-one operators span the various algebras of compacts, it follows that TT is an isomorphism of IC(Hi) 0O- K>(H2) onto JC(H\ 0 H2)' B.3 Other C*-Norms Since the spatial norm on A 0 B is defined using the tensor products of operators on Hilbert space, it is convenient when A and B have natural representations on Hilbert space. Unfortunately, the spatial tensor product A ®a B does not have a universal property like the one which characterizes the algebraic tensor product of vector spaces. In this section we introduce another norm, called the maximal norm (Proposition B.25), for which the associated C*-tensor product is universal for commuting representations of A and B (Theorem B.27). One should immediately wonder whether the spatial and maximal norms are really the same — and if not, how many other norms there are. We insist that all norms || • ||7 on A © B are submultiplicative, in the sense that ||£s||7 < ||£||7l|s||7 for s,t G AQ B; this will ensure that the completion A 07 B is a Banach algebra. To ensure that ^407 B is a C*-algebra, we also require that || • ||7 satisfies the C*-norm identity ||t*£||7 = ||t||7, and then we say || • ||7 is a C*-norm. These C*-algebra tensor products A®1B have some common features. The next proposition says that they are always generated by commuting copies of A and £, and this implies that their representations are always given by pairs of commuting representations of A and B (Corollary B.22). Prom this point of view, the maximal norm of Proposition B.25 is the most natural: the representations of the associated maximal tensor product are in one-to-one correspondence with pairs of commuting representations (Corollary B.28). Proposition B.21. Let A and B be C*-algebras and suppose that || • ||7 is a C*- norm on AQB. Then there are infective homomorphisms iA • M(A) —> M(A M(c)(a0 b) = ca® 6, i£(d)(a® 6) = a 0 db, and iA(a)iB(b) = iB{b)iA(a) = a 0 b foraeA,b^B,ce M(A), and d e M{B). Proof. Fix c G M(A). Since (a, b) 1—> ca 0 6 is bilinear, there is a unique linear map Lc : A 0 B —> A 07 B such that Lc(a 0 6) = ca 0 6. We first want to show that for ailteAQB, ll^(t)||7<||c||||t|| (B.6) B.3 Other C*-Norms 247 a Let t = Y^=i i®bi, and define X = (xij) G Mn(A) by x\j = dj and Xij = 0 for i > 1. Then since Mn(A) is a C*-subalgebra of Mn(M(A)), we have 2 2 (a*c^a^) = X*{c*c ln)X < \\c\\ X*X = \\c\\ (a*a3) th in Mn(i4). Thus there exists Y = {yi3) G Mn(A) such that the (i, j) entry of Y"*y is 2 5^fe=1 VkiVkj = <(l|c|| ~ c*^. th Let Z = (^) := (yi:j 0 bj) G Mn(A 07 S). Then the (i, j) entry of Z*Z is Since Z*Z > 0 in Mn(A07J5), the sum of its entries is positive in A Y?. _, 0 6 ||Lc(i)||2 = ||Lc(*)*Lc(t)||7 = |Ey "i *^ ® ^||7 This proves (B.6), and hence that Lc extends to all of A®1B. The similarly defined map Lc* is an adjoint for Lc as a linear map on the right Hilbert A 07 B-module (A 07 B)A®^B, SO LC G M(A 07 JB) := £((A 07 5)A Corollary B.22. Suppose A and B are C*-algebras, || • ||7 zs a C* -norm on A&B, and 7r : A 07 B —» 5(H) is a nondegenerate representation. (a) TT&ere are unique representations IXA'-A^ B(H) and TTB ' B —+ 2?(W) 5^cA tftat 7r(a 0 6) = 7ivi(a)7rj3(&) = ^(b^Ca) /or all a e A and b € B. 248 Tensor Products (b) Both 7TA and TTB are nondegenerate, and their canonical extensions TTA : M(A) —* B(H) and TTB ' M(B) —* B{TL) also have commuting ranges. (c) If it is the canonical extension of it to M(A®7 B), and if%A and is are as in Proposition B.21, then TTA = TT O iA and TTB = TT o iB; if n is faithful, so are IT A and TTB • (d) If { ei } and { Uj } are approximate identities for A and B, respectively, then in the strong operator topology fl"A(a) = lim7r(a ®Uj), and 7r#(6) = lim7r(ei (g) b). 3 i Proof If we define TTA = TT O iA and TTB = TT oiB, then TTA and TTB satisfy the first part of (c) by definition, are faithful if TT is because iA and is are injective, and satisfy the identity in (a) because iA and is do. We claim that any representations TT'A and -K'B satisfying the identity in (a) must be nondegenerate. Suppose h G H satisfies 7rA(a)h = 0 for all a G A; then 7rB(b)7r'A(a)h = n(a ®b)h = 0 for all a G A and b G B, and h = 0 because 7r is nondegenerate. A similar statement holds for -K'B, so the claim is proved. Since TTA and TTB are nondegenerate, 7rA(ei) and 7r^ (iXj) both converge to 1*^ in the strong operator topology, and part (d) and the uniqueness in (a) follow easily. It remains to prove (b), and we have already seen that TTA and TTB are nonde generate. Suppose that m G M(A) and n G M(B). By nondegeneracy, it is enough to compute on vectors of the form 7rA(a)7TB(b)h for a G A, b G -B, and then 7rA(m)7tB(n)(7rA(a)7rB(b)h) = ^A(ra)^£(n)(7rs(&)7r^(a)/i) = 7rA(m){7rB(nb)7rA(a)h) = 7TA(rn)(TTA{a)7TB(nb)h) = ^(^a) nB{nb)h = 7TB(n6)7r^(ma)/i = nB(n)(7rB(b)7rA(ma)h) = TtB(n)(7TA{ma)7TB{b)h) = 7fs(n)7fA(m)(7rA(a)7TB(^)^), so 7TA(^) commutes with 7f# (n). D Example B.23. If 7r is a tensor product representation p 0 77 of the spatial tensor product .A®a £, as in Corollary B.ll, then TTA is the representation p One attractive property of the spatial norm || • ||a is the equality \\a <8> b\\a = \\a\\ \\b\\ for all a G A and b G B\ (B.7) B.3 Other C*-Norms 249 we sum this up by saying that || • \\a is a cross-norm. We will later prove that every C*-norm on A 0 B is a cross-norm (Theorem B.38), but this is nontrivial, and for the main goals of this book it is enough to see that every C*-norm is subcross: \\a 0 6||7 < ||a|| ||6|| for all a e A and b € B (Corollary B.24). This condition is important because it implies that all such norms are jointly continuous in the A- and B-variables. In fact, we will prove that every C*-seminorm is subcross: a C*-seminorm is a seminorm which is submultiplicative and satisfies the C*-norm identity. Corollary B.24. Let A and B be C*-algebras. Every C*-seminorm \\'\\a onAoB is subcross: \\a 0 6||a < ||a|| ||6|| for all a G A and b € B. Proof. Let* || • H^- be the spatial norm on A® B. Then ||t||7 := max{ \\t\\cr, \\t\\a } defines a C*-norm on A 0 B. If || • ||7 is subcross, so is || • ||Q;. Let || • ||7 also denote the corresponding norm on M{A 07 B). Then, since the norm on any multiplier algebra is submultiplicative, ||o®6||7 = \\iA{a)iB{b)\\, < \\iA(a)h\\iB(b)h = IM| \\H, where the last equality holds because %A and %B are injective and therefore isometric. • Now that we know all C*-norms are subcross, we can define what is obviously the largest possible C*-norm on A 0 B. Proposition B.25. Let A and B be C*-algebras. Then Pllmax • = sup{ ||t||7 : || • ||7 is a C*-seminorm on Ao B} = sup{ ||t||7 : || • ||7 is a C*-norm on AQ B} is a C*-cross-norm majorizing all C*-seminorms on AQB; it is called the maximal norm on A 0 B. The completion of A 0 B in \\ • ||max is a C*-algebra called the maximal tensor product of A and B, and denoted by A 0max B. Proof. Suppose t = ^^cn 0 h and || • ||7 is a C*-seminorm on A 0 B. Because a || • ||7 is subcross, we have ||t||7 < J2i \\ i\\ \\°iII ^y Corollary B.24; thus the sup is well-defined and we have \niai <£-INHMI- The resulting function || • ||max is a seminorm, is submultiplicative, and satisfies the C*-identity because each || • ||7 has these properties. Since the spatial norm || • Her is a C*-norm, || • ||max is a C*-norm, and it is enough to sup over C*-norms; in particular, || • ||max majorizes all C*-norms. Finally, ||a||||6|| = ||a0 6|U<||a0 6||max<||a||||6||, so II • Umax is a cross-norm. D * There is a proof in [92] which does not use multipliers; see also [167, Corollary T.6.2 and Exercise T.J]. 250 Tensor Products Remark B.26. The notation || • ||max is standard, but it should be used with caution. For example, suppose that A0 and BQ are C*-subalgebras of A and £, respectively, and that || • ||7 is a C*-norm on A 0 B. Then the restriction of || • ||7 to the subalgebra AQ 0 BQ is a C*-norm, which one would instinctively continue to denote by || • ||7, and the completion AQ ®7 BQ in this norm is naturally a C*- subalgebra of A 07 B. But if ]| • ||7 is the maximal norm on A 0 JB, its restriction to AQ 0 Bo need not be the maximal norm for AQ 0 BQ; in particular, the inclusion c map Ao 0 B0 —> A 0 B is not generally bounded with respect to the respective maximal norms, and Ao 0max BQ is not naturally a subalgebra of A0max B. This is not a problem for the spatial tensor product, and we can safely identify Ao ®a BQ with a C*-subalgebra of A 0a B (see Corollary B.14); nor is it a problem for the maximal tensor product and ideals rather than subalgebras (see Proposition B.30). Theorem B.27. Suppose A and B are C*-algebras. Then there are nondegenerate homomorphisms %A : A —> M(A 0max B), %B : B —> M(A 0max B) such that (a) iA{o)iB{b) = iB(b)iA(a) = a 0 6 for a G A, b G B; (b) if (f> ®max 4>(iA{a>)iB(b)) = (c) A 0max B = span{ iA(o)^B(b) ' a e A and b G B}. If D is a C*-algebra and JA ' A —> M(D), js ' B —> M(D) are homomorphisms satisfying the analogues of (a), (b) and (c), then there is an isomorphism 0 of A 0max B onto D such that 9{a 0 6) = JA{p)JB{b)- Proof. The existence of the maps i^, %B was established in Proposition B.21. To see that they are nondegenerate, note that if {e^} is an approximate identity for A, then iA(ei)(a 0 6) = (e^a) 0 6 —• a 0 6 in A 0max B because || • ||max is subcross. If 0, ip are commuting representations, the universal property of the algebraic tensor product implies that there is a unique linear map n : A 0 B —> B(H) such that 7r(a 0 6) = 0(a)^(6), and 7r is a repre sentation because (p and ip have commuting ranges. Pulling back the norm from B(H) gives a <7*-seminorm \\t\\ := ||7r(t)|| on A 0 5, so ||7r(£)|| < ||t||max, and we deduce that ir extends to a representation ((TTOJ'A) ®max (TT O jB))(a 0 6) = 7f(jA(a))7f(jjB(6)) = 7r(jA(a)jB(b)) B.3 Other C*-Norms 251 (note that JA(O)JB{O) belongs to D by the analogue of (c)). So we can define _1 0 := 7T o ((jt o jA) 0max (7f o JB)), and 0 is a homomorphism of A 0max B into Z>. On the other hand, by putting A 0max B faithfully on Hilbert space and applying the analogue of (b) to the commuting pair {IA^B)^ we obtain a homomorphism p of D into A0max B such that p(JA(a)JB(b)) = iA{o)iB{b) — a (8) 6. Now (c) and its analogue imply that 0 is an isomorphism with inverse p. • Corollary B.28. Le£ A and B be C*-algebras. Then n \—> (n o iA,n o iB) is a bisection between the nondegenerate representations of A 0max B and commuting pairs of nondegenerate representations of A and B. In particular, we have II *—"'i Umax sup < \\2_]. 0(ai)^(^) : ^V7 are representations with commuting ranges? for every J2i cti^bi e AQ B. Proof We saw in Corollary B.22 that (it o iAln o iB) is always a commuting pair of nondegenerate representations. On the other hand, if (0, ^) is such a pair, the representation n := (j) 0max ^ is also nondegenerate, and satisfies (j) = n o iA, ip = 7toiB by part (b) of the Theorem. Since the norm in any C*-algebra is determined by the representations, and any pair of (possibly degenerate) representations gives a representation of A 0max B by Theorem B.27(b), the last part follows."'" • Corollary B.29. Suppose that A, B and C are C*-algebras. Then there is an isomorphism a of (A 0max B) 0max C onto A 0max (B 0max C) such that a((a 0 b) 0 c) = a 0 (b 0 c). Proof. The previous corollary implies that the maximal norms on (AQB)OC and A©(BoC) are both given by taking sups over commuting triples of representations of A, B and C, so the isomorphism of algebraic tensor products is isometric. • It is important in applications to know how tensor products of C* -algebras interact with ideals and quotients. We have already seen that if I is an ideal in A, then we can identify the spatial tensor product I®aD with a subalgebra of A 0a D (Corollary B.14), and because I © D is an ideal in A 0 D, the subalgebra I 0a D is actually an ideal. The quotient map q : A —> A/I induces a homomorphism q 0a id : A 0^ D —• (A/1) 0a D (Proposition B.13), which is surjective because its range contains the algebraic tensor product. In general, however, ker(g 0^ id) can strictly contain i" 0a D — (kerg) 0a D. We shall see that the maximal tensor product is better behaved in this respect, and then give a sufficient condition for the equality of ker(g 0a id) and / 0a D. tStrictly speaking, the pairs of commuting representations do not form a set, so the least upper bound on the right does not make sense. However, there is no real problem here: the sup is attained by the pair (it o iA, n o iB) corresponding to a faithful representation n of A 0max B. 252 Tensor Products We shall couch these results in the language of exact sequences. By saying that 0 • I —*—> A -J—> B • 0 (B.8) is a short exact sequence of C*-algebras, we usually mean that / is a (closed two- sided) ideal in a C*-algebra A, that i : I —> A is the inclusion map, and that q : A —> B is a surjective homomorphism with kerg = J, so that q induces an isomorphism of the quotient A/1 onto B; sometimes (as in the next Proposition, for example) i will be just an embedding of I onto an ideal in A, and the notation indicates that we can identify / with its image in A. Proposition B.30. For every short exact sequence of C*-algebras (B.8) and every C*-algebra D, there is a short exact sequence 0 • /0max^ • A(g)maxD • B iB o 0 : A -> M(£ ®max £>) and iDo^:B^ M(B (g)max £>). Thus by Theorem B.27 there is a homomorphism (i# o 0) (g)max (i^ o ip) of A 0max C into M(B 0max £)) such that ((%B o(/>) (g)max foo^))Mc) = i5(^(a))iD('0(c)) = (f)(a) ® ip(c); nas this equation implies both that 0 Proof of Proposition B.30. It is easy to see that the image of i®maxid is an ideal in A 0max D. To see that i 0max id is injective, we show that every representation of I ®max D factors through i (S)max id- So suppose TT is a nondegenerate representation of / ®max D. Then 717 : ft o iI and 7T£> := TT oiD are nondegenerate representations with commuting ranges. The extension ftj of 717 to A also has range commuting with that of no (see Corollary B.22(b)), so there is a representation fti 0max TTD of A ®max ^5 and we trivially have (7f/ ®max TTD) ° (* ®max id) = 7Tj 7TA(c)(7TD(d)h) = TT(C® d)h = Tr(i ®max id(c(g> d))h = 0 B.3 Other C*-Norms 253 for all h G 7^, and the nondegeneracy of 7T£> implies TTA(C) = 0. Thus there is a representation p of B such that TTA = p°Q- Since the ranges of TTA and TTO commute, and im7r^ = imp, so do the ranges of p and 7T£>, and there is a representation P ®max TTD of B P ®max 7TD(g(a) ® d) = p(q(d))lTD(d) = 7TA(a)7TD(d) = 7r(a ker(max id) C ker7r = im(z (g)max id), as required. • Proposition B.32. Suppose that (B.8) is a short exact sequence of C*-algebras, and that D is a C* -algebra for which the spatial and maximal norms coincide on BQD. Then 0 > l0aD ^L> A®aD -^ B®aD • 0 is an exact sequence of C*-algebras. Proof We saw in the preamble to Proposition B.30 that i 0 id is an isomorphism onto an ideal in A(g)aD, and g®id is surjective because its image contains q(A)®D = B®D. It remains to see that im(i Let i\) : A ®a D —> Q be the quotient map, and notice that (#(a), d) »—> ip(a<8>d) is a well-defined bilinear map which induces a homomorphism (f) : B O D —> Q satisfying (f>{q(a) 0 d) = ip(a® d). Then ||c||7 :=max{||0(c)||,||c||ff} is a C*-norm on B(J)D dominating the spatial norm; since it is certainly dominated by the maximal norm, our assumption implies that || • ||7 = || • ||a- Thus 0 is bounded with respect to the spatial norm and extends to a homomorphism : B®GD —> Q. Since B.4 C*-Algebras with HausdorfF Spectrum Recall that a C*-algebra A is CCR if n(A) — JC(Hn) for every irreducible repre sentation 7r (see page 220 and Remark 5.17). For CCR algebras with Hausdorff spectrum the general theory of tensor products simplifies, and this simplified ver sion suffices in the main body of the book. In this section, we will work primarily with the spatial norm || • ||a. Since CCR algebras are by definition associated with the algebra JC(H) of com pact operators, to study them we need to know about the representation theory of K(H). Suppose n : A —-> B(H) is a representation of a C*-algebra A and I is a set. Then there is a representation / • TT of A on the direct sum 0ie/ H such that / • 7r(a)(hi) = (jr(a)hi); we call / • IT a multiple of TT. The map h 0 (xi) »—> (xih) ex 2 tends to a unitary isomorphism U o£H®l (I) onto 0i6/ H which intertwines 7T01 with LTT; indeed, a straightforward calculation shows that (I-7r(a))U = f/(7r(a)0l). Our first lemma says that every representation of JC(H) is a multiple of the identity representation id : K(H) —> B(H). Lemma B.34. Suppose thatn : K,{H) —> B(Hn) is a nondegenerate representation of the C*-algebra of compact operators on some Hilbert space H. Then there is a Hilbert space Ho such that n is equivalent to id(8)1 on H Proof. Let e be a unit vector in H, and consider the projection 7r(e0e). The range of n(e 0 e) is a Hilbert space Ho; let {e^ : i £ 1} be an orthonormal basis for Ho- As in Example A.15, the formula Ui(h) := 7r(h 0 e)e^ defines an isometry Ui of H onto the invariant subspace H% := span{7r(T)ei : T £ £(«)}, and Ui intertwines the identity representation id and n]^. For 5, T £ fC(H), the operator (e0e)T*5(e0e) is the scalar multiple (Se | Te)(e®e) of e0e, and hence (7r(S)e, | 7r(T)ej) = (7r(S>(e 0 e)e; | 7r(T)7r(e 0 e)^) = (Se | Te)(ei | e,); thus the subspaces Hi of 7-^ are mutually orthogonal. They span TT{K{H))HO, and also Hn: by nondegeneracy, we can approximate any vector h by one of the form E n (hi 0 ki)h = V^ 7r(hi 0 e)7r(e 0 e)n(e 0 fci)/i, which belongs to TT()C(H))HO- We deduce that [/ := 0 ^ is a unitary isomorphism of 0iG/ W onto Hn = 0iG/ Wi which intertwines / • id and n = 0ie/ 7r|^^. Since 2 2 Ho = ^ (^)> the identification of H 0 ^ (/) with 0i€/ W completes the proof. D Proposition B.35. Suppose that A and B are C*-algebras and that A is CCR with Hausdorff spectrum. Let || • ||7 be a C* -norm on AQB. Then every irreducible representation TT of A 07 B is equivalent to a representation of the form p 0 77 for some p £ A and r\ £ B. Furthermore, \\c\\a > ||c||7 for every c £ AQB, and A®aB coincides with A 0max B. B.4 Hausdorff Spectrum 255 Proof Let TT be an irreducible representation of A 07 B on H-K, and let TTA and TTB be the commuting representations of A and I? such that n(a 0 6) = ^^(0)^^(6), as in Corollary B.22. By the Dauns-Hofmann Theorem, each / G Co(4) determines a multiplier in ZM(A) such that r(f • a) = f{r)r{a) for r G A and a e A. Since the extensions 7TA and 7fs have commuting ranges by Corollary B.22, and since / G ZM(A), the operator 7f^(/) commutes with every 7r(a06) = ^ (a) 71-5 (6), and hence belongs to Cl?^ = TT(A 07 5)'. Thus 7f^ determines a complex homomorphism of Co(A): there exists p G A such that 7TA(/) = f{p)^n^ for / G C0(-A). If p(a) = 0, then because A is Hausdorff we can factor a = / • b for some f E Co (A) satisfying /(p) = 0 (we can do this exactly by Corollary 5.11, or approximately by choosing / = 1 where ||a|| > e), and deduce that 7r(a) = f(p)ir(b) = 0. Thus TTA factors through the quotient A/kerp, which is isomorphic to JC(HP) because A is CCR. By Lemma B.34, we can assume that H^ — Hp 0 Ho and that TTA = p 0 1. If ft G Hp is a fixed unit vector, then Vh : k t-> h 0 k embeds Ho onto the closed subspace ((ft- 0 ft) 0 l)(Wp 0 Wo) of W^. Since (ft 0 ft) 0 1 belongs to IC(HP) 0 1 = TTA(A), each ^#(6) commutes with (ft 0 ft) 0 1. Thus the subspace Vh{Ho) is invariant for 7TB, and 77 : 6 1—> V^7r^(6)V^ is a representation of B on 7^o- Since the operator (ft 0 h\) 0 1 also commutes with everything in the range of TTB and satisfies ((ft 0 fti) 0 l)V^ll = Vh, r\ is independent of the choice of unit vector ft, and we have (1 0 v(b))(h 0 k) = ft 0 77(6)*; = V^ (»?(&)*;) = (7rB(6)14)/c - 7TB(b)(h 0 A;) for every k and every unit vector ft. Thus 7r#(6) = 10 77(6), and 7r(a 0 6)= 7rA(a)7rB(6) = (p(o) 0 1)(1 0 77(6)) = p(a) 0 77(6) = p 0 7?(a 0 6). If TT/(6) = rj(b)T for all 6 G £, then 1 0 T commutes with TT — p 0 77, and hence is scalar; thus 77 is irreducible, and the first part is proved. For the remaining statements, recall that if C is a C*-algebra and c G C, there is an irreducible representation TT of C such that ||c|| = ||7r(c)|| (Theorem A. 14). Thus if c G i075, there exist irreducible representations p G A and n G B such that ||c||7 = \\p 0 77(c)||. Since p 0 77 is norm-decreasing for the spatial norm (Theorem B.9), we deduce that ||c||7 < \\c\\o- for all c G AQ) B. Since this applies to every C*-norm, we deduce that ||c||max < ||c||a, and hence by maximality that IMImax = \\c\\ For the statement of the next lemma, recall that S' denotes the commutant of a subset S of B{H). Lemma B.36. Suppose that TT : A —> B(Hn) and rj : B —> B(HV) are nondegener- ate representations of C*-algebras. If IT is irreducible, then every T G TT^^A^^B)' has the form T = 1*^ 0 R for some R G rj(B)'. In particular, if both TT and 77 are irreducible, then TT 0 77 is an irreducible representation of A®a B. Proof For any T in jB(W7r 0 Hv) and k, I G Hv, (9, ft) ^ (T(# 0 fc) | ft 0 Z) is a bounded sesquilinear form on H^ so there is a bounded operator Sk,i on H^ such 256 Tensor Products that (SkAd) I h) = iT(9 ®k)\h®l) for all (ir(a)Skil(g) \ ft) = ((ir(a) 0 l) (T( = (T(7r(a)g®k) \h®l) = (Skil(n(a)g) | ft), and hence 5^,/ G 7r(^4)'. Since 7r is irreducible, so that n{A)' = Clft^, there is a bounded sesquilinear form B{-, •) on Hv such that Sk,i = B(k, tyl-H^i which in turn is given by a bounded operator R: B(k,l) = (Rk \ I). But then (T(g 0 /c) | ft 0 Z) = (g®Rk\h®l) for all g, ft G W^ and k,l EHV. Since T = lft^ 0 i? commutes with 1^ 0 77(6) for all 6G5, this implies that i? G rj(B)f. D Theorem B.37. //A and B are CCR C*-algebras with Hausdorff spectrum, then (7r, rj) i—> 7r 0 rj induces a homeomorphism of A x B onto (A 0a B) . Proof. If 7r = AdC/ o 7r; and rj = AdF o 77', then the unitary operator U 0 V implements a unitary equivalence between 7T077 and 7rr<8>r)'. Therefore (7r, 77) h-> 7T077 gives a well-defined map $ on equivalence classes, and Lemma B.36 implies that <3> has range in (A 0a B) . On the other hand, the uniqueness in Corollary B.22 implies that (TT 0 77)A = 7r 0 1^ and (-zr 0 77)5 = lft^ 0 77. If 7r 0 77 is equivalent to 7r' 0 77', then 7r 0 lft is equivalent to -K' 0 lft^ and ker7r = ker7r'; similarly, ker 77 = ker 7/. Since irreducible representations of algebras with Hausdorff spectrum are equivalent if and only if they have the same kernel, it follows that 3> is injective. At the level of states, we have a map \I> : (p, r) >-> p (g> r of 5(A) x 5(JB) into 5(A 0a 5). Since all these functionals have norm 1, a calculation on elementary tensors is enough to show that \I> is continuous for the weak-* topologies. Another straightforward calculation shows that, if np and 7rT are the GNS-representations with cyclic vectors hp and ftT, then p If p and r are pure, the representations np and 7rr are irreducible, and hence so is TTP 0 7rr. Thus (B.9) implies that 71-p 0 7rT is equivalent to the GNS-representation 7rp(3T (Corollary A.8); since np 0 TTT is irreducible, this implies both that p 0 r is pure (Lemma A. 12), so that \I> maps P(A) x P(#) into P(A 0^ I?), and that we have a commutative diagram P(A) x P(B) • P(A 0, S) AAXAB / (i4®(rS) B.5 Tensor Products of General C*-Algebras 257 in which A^, A# and A^®^, denote the GNS-maps. Recall that the maps Ac are continuous, open surjections for the weak-* topology on P(C) and the usual topology on C (Theorem A.38). Thus if U is open in (A 0a B) , then 1 1 $- (C7) = Aj4xAB(*- (A^B(f/))) is open, and $ is continuous. Proposition B.35 implies that <£ is surjective, and it remains only to see that the inverse 3>_1 is continuous. Suppose that $(7^, 77^) = 7r; 07?^ —> 3>(7r, rj) = 7T077 in (A 0a B) ; by symmetry, it will be enough to show that 7T; —» n in A Since the open neighbourhoods of 7r have the form Oj = {pe A: J^kerp} for some ideal J of .A such that n(J) 7^ 0 (Corollary A.28), it is enough to prove that eventually 7r^(J) ^ 0. Since J 0a B is an ideal in A 0a B (see the discussion preceding Proposition B.30), the set A U:={re {A 0a B) : r(J 0, B) ^ 0} is an open neighbourhood of IT 0 77. Thus 71^ 0 77^ G U for large i, and this is only possible if 7r^( J) ^ 0. This completes the proof. • B.5 Tensor Products of General C*-Algebras In the previous section, we identified the spectrum of the spatial tensor product A 0a B when A and B themselves have Hausdorff spectrum, and much of what we have proved holds more generally. It is not easy to dig detailed proofs of the resulting theorems out of the literature, so we shall provide them here. Since we have not used these results in the book and our goals are archival, we have settled for quoting some highly nontrivial but relatively well-archived material. Thus the pace in this section is a little faster. The first main theorem says that the spatial norm is minimal among all C*- norms on the tensor product. For this reason, the spatial tensor product is often called the minimal tensor product. Theorem B.38. Suppose that A and B are C*-algebras and that \\ • \\a is a C*- norm on AO B. Then \\ • ||a is a cross-norm which dominates the spatial norm: \\t\\a > \\t\\a forte A OB. Recall that if 0 G S(A) and ip G S(B), then there is a linear functional $ 0 ip on A 0 B which satisfies 0 i/>(t*t) > 0 for all t G A 0 B (Theorem B.9(a)). If || • ||a is a C*-norm on A 0 B, it is not a priori clear that 0 0 ip is a-continuous; that is, continuous for the norm || • ||a on A 0 B. If Proposition B.39. Suppose that A and B are C*-algebras and that || • ||a is a C*-norm on A 0 B. Then 4> 0 ip is a-continuous for every \\t\\a for all t G AG B. Proof The (<=) direction is provided by Corollary B.12. For the other direction, notice that if s, t G A &a B, then s*t*ts < \\t\\ls*s in A 0a B. (B.10) Thus if s, t G A 0 JB, we can apply (p 0 ip to (B.10) to conclude that „ ||2 > te^(s*t*ts) 11 !la~ S$ := {ip G S(B) : (f) 0 ij) is a-continuous }. For each , S% is a convex, weak-* compact subset of S(B), and consists of those I/J G S(B) such that 0 0 ip extends to a state 0a ip on ^4 0a JB. Lemma B.40. Suppose that A and B are C*-algebras with identities, and that || • ||a is a C*-norm on A® B. (a) Suppose that b is a self-adjoint A-nuclear element of B. Then for every pure state 4> of A, we have ||6||=sup{hK&)|:V€^}. (b) If every self-adjoint element b of B is A-nuclear, then (f> 0 ip is a-continuous for every 4> G S(A) and every ip G S(B). Proof Suppose that b G B is self-adjoint and .A-nuclear. Then C*(b, 1) is commu tative, and there is a complex homomorphism h on C*(b, 1) such that \h(b)\ — \\b\\. Let (/> G P(A). Then cj> 0 h is cr-continuous: the easy part of Proposition B.39 says that every product state is continuous with respect to the spatial norm. By assumption on 6, <\> 0 h is also a-continuous on A 0 (7* (6,1), and extends to a state <\> 0Q h on the closure A 0a C*(6,1) of A 0 C*(6,1) in A 0a B; this in turn B.5 Tensor Products of General C*-Algebras 259 extends to a state £ G S(A 0a B) by [111, Theorem 3.3.8]. Define Vi G S(B) by ^i(c) := £(1 0 c). We claim that f (a 0 c) = (a)i/>i(c) for all a G J4 and c G 5. (B.ll) (This assertion would be obvious if £ were multiplicative; since it need not be, we need to use that 0 is pure.) For 0 < c < 1, define c(a) := £(a (8> c). Then for any fixed c, (/>c and ||0c|| + Ul-C\\ = 0C(1) + 01_c(l) = 0(1) = 1, we must have (j)c = f (a 0 c) := 0c(a) = 0(a)0c(l) = 0(a)£(l 0 c) = 0(a)^i(c), justifying the claim. We deduce that (/>Gipi is the restriction of a state £ to A 0 i?, and hence that ^i G^. Since |Vi(6)| = im®6)l = l^(l)^)l = ll&ll, part (a) follows. Since S^ is convex and weak-* closed, and determines the norm on self-adjoint elements of B by part (a), Lemma B.6 implies that S% = S(B) for all 4> G -P(-A). For each ip G S(B), define 5^ := { 0 e S(A) : 0 0 ^ is a-continuous }. Again, S^ is convex and weak-* compact. Since the statement 5? = S(B) means 0 G 5^ for all V € 5(5), we have just proved that P(A) C S^ for every ^ G 5(B). Thus it follows from the Krein-Milman Theorem that S^ = S(A) for every ^, and this is what we wanted to prove. • To apply this lemma, we need to show that every self-adjoint element b G B is A-nuclear with respect to every C*-algebra A. Since C*(6,1) is commutative, it is enough to prove that AoC always has a unique C*-norm when C is commutative — or, in other words, that every commutative C*-algebra C is nuclear. Recall that the irreducible representations of a commutative C*-algebra C are all 1-dimensional, and the spectrum C is the space of non-zero complex homomorphisms on C with the weak-* topology; if C is already of the form CQ(T) for some locally compact space T, then t i—> et is a homeomorphism of T onto C. Since the only state associated to a 1-dimensional representation is the representation itself, t i—> et is also a homeomorphism of T onto P(A). Lemma B.41. Suppose that S and T are locally compact Haus dorjf spaces. Then there is a unique C*-norm on Co (5) 0 CQ(T). 260 Tensor Products Proof. Suppose that || • ||a is a C*-norm on Co(S) 0 Co(T). Corollary B.22 implies that every irreducible representation of Co(5) 0a CQ(T) has the form es 0 et; thus A (C0(S) Ga C0(T)) = £ := { (s, t) G S x T : es 0 et is a-continuous }, and for every b G Co(S) Ga CQ(T) we have ||6||a = sup{ |es 0a et(b)\ : (s,t) G £}. (B.12) We claim that E is a closed subset of S x T. Suppose that { (si,ti) } C E1 and that (si,ti) —> (s,£). We need to verify that es 0 e$ is a-continuous. But eSi —> es and e^. —> et in the weak-* topologies. Thus eSi 0 eti(v) —> es 0 et(v) for all v G Co(S) 0 Co(T), and by assumption K 0^W| It follows that es 0 et is a-continuous. This establishes the claim. If there is a point (so?^o) ^ S x T \E, then there is a neighbourhood U x V of (SOJ^O) which does not meet E. There are non-zero functions / G CQ(S) and g G C0(T) such that f(s) = 0 if 5 ^ [7 and #(£) = 0 if ^ V, and then ||/ 0 g\\a = sup{ |es 0 etU <8> ^)| : (s, *) G £? } = 0; since / 0 # 7^ 0 in Co(5) 0 CQ(T), this is impossible, and E must be all of S x T. Thus the right-hand side of (B.12) is independent of || • ||a, and all the C*-norms on C0(S) 0 C0(T) coincide. • Our next lemma allows us to reduce questions about nuclearity to algebras with identity. Recall that if A is a C*-algebra, then we write A for the C*-subalgebra of M(A) generated by A and 1; thus A = A if A has an identity, and A = A1 otherwise. Lemma B.42. I£A and B are C*-algebras, then every C*-norm on AoB extends to C* -norms on A® B, AG B, and AG B. Proof. It is enough by symmetry to prove that if A has no identity and || • ||a is 1 a C*-norm on A 0 B, then || • ||a extends to A G B. Let n : A 0a B —> B(H) be a faithful nondegenerate representation. By Corollary B.22 there are faithful nondegenerate representations TTA : A —>• B(H) and TTB : B —> S(W) such that 7r(a 0 6) — 7rA(a)7r^(6) for all a G ^4 and b £ B. Since A does not have an identity, A1 embeds naturally in M(A) (see Example 2.41), and hence It A is faithful on A1. Thus TTA 0 7T^ is a faithful representation of A1 0 B (see Remark B.4), and we can 1 define \\t\\a := \\7tA G 7rB(t)\\ for t G A 0 B. • Proposition B.43. Every commutative C*-algebra is nuclear. Proof. By the previous Lemma, it is enough to prove that if X is a compact space and B is a C*-algebra with identity, then all C*-norms on C(X) 0 B coincide. We shall do this by finding a formula for the norm of an arbitrary element which is independent of the given C*-norm. So suppose that || • ||a is a C*-norm on C{X)GB. B.5 Tensor Products of General C*-Algebras 261 We first claim that if £ is a pure state on C(X) 0a B, and 0(/) := £(/ 0 1) and ip(b) := £(1 0 6), then 0 and ^ are states such that £ = 0 0a -0. They are an positive functionals satisfying uj(za) = (7TLJ(z)7Tu;(a)hu; \ hu) = Lu(z)(iru;(a)hUJ \ hu) = u(z)u(a). (B.14) Since / 0 1 belongs to the center of C(X) 0a B and £ is pure, (B.13) follows from (B.14), and we have verified the claim. If t G C(X) 0a B is self-adjoint, then there is a pure state £ such that ||t||Q = |£(£)| (Lemma A. 10); in other words, \\t\\a = sup{ |£(t)| : £ G P(C(X) 0a B) }. (B.15) Thus the claim in the previous paragraph implies that ||*||a < sup{ |0 0a rl>(t)\ : 0 G S{C(X)) and ^ G S(B) } < ||t||a. (B.16) For any t G C(X)QB, we can apply this to the self-adjoint element t*t, and deduce that ||*||2 = ||^||a = sup{ 0 0 rl>{t*t) : 0 G S(C(X)) and ^ G 5(B) }. Since this last expression is independent of || • ||a, the Proposition follows. • Proof of Theorem B.38. Once we see that the spatial norm is minimal, it will follow that || • ||a is a cross-norm: since both the spatial norm and the maximal norm are cross-norms (Theorem B.9(b) and Proposition B.25), we have ||a|| ||6|| = \\a 0 b\\a < \\a 0 6||a < \\a 0 6||max = ||a|| ||b||. Therefore, in view of Proposition B.39, we only need to prove that 0 0 ip is a- continuous for every 0 G S(A) and if) G S(B). Lemma B.42 implies that || • ||a extends to a C*-norm on A 0 B, and Lemma A.7 that 0 and ip extend to states on A and B; thus we may as well assume that A and B have identities. Now Proposition B.43 implies that each self-adjoint element of B is A-nuclear, and the result follows from Lemma B.40. • 262 Tensor Products Corollary B.44. Every C*-algebra with Hausdorff spectrum is nuclear. Proof. This follows immediately from Proposition B.35 and Theorem B.38. • Theorem B.37 says that for CCR algebras with Hausdorff spectrum, the map (7r, 77) i-» 7r 0 77 is a homeomorphism of A x B onto (A Theorem B.45. Suppose that A and B are C*-algebras. (a) The map (^,77) 1—> 7T0 77 induces a homeomorphism $ of Ax B onto its range A in (A®aB) . (b) If either A or B is GCR, then this homeomorphism is surjective. (c) If A and B are separable and if either A or B is nuclear, there is a home omorphism of Prim A x Prim B onto Prim(A 0CT B) which takes (P, Q) to P0aP + A0aQ. Remark B.46. Separability is used in (c) to ensure that prime ideals are primitive (see Theorem A.49). Since this is the only way separability is used, the Theorem also applies to other algebras in which prime ideals are known to be primitive: for example, to algebras whose primitive ideal space is second countable or almost Hausdorff (Theorem A.50). To prove (c), we also need to know that quotients of nuclear C*-algebras are nuclear; this is a deep result, but in view of the current relaxation of our policy on prerequisites, we refer to [17, Corollary 4] for the details. For the proofs of the first two parts we need some facts about von Neumann algebras: C*-subalgebras of B(H) which are closed in the weak operator topology. The commutant of a subset S C B(H) is the set S' := {T e B(H) :TS = ST for all S e S}; if S is closed under taking adjoints, then Sf is a von Neumann algebra. Von Neumann's Double Commutant Theorem [111, Theorem 4.1.5] says that if S is a nondegenerate self-adjoint subalgebra of B(H), the double commutant S" := {S')' coincides with the weak operator closure of S. The tensor product of von Neumann algebras A C B(H) and B C B(V) is the von Neumann algebra A 0 B on H 0 V generated by A 0 B. If .A is a von Neumann algebra on Hi, then A®1H2 :={T0 1GP(WI0W2) :TeA} B.5 Tensor Products of General C*-Algebras 263 is a von Neumann algebra on Wi 0 W2 with (A®l (see, e.g., [29, 1.2 Proposition 4]). More generally, we have (A 0 B)' = A' 0 B' for any von Neumann algebras A and S [159, Theorem 5.9]. A representation TT of a C*-algebra A is irreducible iff IT (A)' = Cl^ (Lemma A.l), and hence it follows from the Double Commutant Theorem that TT is irreducible iff it (A)" = B(Hn). A representation 7r : A —> B(Hn) is called a factor representation if the von Neumann algebra ^(A)" generated by its range is a factor; that is, if the center TT{A)" C]TT{A)' of n(A)" contains only scalar multiples of the identity operator 1?^.. We will need the nontrivial result that GCR algebras are Type I [28, Theorem 5.5.2], which implies that every factor representation of a GCR algebra is equivalent to one of the form TT 0 In for some irreducible representation TT [28, Proposition 5.4.11]. We proved parts (a) and (b) of Theorem B.45 for CCR algebras A and B with Hausdorff spectrum in Theorem B.37. We used the extra hypotheses in two ways: to show that if 7r®l^1 is unitarily equivalent to 77(8)ln2, then TT and 77 are equivalent, and to allow application of Proposition B.35 to deduce surjectivity of 4>. Therefore parts (a) and (b) will follow from the argument used in Theorem B.37 and the next two lemmas. Lemma B.47. Suppose thatir andrj are irreducible representations of a C* -algebra A. If there are Hilbert spaces Hi andH2 such that TT<^ln1 andrj<^ln2 are unitarily equivalent, then TT and 77 are unitarily equivalent. Proof. Let U : H^ 0W1 ^ Hv 0% be a unitary intertwining TT0 \nY and rj0 1^2. Let p be a rank-one projection on Hi, and set P := lnn 0p, Q '•= UPU*. Since n TT and 77 are irreducible, we have TT{A)" = B(Hn) and r)(B) = B(HV), and it A l follows from (B.17) that (TT® lWl)(A)' = lHir ® B{Hi) and (77® ln2)( Y = nv ® / B(H2)- Thus P is a minimal projection in (TT 0 1^1)(T4) ; conjugation by U is an isomorphism of TT 0 1(A)' onto 77 0 1(A)', so this implies that Q is a minimal / projection in (77 0 1^2)(A) . Thus Q — \n 0 ^ for some rank-one projection q in an B(H2)- Notice that UP implements an equivalence between 7T0 ln1 \p{n-n^>Hi) d smce V ® I^IQCW ®7i2)» these representations are equivalent to TT and 77, the result follows. • Lemma B.48. Suppose that A and B are C*-algebras and that either A or B is a GCR algebra. If \\ •1|7 is a C* -norm on Ad)B, then every irreducible representation of A(g>7 B is equivalent to one of the form TT 0 77 for some TT £ A and rj G B. Proof. Suppose that A is GCR and \i G (A 07 B) ; let \±A and /x# denote the corresponding commuting representations of A and 5, as in Corollary B.22. Ev erything in HA{A)N'^IIA{A)' commutes with both HB(B) and HA(A), and therefore n with everything in fi(A 07 B) = B(Hfj)\ thus 11 A is a factor representation of A, and is equivalent to TT 0 ln0 for some TT G A and some Hilbert space WQ. Since 264 Tensor Products /J>B(B) C /J,A(A)', this equivalence takes ^B(B) into (TT(A) 0 1)' = lHir 0 B(Ho), and iiB into 1^0?] for some representation r\ of B\ since / 1 0 r^B)' = (£(«„) 0 v(B)Y C (1 0 r?(5)) , 77 is a factor representation. Thus we can assume that /J, = ix ) 77 where TT is irreducible and 77 is a factor representation. But then /f »(A 07 B)" = (TT(A) 0 rj(B)) = n(A)" 0 77(B)" = B(Hn) < 77(5)". Since // is irreducible, H(A 07 B)" = B(H* 0 Wo) = B(WTT) ® S(W0). Thus 77(B)" = B(Ho), and 77 is irreducible. • Corollary B.49. Every GCR C*-algebra is nuclear. Proof. Since the norm in a C*-algebra is determined by its irreducible representa tions (Theorem A. 14), the corollary follows from the Lemma, Theorem B.9(c) and Theorem B.38. • It remains to prove part (c) of Theorem B.45. The idea is to prove that the homeomorphism of part (a) descends to a map on primitive ideals by identifying the kernel of TT 0 77 in terms of ker7r and ker77. This is a special case of the next lemma; for it, recall that the sum / 4- J := {a + b : a G /, b G J} of two ideals is itself a closed ideal. Lemma B.50. Suppose that A or B is nuclear and that TT : A 0a B —• B(H7r) is an irreducible representation. Let TTA : A —> B[^H^) and TTB : B —» B(Hn) be the commuting representations such that 7r(a0 6) — TTA{O)7TB(O), as in Corollary B.22. Then ker7r = (ker^) 0a B + A 0CT (ker^s). Proof. Let I = (kernA) 0a B + A 0a (kerTTB)- Then / is a closed ideal in A II \ ^—'2=1 / II II \Z-^Z=1 / || ||Z—'l=l || The map (TT^ (a), 71-5(6)) «—> w(a 0 b) is well-defined and bilinear, so it induces a linear map v : TTA(A) 0 TTB(B) —» (A 0CT B)//, which is easily seen to be a *-homomorphism. Thus TTA(ai) <8>7rj3(fei) := itf(y^ a*0M E2 111; II \Z—'i / defines a C*-seminorm || • ||^ on TTA(A) 0 TTB(B). B.5 Tensor Products of General C*-Algebras 265 On the other hand, there is a *-homomorphism We claim that 0 = 5Z A^TTA(ai) for j = 1,..., fc, and 7rs(^) = 5Z._ KjKB(bj) for i = l,...,fc. The first equation implies that A^ = 0 for all i, j, and the second that E TTAK) 0 nB(bi) = Y^. . i Aij7TA(oi) <8) TTB(^) = 0; this proves the claim. Since 0 is injective, we can use it to pull back the norm from B(Hn), and obtain a C*-norm || • \\p on TTA(A) 0 TTB(B) such that i k II II *fc . 7rA(ai)0 7rB(6i) := V\ i ^(a^Tr^) 2=1 N/3 II*—^z=l E J5(WT) If .A is nuclear, then 7TA(-A) = A/(ker7r^) is nuclear [17, Corollary 4], and || • \\p must coincide with the maximal norm. Since the maximal norm dominates every (7*-seminorm (Proposition B.25), we deduce that for J2 ai ® bi € A 0 B, a b >(yi i® i)\\ = y^.7TA(az)0 7rB(6i) < \\J2.nA(a>i) ® 7rB(bi) \ ' •" / II IK •** lli> IK '^ as required. The same argument applies if 5 is nuclear, and hence the lemma is proved. • Lemma B.51. Suppose that IT : A —* B{TL) is a factor representation of a C*- algebra. Then the kernel of TT is a prime ideal Proof. Suppose that / and J are ideals in A with IJ c ker7r and / (£_ ker7r. Then V := ir(I)H is a non-zero invariant subspace for 7r, and the orthogonal projection P of H onto V belongs to n(A)1'. If { e^ } is an approximate identity for /, then 7r(e^) —> P in the strong operator topology. Thus P belongs to 7r(A),fn7r(Ay = Cl-^, and must be a scalar multiple of the identity. In particular, this implies that V = H. But then JI = IJ C ker-zr implies that {0} = 7r(J)7r(I)H = 7r(J)W, and we have Jcker?r. • 266 Tensor Products Proof of Theorem B.45(c). If IT G A and 77 G P, then (irt&rfJA = 7T01 and (TT^T^B = 1 0 77. Thus we can apply Lemma B.50 to the representation TT 0 77 to deduce that ker(-7r 0 77) = (ker 7r) 0a P + A 0a (ker 77). This equality shows that the map $ of part (a) induces a well-defined map ^ of Prim A x PrimP into Prim(A 0a P) such that #(P,Q) = P0(7P + .A0a<2 and A AxB > {A®aB) Prim A x PrimP • Prim(A 0a B) commutes. Because the vertical arrows are continuous open surjections, the conti nuity of $ implies that \I> is continuous. To see that \I> is injective, suppose that ker(7r 0 77) = ker(7Ti 0 771). If ker-zr / ker7Ti, we may suppose there exists a G A such that 7r(a) ^ 0 and 7Ti(a) = 0 (otherwise swap n and 7Ti). But now if 77(6) 7^ 0, then a 0 b G ker(7Ti 0 771) and a 0 6 ^ ker(7r 0 77), which is a contradiction. So we must have ker7r = ker7i"i, and A similarly ker 77 = ker 77!. To establish surjectivity of \I/, suppose that TT G (A 0a B) , so that ker7r is a typical element of Prim (A 0a5). Let 7r^ and ^5 be the commuting representations such that 7r(a 0 6)= ^A{O)TTB{^)' Then f 7 7TA(A)" H 7rA(A)' C 7TB(B) H TTA (A) = 7r(A 0a P)' - Cl^ and similarly for 7TB, SO TTA and 7r^ are factor representations. Thus ker^ and ker TiB are prime by Lemma B.51, primitive by Theorem A.49, and the equality ker-zr = (ker^) 0a B -f A 0CT (ker 7TB) of Lemma B.50 says that \I> is surjective. We have now seen that ^ is a continuous bijection. To see that its inverse is continuous, suppose ty(Pi,Qi) —» \£(P, Q) in Prim(A 0CT P); it is enough to show that this implies Pj —• P in Prim A. Let J be a closed ideal in A which is not contained in P, so that Oj = {K ePrim A: J (£ K} is a typical open neighbourhood of P. Then Oj^aB is a neighbourhood of \£(P, Q) in Prim(A 0a P), so eventually *(Pi, Q*) G Oj®aB- But J C P; implies J 0a P C x Pl®a B C 9(Pi,Qi), so tf(Pi,Qi) G 0J<8)t,B implies P; G Oj. Thus ^~ is continuous, and ^ is a homeomorphism. • B.5 Tensor Products of General C*-Algebras 267 Hooptedoodle B.52. The first part of Theorem B.45 can be found in [172, 173]. (Wulfsohn has pointed out that his proof of Theorem B.45(c) in [174] is not valid in the generality claimed there, though it is valid for GCR algebras.) One can deduce from Theorem B.45 that A®aB is GCR (respectively, CCR) if and only if both A and B are GCR ( respectively, CCR); see [172, 173], [162, Theorem 2], and the references given in [162]. It is also proved in [162, Theorem 2] that A 0CT B has continuous trace if and only if both A and B do; the "if" direction is contained in Corollary 3.37, Proposition 5.15, and Lemma 5.30. Hooptedoodle B.53. The class of nuclear C*-algebras occurs often in applications, partly because their tensor products are tractable, but also because the class is stable under standard constructions. Thus, for example, ideals and quotients of nuclear algebras are nuclear. Conversely, it is closed under taking extensions: if 0 >I—^A—q—>A/I >0 is exact and both A/I and I are nuclear, then so is A. To see this, let B be a C*-algebra. Then we have a commutative diagram 0 • I i(8)id <7<8>id , . _ 0 > I®aB > A®aB • A/1 in which the top row is exact by Proposition B.30. A quick diagram chase shows that the middle vertical arrow is injective, and hence isometric. Other constructions which preserve nuclearity include taking tensor products with nu clear algebras, forming inductive limits of nuclear algebras, and taking crossed products of nuclear algebras by actions of amenable locally compact groups; these stability properties are surveyed in [93]. More recently, it has been shown that crossed products of nuclear C*-algebras by coactions of arbitrary locally compact groups are nuclear [133]. Theorems of Connes [19] and Haagerup [68] show that the nuclear C*-algebras are precisely the C*-algebras which are amenable in the sense of Johnson [80]. Appendix C The Imprimitivity Theorem In this appendix we shall give a more complete treatment of the fundamental exam ples which we discussed briefly at the ends of sections 2.1 and 2.4. Specifically, we shall show that if H is a closed subgroup of a locally compact group G, then CC(G) over can be completed to form a Hilbert module XC*(H) the group G*-algebra of if, and identify K>(XC*(H)) with the transformation group G*-algebra CQ>(G/H) xir G. We shall then show how to recover the Mackey/Blattner process for inducing uni tary representations from H to G, and derive the classical imprimitivity theorem characterizing these induced representations. This last result explains why /C(X) is often referred to as an imprimitivity algebra and why the isomorphism of /C (X^* (#)) with Co(G/H) xir G is sometimes called the (abstract) Imprimitivity Theorem. We postponed this discussion because we did not want to assume background knowledge of locally compact groups and vector-valued integration, and here we will begin with a brief review of this necessary background. In Section C.l we discuss Haar measure and quasi-invariant measures on homogeneous spaces, and in Section C.2 we consider G*-algebra-valued integrals of continuous functions with compact support. In Section C.3, we construct the group G*-algebra C*(G) of a locally compact group G, and prove that there is a one-to-one correspondence between unitary representations of G and representations of C^(G). We prove the Imprimitivity Theorem in Section C.4, relate it to the Mackey/Blattner formulation in Section C.5, and prove the Stone-von Neumann Theorem in Section C.6. C.l Haar Measure and Measures on Homogeneous Spaces A measure ^ on a locally compact group for which every open set is mea surable is called a Borel measure. If for each open set V C G, /i(V) = sup{ n(C) : G is a compact subset of V } and if for each measurable set A C G, l^(A) = inf{^(V) : V is a open neighbourhood of .A}, then /i is called regular. A measure is said to be left invariant if /j,(g • A) = fi(A) for all measurable sets A and all g G G. A regular left-invariant Borel measure on G is called a Haar 269 270 The Imprimitivity Theorem measure. Every locally compact group G has a Haar measure, and it is unique up to multiplication by a positive scalar. This Haar measure assigns strictly positive measure to every nonempty open subset of G and finite measure to every compact subset. These properties of Haar measure guarantee that for every / G CC(G), 11/11!:= f \f{s)\d^s) JG is finite.* To see that Haar measure exists, one uses the Riesz Representation Theorem, and constructs instead a linear functional / on CC(G) which is positive in that 1(f) > 0 whenever / G C+(G), and left invariant in that l(rr(f)) = 1(f) for all l / G CC(G) and r G G. (Recall that rr(f)(s) := f(r~ s) and C+(G) is the set of functions / G CC(G) satisfying f(s) > 0 for all s G G.) Such a functional is called a Haar functional Details of this argument^ and proofs of our assertions about Haar measure can be found in [55, §2.2], [54, Chap. Ill §7] or [71, §15]. Examples of Haar measure include Lebesgue measure on Rn and Tm, as well as counting measure on a discrete group. In these examples the Haar measure /x is also right invariant: JJL(A • g) = /JL(A) for all g G G. In general, a left-invariant Haar measure need not be right invariant, but the uniqueness of Haar measure implies that there is a continuous function A : G —• E such that A(t) f f(st)d^s)= [ f(s)d,x(8) JG JG for each / G CC(G). This modular function A is a continuous homomorphism [55, Proposition 2.24]. When A = 1, the left Haar measure is right invariant, and we say the group is unimodular, all abelian and compact groups are unimodular, but many important examples such as the ax + b group are not [55, §2.4]. The need to decorate formulas with modular functions will cause us some pains later*. Suppose that H is a closed subgroup of a locally compact group G. The ho mogeneous space G/H will play a pivotal role. Even in simple examples, such as G = E and H — Z, there need be no continuous cross-section c : G/H —> G for the orbit map q : G —± G/H satisfying q o c = id. However, there is always a kind of approximate cross-section which will be prove useful. Here and later, we often write s for the coset sH G G/H. Proposition C.l. Let G be a locally compact group and H a closed subgroup. Let v denote Haar measure on H. Then there is a bounded continuous function *Of course, || • ||i is a norm on CC(G) and the completion of (CC(G), || • ||i) is the Banach space LX(G) of (equivalence classes) of integrable functions on G. Since we wish to allow arbitrary locally compact groups G, but wish to avoid the associated measure-theoretic difficulties, we will work primarily with CC(G). ^Haar showed in 1933 that every second countable locally compact group had a left-invariant measure [69]. Weil and Kakutani independently extended his result to arbitrary locally compact groups, and uniqueness was proved by von Neumann. Further discussion and references can be found in the notes in [55, §2.7] or at the end of [71, §15]. •t-We have elected to include modular functions so that our treatment will be useful for reference. Someone looking at these constructions for the first time might be well-advised to ignore the modular functions. C.l Haar Measure 271 b : G —> [0, oo) such that (supp b) fl CH is compact for every compact set C in G, and such that J b(st) dv{t) = 1 for all seG. JH Such a function is called a Bruhat approximate cross-section for G over H. Proof We claim that it suffices to prove the result when G is cr-compact. Let Go be a / b(st) dv{t) = [ b0(c(s)st) dv{t) = 1 for all seG. JH JH If C is compact then there are SI,...,SJV G G such that Si — c(s~i) and C C s an Ui=i iGo- Therefore supp(6) C [ji=1 s^supp(fro), d it suffices to consider the case where G is cr-compact, as claimed. Suppose that G = U^Li Kn with each Kn compact and contained in the interior Kn+i °f -^n+1- (For notational convenience, let K$ — K-\ — 0.) Let Cn = Kn \ K%_1H and Un = K^+i \ Kn-2H. Note that Cn is compact, that Un is a relatively compact open^ neighbourhood of Gn, and that C := U^Li Cn is closed. Using a partition of unity^ on G subordinate to the cover Cc U U^Li Un, we can find functions fn G C£(G) such that supp(/n) C t/n, such that / := ^2nfn is in + Gb (G), and such that f(s) = 1 for all seC. Now suppose that A C Kn and AiJ n Um ¥" 0- Then A fl UmH ^ 0 and AD (Km-2H)C T^ 0. It follows that m < n +1. Therefore the saturation G# of any compact set C has compact intersection with the support of /. We claim that if seG, then there exists h G H such that sh G C. For we certainly have s G KnH for some n. If n = 1, we're done; if s ^ Gn, then s G Kn-\H, and we proceed by induction. It follows from the two assertions in the previous paragraph that 6(s):= / f(st)du(t) JH is finite and nonzero for all s G G/H. We obtain a function b with the required properties by defining b(s)=b(sr1f(s). D We would like to choose a measure on the quotient G/H which is compatible with Haar measures on G and H. If H is normal in G, then things are relatively §The product of a compact set and a closed set is always closed. ^Every locally compact group is paracompact by [71, Theorem 8.13] 272 The Imprimitivity Theorem straightforward. For then G/H is a locally compact group and has a Haar measure a; the map / t—> fG,H JH f(st) dv(t) da(s) is a Haar functional associated to a Haar measure \i on G such that / f(s) dfi(s) = f [ f(st) du(t) da(s) for all / G CC(G). (C.l) JG JG/H JH In general, G/H is not a group, and may not have an invariant measure. However, it always has a quasi-invariant measure: a measure a on G/H such that the measure l ar defined by ar(E) := a(r~ • E) is equivalent to a for each r G G. (Two measures are equivalent when they have the same null sets.) Our goal is to produce such a measure which satisfies a formula as close to (C.l) as possible. The obstruction to producing an invariant measure has to do with how the Haar measures \i and i/onG and H fail to be right invariant. If H is not normal in G, then the modular function 6 on H may not coincide with the restriction of the modular function A of G to if." Our construction of a quasi-invariant measure requires a continuous function p such that p(st) = (6(t)/A(t))p(s) for all s G G and t G H\ such a function p is sometimes called a rho-function. If 6 = A|#, then we can take p = 1, and the measure constructed below will be invariant for the left action of G on G/H. Lemma C.2. Suppose that G is a locally compact group with a closed subgroup H, and let p, v be Haar measures on G, H, respectively. Given any continuous function p : G —> (0, oo) such that p(st) = (8(t)/A(t))p{s) for allseG and t G H, (C.2) there is a quasi-invariant measure a on the Borel sets of G/H such that f f(s)p(s) dp(s) = [ I f(st) du(t) da(s). (C.3) JG JG/H JH Furthermore, the Radon-Nikodym derivative of the translate ar with respect to a is given by dar ,.v _ pjr^s) da [S) ~ p(s) ' There always exist continuous functions p satisfying (C.2). Proof Suppose that p satisfies (C.2). Define $ : CC(G) -+ CC(G/H) by *(/)(*):= / f(st)dv(t). JH An application of Proposition C.l shows that $ is surjective: if 0 G CC(G/H), then (4>oq)b belongs to CC(G) and satisfies <&(( oq)b) = (f). We will construct a positive "If H is normal, then evaluating both sides of (C.l) on hf : s i—>• f(sh) shows that the modular functions on G and H coincide on H. C.l Haar Measure 273 linear functional F on CC(G/H) and apply the Riesz representation theorem [125, Theorem 6.3.4] to obtain a. We would like to define F by *W))= [ f(s)p(s)d^s); (C.4) JG since <3> is surjective, it is enough to show that the right-hand side of (C.4) depends only on $(/) rather than /. Equivalently, we need to show that if $(/) is identically zero, then the right-hand side of (C.4) is zero. Suppose that for some / G CC(G) we have f f(st) dv(i) = 0 for all seG. (C.5) JH By multiplying a Bruhat approximate cross-section b by a compactly supported function which is identically one on supp(/), we obtain a function b\ G C^{G) such that / bi(st) dv(t) — 1 for all s G supp(/). JH (Such a function is sometimes called a cut-down Bruhat approximate cross-section.) 1 1 Define g G CC(G) by g(s) = ^(s^A^- )/^- ). If (C.5) holds, then J:= [ g*f(t)dv(t)= I [ g{r)f{r-H)d^{r)dv{t) JH JHJG = [ 9(r) [ f(r-H)dv(t)d»(r) = 0. JG JH Consequently, 0 = 7= f g*f(t)du(t)= f f gMfir-^d^d^t) JH JHJG = / [ hir-H-^Air-H-^pir-H-^fir-^d^duit) JH JG = [ [ 61(rt)(A(t)/^(t))Krt)/(r)^(t)^W JG JH = / f(r)p(r)dn(r). JG Thus the functional F is well-defined on CC(G/H). To see that F is positive, note that if 0 G CC(G/H) is positive, then f := ($ o q)b is a positive function satisfying $(/) = , and this choice of / makes the right-hand side of (C.4) positive. We deduce that there is a measure a satisfying (C.3). Notice that if 0 G CC(G/H), then / (f>(s) dar(s) = / (f)(r • s)da(s). JG/H JG/H 274 The Imprimitivity Theorem Since we can write 0 = $(/), we have / 4>(r • a) da(s) = / f(rs)p(s) dp(s) = / /(rs)-^\p(r«) dp(s) JG/H JG JG P{rs) = lGf(s)^^p(s)dp(s) I ~^ f f(st)du(t)da(s) JG/HIG/I P(S) JH f p{r-xs) -(j)(s)da(s), JG/H p(s) and the statement about Radon-Nikodym derivatives follows. We have proved everything except the existence of a suitable function p. For this, choose / G C+(G) with /(e) ^ 0 and JH f(t) dv(t) — 1, and define u;(s) = / fists'1) dv(t) for all s e G. JH We then have uu(st) = 6(t)u>(s), and we can take p(s) := A(S~1)UJ(S). • C.2 Vector-Valued Integration on Groups The theory of integration for functions with values in a Banach space can be quite complicated, but the theory simplifies considerably when one only cares about continuous functions of compact support with values in a C*-algebra A. We shall further simplify things by only integrating functions with respect to a Haar measure juona locally compact group G. For the general theory, we refer to [54, Chap. II]. If / E CC{G,A), then s ^ ||/(s)|| belongs to CC(G), and := f \\f(s)\\dp(s) JG is finite and satisfies ||/||i < ||/||oo /i(supp/). Lemma C.3. Let A be a C*-algebra. There is a unique linear map I : CC(G, A) —> A such that for any representation TT : A —> BiTi^), we have (n(l(f))h \k) = f (ir(f(s))h \ k) dp(s) for all h,ke H*. (C.6) JG We write JG f(s) dp(s) := /(/). Then L f(s) d(j,(s) < ll/lli, (C7) (J f{8)d»(8)) = f f{sY d^s), (C.8) C.2 Vector-Valued Integration on Groups 275 and for all a G M(A) we have I / f(s) dfi(s) J a = / f(s)ad/j,(s) and ai f(s)dfi(s)j = / af(s)dfx(s). Finally, if L : A —> B is a bounded linear map into a C*-algebra B, then L ^J f(s) dp(s)) = J L{f(s)) d/x(s). (CIO) Proof The uniqueness is clear: if 7r is faithful and b G A, the numbers (71*(6)ft | /c) determine 6 uniquely. Let 7r : A —> B(HTT) be a representation. The right-hand side of (C.6) is a sesquilinear form on H^ which is bounded by ||/||i. Therefore there is an operator TJ G B(H7T) with ||T/|| < ||/||i satisfying (TJh\k) = [ {n{f(s))h\k). (C.ll) JG Note that / H-> TJ is linear in /. We claim that TJ G TT(A). Let e > 0, K = supp(/), and let C be a compact neighbourhood of K. Since K is compact, there is a symmetric open neighbourhood V of e G G such that IfV C C and r_1s G V implies ||/(r)-/00|| r There are ri,...,rn G K such that K C UlLi ^- Using a partition of unity c subordinate to the finite cover { K } U { r^V }, we can find fa G CC(G) such that 0 < fa < 1, supp(^) C nV, X^ ll/-«fcl|l= / H/(*)-Se(*)||dM*) JG < f E"_, H/(s)" fMWMs)Ms) (C.12) < {s)Ms) e ^)jGK^ - - Thus ||T£ - TJ|| < e. But if c* := JG &(s) dfj,(s), then Since e was arbitrary, it follows that TJ G TT(A), as claimed. _1 Now let p : .A —> B(HP) be a faithful representation and define /(/) := p (T?). -1 Notice that p (T£) — J^ Q/(r^), and since p is isometric, \\I(f) — ^2i Q/(r^)|| < e. Therefore ||7r(/(/)) - TJ\\ < 2e. Since e is arbitrary, this establishes (C.6). 276 The Imprimitivity Theorem Properties (C.7), (C.9) and (C.8) follow from the defining property (C.6). Fi nally, (CIO) follows from the estimates ||L(/(/)) — ^2ici^{f(ri))\\ < ell^ll and ||£°<7e-£°/||i Definition C.4. Suppose that A is a C*-algebra and a G A. Let || • ||a be the seminorm on M(A) defined by ||6||a = ||a6|| + \\ba\\. The strict topology on M(A) is the topology generated by the seminorms {||-||a:aGA}. Example C.5. Since B(H) = M(/C(W)), it has a strict topology. A net {Ti }ieA in B{H) converges to T strictly if and only if T{K -> Tif and KT{ -> XT for every if G JC(H). The *-strong topology on 5(W) has subbasic open sets N(T, h,e):={Se B{H) : ||(S - T)h\\ + \\(S* - T*)h\\ < e } parameterized by T G B(H), h GW, and e > 0. Therefore a net { Ti }i^A converges to T *-strongly if and only if both Ti —> T and T* —> T* in the strong operator topology. As the next result shows, these topologies agree on bounded subsets. Lemma C.6. On (norm) bounded subsets of B(7i), the strict topology coincides with the *-strong topology. We actually need a more general version of Lemma C.6 in which we replace the Hilbert space H with a Hilbert A-module X. Recall that £(X) = M(/C(X)) (Corollary 2.54) so that £(X) has a strict topology. It also has a *-strong topology: by analogy with the above, we say that { Ti } in £(X) converges to T *-strongly if and only if both T^x) -> T(x) and T?(x) -+ T*(x) in X for all x eX. Lemma C.6 is a special case of: Proposition C.7. Suppose that X is a Hilbert A-module. Then strict convergence in C(X) = M(/C(X)) implies *-strong convergence, and the strict topology coincides with the *-strong topology on (norm-) bounded subsets of C(X). Proof. Suppose that Ti —• T strictly. Given XGX, Lemma 2.31 implies that there exists j/GX such that x = y • (y , y) = (y , y){y). By assumption, both T T y and y y Tl yS)T in norm In articular V(x)^ ' ^ "^ >cW( ' ^ K(x)( ' ^ ~* K(XP ' - P > Ti(x) = T^Jy , y)(y)) - T(KJy , y)(y)) = T(x). Since the adjoint operation is norm continuous on £(X), the strict convergence also implies that T* (y , y) -> T* (y , y), from which it follows that T* (x) -> T* (x). /C(X) /C(X) On the other hand, suppose that Ti —> T *-strongly and that ||T^|| < M for all i. For any fixed x,j/GX, Ti(x) —• T(x) implies that T^ (x , y) —> T (x , y) in norm in £(X). Similarly, we have T? (x , y) —> T* (x , y) in norm. It follows /C(X) /C(X) C.2 Vector-Valued Integration on Groups 277 that for any finite sum F of such basic operators, T{F —• TF and T*F —> T*F in norm. If K G /C(X) and e > 0, there is a finite sum F such that \\K - F\\ < e/SM. Choose io such that i > io implies \\TiF — TF\\ < e/3. Then for i > z'o, we have ||T-if - TitCH < ||r2i^ - TiF\\ + ||TiF - TF|| + ||TF - Tif || < e. We can similarly deduce from T*F->TF that || AT. - KT\\ = \\T*K* - T*K* || -• 0. D For homomorphisms u : G —> t/£(X), there are apparently several different different notions of continuity We can ask that u be strictly continuous in the sense that s \—> us is continuous from G into C(X) with the strict topology Or we could, by analogy with unitary representations, ask that u be strongly continuous in the sense that s i—• us(x) is continuous for all x G X. Fortunately these notions coincide. Corollary C.8. If u : G —> UC(X) is a homomorphism into the unitary group of C(X), then u is strictly continuous if and only if it is strongly continuous. Proof Since each us has norm one, Proposition C.7 implies that is is enough to prove that if s i—> us{x) is continuous for all xGX, then s H-> u*(x) is continuous. The proof is similar to that of Lemma 1.3(a): u x u x u x IK0) - <(Z)IIA = «0) - t( )» *8( ) ~ t( ))A = 2||x||^ — (usul(x) , x) — (x , usul(x)). (C.13) If s —> t, then by assumption ^s^(x) —> utul(x) = x\ therefore (C.13) converges to 0 as s —> t. D Remark C.9. Proposition C.7 says that strictly convergent nets in B(H) or C(X) always converge *-strongly. When H is infinite-dimensional, the next example shows that the converse can fail for unbounded nets. Since the principle of uni form boundedness implies that a strongly convergent sequence is bounded, and hence strictly convergent by Proposition C.7, the example also shows that even on separable Hilbert space the *-strong topology is not first countable, so that it is necessary to use nets instead of sequences. ( Example CIO. Let {en} ^L1 be an orthonormal set in H. For each n > 1, define Tn G B(H) by Tnh = y/n(en 0 e^)(h) = y/n(h | en)en. We will show that 0 belongs to the *-strong closure of S := {Tn : n > 1}. Since the operators in S are self-adjoint, it is enough to prove that for every e > 0 and hu...,hk G H, S meets the neighbourhood U = {T G B(H) : \\Thi\\ < e for i = 1,..., k } of 0. If Tn £ £7, then r 2 c2 (ai4) E*=iii ^ii ^ ' 278 The Imprimitivity Theorem which implies that J2i=i \(hi I en)\2 > e2/n. But e 2 e 2 E* i INI2 ^ E* i E°°, i(fc i «)i = E°°, E" i ic* i «)i ' so (C.14) can't hold for all n, and S must meet £/. 1 Now define K to be zero on ({cn}^-^- , and satisfy Ken = (l/y/n)en. Then jRT is compact, but ||TnX|| = 1 = ||i£Tn||, so the sequence {TnK} cannot converge in norm. Thus 0 is not in the strict closure of S. We will write MS{A) for the space M(A) with the strict topology. If f e CC(G, MS(A)), then for each a G A the function 5 i—» /(s)a belongs to CC(G, A). In particular, {||/(s)a||:s€G}is bounded for each a G A The principle of uniform boundedness implies that {||/(s)|| : s G G} is bounded, so that CC(G,MS(A)) consists of norm-bounded functions.* Since any norm-decreasing nondegenerate representation 7r of CC(G) extends to a nondegenerate representation of C*(G), n has a canonical extension n to M(C*(G)) (Proposition 2.50). Lemma C.ll. Let A be a C*-algebra. There is a unique linear map f h-> fGf(s)d/j,(s) from CC[GJMS(A)) to M(A) such that for any nondegenerate representation n : A —> B(HTT) (7r(ff(s)dfi(s)^h\k^= f (n(f(s))h \ k) for all h,k eH*. (C.15) We have [ f(s)dp(s) < H/lloo '/i(supp/). JG Equations (C.8) and (C.9) are m/za7 m £/ws context. If L : A -^ B is a nondegen erate homomorphism into a C*-algebra B, then (CIO) holds as well. Proof. Since 7f(/(s))7r(a)/i = 7r(/(s)a)/i, the function 5 i—> 7r(f(s))k is continuous for all fc G 7r(^4)W7r. Since 7r is nondegenerate and / is norm-bounded, it follows that 5 i—> 7f(/(s)) is strongly continuous. Therefore 5 i—> (#(/($))/i | fc) is continuous and bounded by ||/||oo • MSUPP/)INI ll^ll- Thus, as in the first part of the proof of Lemma C.3, there is an operator TJ in B(7i7T) satisfying (C.ll) (with n in place of 7r). If p is faithful as well as nondegenerate, then Proposition 2.53 implies that p is an isomorphism of M(A) onto C = {Te B(HP) : Tp(A) C p{A) and p{A)T C p(A)}. *It is not true that s v-+ ||/(s)|| need be continuous in this case. For example, consider the algebra A = {a G C([0,1]) : a(0) =0}, and let /(s) be the function which is 1 at 0, identically zero if t > s, and linear from 0 to s. Then multiplication by f(s) is a multiplier on A, and f(s) converges strictly to 0 as s —• 0, but ||/(s)|| = 1 if s ^ 0. C.2 Vector-Valued Integration on Groups 279 But (Tfp(a)h | fc) = / (p(f(s))p(a)h | fc) dfi(s) JG = / {p(f(s)a)h | fc) dp(s) JG = (p(/ f(s)adp(s))h\k), where the last equality follows by applying Lemma C.3 to g(s) := f(s)a. Therefore Tfp(a) G p(A). Since a similar argument implies that p(a)Tr G p(A), we have p l p T f eC = p(M(A)). Thus, we can define fG f(s) dp(s) := (p)- (T f). Since p is an isomorphism of M(A) onto C, the argument above shows that (C.9) holds. If n is any nondegenerate representation, then we can use Lemma C.3 to show that (it^J f(s)dp(s))7r(a)h I fc) = (TT(J f{s)ad^{s))h | fc) - /\ix(f(s)a)h\k)dp{s) JG = I (7t(f(s))7r(a)h\k)dp(s). JG Now (C.15) follows from the nondegeneracy of 7r, and (C.8) is straightforward to check. Finally, if L : A —> B is a nondegenerate homomorphism, then l(j f(s)dii(sj)L(a)b = L(J f(s)dp(s)a)b = L(J f(s)adfjL(s))b by (C.9) = (JL(f(8)a)d»(8J)b by(C.lO) = j L(f(s))L(a)bdp(s) = (J L(f(s))dp(s))L(a)b. This suffices. • As an application of the ideas in this section, we see that our C*-algebra-valued integrals behave nicely with respect to Hilbert-module inner products. Corollary C.12. Let X be a Hilbert A-module and f a compactly supported func tion of G into C(X) which is continuous for the strict topology. Then for each x,y G X, s I—> (x , f(s)(y)) is in CC{G,A) and x , (y f(s)dv(s))(y))A = J(x, f(s)(y))Adti*)- 280 The Imprimitivity Theorem Remark C.13. Since T i-> (x , T(y)) is a bounded linear map from £(X) to A, this Corollary would be a straightforward consequence of property (CIO) of Lemma C.3 if / were continuous for the norm topology. Proof. Since s i—» f(s)(y) is continuous by Proposition C.7, the first assertion is a consequence of the Cauchy-Schwarz inequality (Lemma 2.5). Left-handed versions of Proposition 2.31 and Lemma 2.30 imply that y = R(z) for some R G /C(X) and z G X. Since T *-* (x , T(y)) is a bounded linear operator from /C(X) to A and s»->/(s)JRis norm continuous, Lemma C.3 implies that , [J f(s)dKs))(y))A = (*,(/ /(s)i?d/i(s))(*) = I (x , f(s)R(z))Ads L(x , f(s)(y))Ads. D C.3 The Group C*-Algebra Let G be a locally compact group with left Haar measure JJL and modular function A. The group C*-algebra is meant to be a C*-algebra whose representations correspond to the unitary representations of G in a natural way. We start with the space CC(G) of compactly supported continuous functions on G, which is a *-algebra with multiplication given by convolution: z*w(s) = / z(r)w(r~1s)dfA(r), JG and involution given by z*(s) := A(s 1)z(s~1). Using Lemma C.ll, each unitary representation U gives a ^representation TTU : CC(G) —> B(H) via TTU(Z) = / z{s)Usdii{s)] JG thus {iru(z)h \k)= [ z(r)(Urh \ k) dfi(r) for all z G CC(G) and h,keH. JG Notice that ||7rc/(;z)|| < ||^||i. Since one of the standard properties of the modular function says / z(s)dn(s) = [ zis-^Ais'^d^s) JG JG for any z G CC(G) [55, Proposition 2.24], we always have ||z*||i = ||z||i. A *- representation -K of CC(G) is called norm-decreasing if ||7r(z)|| < ||z||i, and cyclic if span{ 7r(z)h : z G CC(G) } = Hn for some h G Hn. C.3 The Group C*-Algebra 281 Our first task is to see that every norm-decreasing ^representation of CC(G) arises from a unitary representation U (Corollary C.18). We start by defining the universal C* -norm of z £ CC(G): \\z\\ := sup{ ||7r(2;)|| : n is a cyclic norm-decreasing ^representation of CC(G) }. Notice that \\z\\ < \\z\\i, and, since every *-representation is the direct sum of cyclic ^representations, ||7r(z)|| < ||z|| for every norm-decreasing ^representation of CC(G). (One takes cyclic representations in the definition of || • || merely to guarantee that we are taking the supremum over a set.*) It is not hard to see that || • || is a C*-norm on CC(G), and the completion of CC(G) with respect to || • || is a C*-algebra C*(G) called the group cstar@group C* -algebra of G. Notice that any || • ||i-norm-decreasing ^representation of CC(G) is necessarily || • ||-continuous and therefore extends to a representation of C*(G). Hooptedoodle C.14- The norm || • || on CC(G) is called the universal norm to distin guish it from the reduced C*-norm obtained from the left-regular representation of G. The left-regular representation is the unitary representation A : G —• U(L2(G,n)) de 1 fined by Xr(h)(s) = h(r~ s). The integrated form TT\ is a norm-decreasing injective ^representation of CC(G), and we can define the reduced norm || • ||r by Mr ~ ||7TA(S)|| Z€CC(G). We always have || • ||r < || • ||, but they do not always agree; indeed, the reduced norm equals the universal norm if and only if the group G is amenable [124, Theorem 7.3.9]. Using this characterization, one can see that all abelian and all compact groups are amenable. On the other hand, the free group F2 on two generators is not. In general, therefore, the completion C*(G) of CC{G) with respect to the reduced norm is a proper quotient of the group C*-algebra C*(G); C*{G) is called the reduced group C* -algebra. Since the left-regular representation separates points of CC(G), the universal norm is actually a norm, and CC(G) embeds faithfully in the completion C*(G). Thus we can and will view CC(G) as a *-subalgebra of C*(G). We are going to use the inductive limit topology on CC(G), which is discussed in Appendix D.2. One of the reasons the inductive limit topology is important is that it implies convergence in the || • ||i-norm, and therefore in the C*-norm on CC(G) (Corollary D.6). In working with this topology, it is often enough to know that if { Z{ }iej is a net of functions, zi —> z uniformly, and all the supports of the zi are eventually in a fixed compact set, then z^ —> z in the inductive limit topology.^ For such nets it is clear that \\zi — z\\\ —> 0, and hence ||z^ — z\\ —> 0 also. One of the difficulties in working with group C*-algebras is that not all the elements in C*(G) are represented by functions on G. This leads to interesting but * Formally, we should be looking only at cyclic representations on subspaces of a fixed Hilbert space of suitably large dimension. Since every cyclic representation is unitarily equivalent to one of these, our formulation should not be misleading. 1"Notice that we did not say that these are precisely the convergent nets. It is common in the literature to use the phrase "converges in the inductive limit topology" to describe nets which converge uniformly and are eventually supported in a fixed compact set, and it would be easy to come to the conclusion that such nets describe the topology. That this is not so is illustrated in Example D.9. (See also Remark D.ll.) 282 The Imprimitivity Theorem counterintuitive situations, such as that described in Hooptedoodle C.14, where a faithful representation of CC(G) need not be faithful on C*(G). Another such subtlety is that "evaluation at s G G" on elements of CC{G) is not continuous with respect to the universal C*-norm. If H is a locally compact group and / : G —> CC(H) is a sufficiently nice function, then we can view / as being either C* (iJ)-valued or Cc(i7)-valued, and we can try to make sense of JG f d^i both as an element of C* (H) (using Lemma C.3) and as an element of CC(H). Naturally, we'd then like to know that these elements coincide in C*(H). (If evaluation was a bounded operator on C*(H), we could see this by applying (CIO) of Lemma C.3.) Our next result accomplishes this in sufficient generality for our purposes. * Lemma C.15. Let G and H be locally compact groups. Let F G CC(G x H) and define f : G -> CC(H) by f(s)(t) = F(s,t). Then JG defines an element of CC{H). On the other hand, we can view f as an element of CC(G,C*(H)) and obtain an element of C*{H) via Lemma C.3: [ f(s)dii{s). JG In C*{H), we have b = b. Proof Since the inductive limit topology is stronger than the C*-norm topology, / is certainly in CC(G,C*(H)) and b is well-defined. On the other hand, it is not hard to see that b G CC(H). Let L be the inclusion of CC(H) into C*(H). Then the composition F i—• b i-» t(b) defines a linear map <£ : CC(G x H) —> C*(H). Similarly, F \—> b is a linear map \fr : CC(G x H) —> C*(H). Since these maps clearly agree on functions of the form F(s,i) — g(s)k(t) for g G CC{G) and k G CC{H), and since functions of this form span a dense subspace of CC{G x H) in the inductive limit topology, it suffices to prove that $ and ^ are continuous for the inductive limit topology on CC(G x H). Let v be a Haar measure on H. Suppose that supp(F) C C x K with C compact in G and K compact in H. Then the continuity of $ and \£ follows from the following estimates: ||*(F)|| < H/IK = / \Ms)\\dv(s) HWH < IHIi = / \W)\\dp{t) < i/(JOsuP| / f(s)(t)dn(s) *We thank Wojciech SzymanskJHi for simplifying a previous argumentteHUG . 1 For z G CC{G) and r G G, define rr(z){s) = z(r~ s). Then using the left invariance of Haar measure, it is not hard to check that rs(z)* *rs(w) = 2* *w, so that rr is isometric for the universal C*-norm, and that rr(z) * w = rr(z * w), so l that rr extends to an adjointable operator on C*{G)c*{G) with adjoint rr-\ — r~ . Thus there is a unitary element IG{T) of £(C*(G)) = M(C*(G)) characterized by IQ{T)Z = rr(z) for 2 G CC(G). Since 5 H-» TS(Z) is continuous from G to CC(G) with the inductive limit topology, and since each ic(s) has norm 1, it is not hard to see that IQ is a strictly continuous homomorphism. Corollary C.16. If z,w G CC(G), then as elements ofC*(G) we have L z(s)ic(s)w dfjb(s) = z *w. Proof. Apply Lemma C.15 with H = G and F(s,t) = z(s)w(s 1t), so that f(s) = Z(S)IG(S)W = Z(S)TS(W). • Proposition C.17. Suppose that X is a Hilbert A-module and that u : G —> UC(X) is a strongly continuous homomorphism of G into the unitary group of C(X). Then for each z G CC(G), Lemma C.ll implies that Ku(z) = / z(s)usdn(s) JG is a well-defined operator in C(X), and nu extends to a nondegenerate homomor phism ofC*(G) into C(X), called the integrated form of u. Conversely, ifir : CC(G) —> C(X) is a norm-decreasing nondegenerate ^-homo morphism, then there is a strongly continuous homomorphism u : G —* UC(X) such that Tr — 7TU. In fact, if TT is the canonical extension of TT to M(C*(G)), then we can take us = 7t(iG(s))- Proof The map s \-> us is strictly continuous by Corollary C.8; thus TTU(Z) is defined by Lemma C.ll as claimed. Using Corollary C.12 and a mild form of Fubini's Theorem^, (x , nu(z * w){y)) = / z*w{s)(x , us(y)) dfi(s) A JG 1 = z(r)w(r~ s)(x , us(y)) dfi(r) dfi(s) JG JG II z(r)w{s)(x , urs{y)) dfi(s) d(i{r) JG JG = z(r)w(s)(u*(x) ,u8(y)) dfj,(s)dij,(r) JG JG A = / z(r)(u*(x) ,TTu{w)(y))Adfi(s) J G = (X ,7Tu(z)7Tu(w)(y)) . %A suitable version of Fubini can be obtained representing everything on Hilbert space, and then invoking the result for scalar-valued functions. 284 The Imprimitivity Theorem A similar computation shows that 7ru(z*) = TTU(Z)*. Now let p be a state on A. Then (x \ y)p := p((y , x) ) is a positive sesquilinear form on X. The completion of X with respect to the norm || • \\p = (• | -)p is a Hilbert space Hp. Each 7ru(z) defines an operator TT(Z) in B(HP) (bounded by ||z||i), and 7ir : CC(G) —> B{HP) is a norm-decreasing ^representation of CC(G). Therefore ||7r(^)|| < ||2:|| by definition of the universal norm. Thus 2 2 p((nu(z)(x) , nu(z)(x))A) = (n(z)(x) \ *{z)[x))p = \\*{z)(x)\\ p < \\z\\ \\xfp 2 <\\z\\ p((x,x)A). Since p is arbitrary, it follows that ||7rn(z)|| < ||z||, and TTU can be extended to C*(G) as claimed. The nondegeneracy is straightforward — consider nonnegative functions / G CC(G) with small support and integral one.^ Now let 7r be a norm-decreasing nondegenerate *-homomorphism of CC(G) into £(X). Extend n to C*(G), let it be the canonical extension to M(C*(G)), and take us := Tt^icis)). Using the nondegeneracy of n and the strict continuity of ic, it is not hard to check that s \—> us is strongly continuous. Since z*w — JG z(s)io{s)w dp, (Corollary C.16), we have (y , TT(Z * W)(X))A = / z(s)(y , 7r(iG(s)w)(y))A dp(s) = / z(s)(y , (us7r(w))(y))Adp(s) = (x ,7ru(z)<7r(w)(y))A. Since TT is nondegenerate, it follows that TT = 7ru, as required. D Since a Hilbert space is a Hilbert C-module, the following result is immediate from Proposition C.17 and the definition of the norm on CC{G). Corollary C.18. Suppose that n : CC(G) —> B^K) is a norm-decreasing nonde generate *-representation of CC(G). Define Us = 7t{iG(s)). Then U is a unitary representation of G and TT = TTJJ. In particular, for z G CC(G) we have \\z\\ = sup{ ||7Tc/(^)|| : U is a unitary representation ofG}. Since every representation of C*(G) defines a unique norm-decreasing *- representation of CC(G), the following proposition is a straightforward consequence of the previous corollary. Proposition C.19. The map sending a unitary representation U of G to its inte grated form TTU establishes a one-to-one correspondence between the unitary repre sentations of G and the nondegenerate representations ofC*(G). ^The more sophisticated could use an approximate identity for CC{G) in the inductive limit topology. C.4 The Imprimitivity Theorem 285 Example C.20. Suppose that G is abelian and that G is the Pontryagin dual of G; that is, G is the locally compact abelian group of continuous homomorphisms 7 : G —• T with the topology of uniform convergence on compact sets [55, §4.1]. Then C*(G) can be identified with Co(G) via the Fourier transform. To see this, note that by Theorem A. 14, the C*-norm of z G CC{G) satisfies M|=sup{ ||7r(*)|| :ir€C*(G)A}. (C.16) Since C*(G) is commutative, each n G C*(G) is one-dimensional (Example A. 16), and is therefore the integrated form of a one-dimensional unitary representation, which is a continuous homomorphism 7 G G. In other words, *(z) = 5(7) := / *(sh{s)diJ,(s), JG which is the integral defining the Fourier transform z : G —> C of z. Equation (C.16) therefore says that ||z|| = ||^||oo- From abelian harmonic analysis we know that z is continuous and vanishes at infinity on G, and that the image of CC(G) is dense in Co(G) [55, Theorem 4.13]. Since the Fourier transform converts convolution to pointwise multiplication and the involution to conjugation, the map z 1—» z extends to an isomorphism of the C*-algebra C*(G) onto Co(G). C.4 The Imprimitivity Theorem Let H be a closed subgroup of a locally compact group G. We want to construct a Hilbert C*(iJ)-module X from CC(G), and to identify the imprimitivity algebra /C(X). The main result says that /C(X) is isomorphic to an algebra built from ^0 •= CC(G x G/H) in much the same way as C*(G) is built from CC(G), where £?o has multiplication and involution given by M(rr(F)) = UrM{F)Wr for all r G G and F G C0(G/H). We will show in the next Proposition that each covariant pair (M, U) determines a representation L = M x U of Eo via the formula L( Once we have this, we trivially have \\L( H0II := sup{ \\M x I7(0)|| : (M, U) is a covariant representation}. (C.17) The completion of E0 in this norm is a C*-algebra; we shall denote it by CQ(G/H)X\T G, because it is an example of the crossed products discussed in Section 7.1. Be cause the algebra is determined by the action of G on G/H, or equivalently by the transformation group (G,G/H), it is sometimes called a transformation group C*-algebra . We now give the promised proof that M x U is a *-homomorphism. The extra generality obtained by considering covariant representations with values in C(X) (called covariant homomorphisms) will be useful later. Proposition C.21. Let X be a Hilbert A-module. Suppose that u : G —> UC(X) is a strongly continuous homomorphism, and that M : CQ(G/H) —» C(X) is a homomorphism satisfying the covariance condition M(rr(F)) =urM(F)u*r for all r e G and F G Co(G/H). Then the map L : E0 —• C(X) defined by L(4>):= [ M(4>(rr))urdfi(r) JG extends to a homomorphism M x u ofCo(G/H) xr G into C(X). If M is nonde- generate, then so is M x u. Proof. It is not hard to see that r i—• 0(r, •) is continuous from G into Co(G/H), so r i—> M(cj){r, •)) is norm continuous. Since r i—• ur is strictly continuous, the function r i—> M(0(r,-))txr belongs to Cc(G, Ms(JC(X))), and the integral defining L(4>) makes sense by Lemma C.ll. To see that L is a homomorphism on EQ, let ^, (j) G E'o- Since point evaluation is a homomorphism on Co(G/H), it follows from Lemma C.3 that in C0(G/H) 1 <\> * ^(s, •) = [ 0(r, -)rr (^(r" ^ •)) d^(r). (C.18) C.4 The Imprimitivity Theorem 287 Therefore (a; , L( 2 p«L(0)(x) , L(0)Or))J = (iiW(a:) | Li(M*))„ < \\4>\\ {x | *)p = IMlM<*,*>J. The required estimate follows because this holds for all states p. Now suppose that M is nondegenerate. To show that L = M x u is non- degenerate, it suffices to see that M(CQ(G/H)) • X c L(CC(G x G/ff)) • X. Fix F G Co(G/H), x G X, and e > 0. Let iV be a neighbourhood of e in G such that ||Mr0*0 - z|U < ^IIFH^1 for r e N. Choose g e C+(G) such that supp(#) C N and fGgdp = 1. Then if we define 4> 6 CC(G x G/H) by (r, s) := g(r)F(s), we have ||L(0)(x) - M(F)(x)\\ = II / M(F)s(r)(^(z) - x) dp(r) UJG as required. • Remark C.22. In Proposition C.30, we will show that every representation of Co(G/H)y\TG is the integrated form of a covariant representation, so that (M, U) i—> M xi U is a one-to-one correpondence between the covariant representations of (CO(G/H),G,T) and the nondegenerate representations of CQ{G/H) xr G. Our main theorem concerns a closed subgroup H of a locally compact group G. Recall that we are writing fi and v for Haar measures on G and iJ, and A and 6 for the modular functions. In addition, we define j(t) = (A(t)/<5(£)) 2 for t G H. 288 The Imprimitivity Theorem Theorem C.23. View E0 = CC(G x G/H) and B0 = CC(H) as *-subalgebras of the C*-algebras E = C0(G/H) xr G and B = C*(H), respectively. Let X0 = CC(G). For f,g € Xo, b G Bo, and (p € EQ, define / • 6 = / /OOA^hWfeW dv(t), (C.19) JH (f , *)fl (*) - 7(0 / W)9{rt) dfi(r) (C.20) 0 • /(s) - / 0(r, ^/(r-^) dM(r), and (C.21) JG (f , T/ien Xo is an EQ - Bo-pre-imprimitivity bimodule. The completion X is a C(G/H) xir G -C*(H)-imprimitivity bimodule, and C(G/H) xr G and C*(H) are Morita equivalent. Just as for CC(G), convergence in the inductive limit topology in EQ implies convergence in the || • || i-norm, and therefore in the C*-norm. The following technical lemma on convergence in the inductive limit topology will be useful. Lemma C.24. Suppose that {4>N,K } is a net of functions in EQ, indexed by de creasing neighbourhoods of e G G and increasing compact subsets K of G/H, satis fying Proof Fix Then if N C V and K D N • K2, we have supp(^jv,K * 0) C VQKI x VQ • K2 and |0JV,K * (f>(r,s) — 0(r, s)| < e everywhere. A similar argument gives the second part. • Lemma C.25. There is a net {(pm}meM of functions in EQ such that each m is a finite sum of the form J27=i (/T1 > fi1) for fT ^ *o? and such that for each Eo /GX0 and (/) e EQ Similarly, there is a net {bj }J€J of functions in Bo of the form bj = (gj , gj) Bo such that for each / GXQ and b G Bo, f • bj —> / and bj * 6 —> b in the appropriate inductive limit topologies. m Proof We construct a net { 4>N,K } EQ satisfying (C.23) and (C.24) consisting of functions of the required form; then the previous Lemma completes the proof of the first part. Fix a neighbourhood N of e and a compact subset K of G/H. Let q : G —> G/H be the quotient map, and choose a compact subset C of G with q(C) = K. Using Proposition C.l, we can find a cut-down Bruhat approximate cross-section h G C+(G) such that / h(st) dv(t) = 1 for all s G CH. JH Let D — supp(/i), and let V be a symmetric neighbourhood of e G G such that 2 5 V C iV. Then there are s\,..., sn G G such that D C UILi ^ *- Multiplying by a partition of unity subordinate to the finite cover { Dc } U { Vs^ } gives functions hi G C+(G) such that supp(/ii) C Vs; and V"1 / fc^st) di/(*) = 1 for all s G CH. _1 Let Ci — {JG h{(r) d/x(r)) and take fa-.— c/ hi. Then P Therefore (fa , fa)(r, s) = 0 if r £ V2. Furthermore, Eo £* [ B(fi>fi)(r>*)Mr) E ^^*=i JG ° n 1 1 = y^ / / hi(st)cihi(r- st)A(r- st)di'(t)dtJL(r) ^t=i JG JH = XT_ / h*(st) / Q^(^_15t)A(r_1st)d/x(r)di/(t) = Y.n , I hi(8t) I Cihi(r)dtJL(r)dv(t) ^l=x JH JG which equals 1 when s G Cfi, or equivalent ly when s G if. Thus we can take 0iv,K := ^ (fa , /i>. E0 The construction of the functions bj is similar, but considerably less messy. • 290 The Imprimitivity Theorem Proof of Theorem C.23. In view of Proposition 3.12, it will suffice to verify the axioms of Definition 3.9. Straightforward computations, using that A and 8 are homomorphisms into an abelian group, show that f-(b*c) = (f-b)-c, (i>*4>)- f = ip- (•/), and 4>-(f •*) = (• f)-b for all / G Xo, 0, ip G E$, and 6,cGBo. In other words, Xo is an EQ - ^o-bimodule. Similar computations show that condition (d) of Definition 3.9 holds. To verify condition (a) of Definition 3.9, we do more calculations. It is relatively easy to check that (• , •) and (• , •) satisfy the requirements of Lemma 2.16, with the exception of the positivity (condition (d) of Definition 2.1), which is rather subtle. For it, we observe that (• , •) is continuous in the inductive limit topology: if fi -^ f m the inductive limit topology in XQ, then for any g G XQ, (fi , g) —> E0 (f , g) in the inductive limit topology, and therefore in the C*-norm, on EQ. A E0 similar statement holds for (• , •) . If we now fix / G XQ and take { 0m } as in Bo Lemma C.25, then (C.25) implies that in the C*-norm nm (/,/) =lim<^-/,/) =lim£ < E(/r,/D-/,/)B X BQ m B0 rn ^—^%—l E0 'B0 = Hm^nm * • N B B B m *•—'i=l B0 'B0 rn ^—^%—\ B0 B0 Since elements of the form a* a are positive and (/ , /) is the norm limit of positive Bo elements, it follows that (/ , /) > 0 for all / G Xo. The proof of the positivity of Bo (• , •) is similar. This establishes condition (a) of Definition 3.9. EQ Condition (b) of Definition 3.9 is a consequence of (C.26) and its counterpart for BQ. Thus we need only verify condition (c): that the module actions are bounded. As in the previous paragraph, this will boil down to the continuity of the actions and inner products in the inductive limit topologies. For the moment, let Xi denote the completion of XQ as a Hilbert C*(iJ)-module, as in Lemma 2.16. We will usually suppress the map q : Xo —> Xi. Let F G Cb(G/H), define F • /(*) = F(s)f(s) 2 for / G X0, and observe that (F-f , g)B = (f , F-g)B . Thus if F0 := (\\F\\ l-FF)i in Cb(G/H), then \\F\\oo(f , f)B - 3c0(G/H)(F)(f) = F • / (really, Jc0(G/H)(F)(q(f)) = q(F • /), but we have agreed to suppress q). Since a computation shows that (r3(f) , rs(g)) = (/ , g) for all s G G and f,ge X0, setting JG(s)(f) := rs{f) gives a strongly continuous homomorphism JG ' G • UC(Xi). The pair (JC0(G/H)->3G) is covariant in the sense of Proposition C.21: Jc0(G/H)(TS(F)) = JG(s)jc0(G/H)(F)jG(sy for all 3 G G and F € CoCG/ff). Therefore there is a homomorphism L := JC0(G/H) * Jc of CQ{G/H) XT G into £(Xi), which is automatically norm-decreasing. Thus by Corollary 2.22 we have* 2 To show that 2 (^•/, (g , L(M/)>fl = / fa , jCo(G/H) (0(r, -))iG(r)(/)>fl ^(r). (C.28) J G The integrand of (C.28) is Cc(iJ)-valued and continuous in r with respect to the inductive limit topology. By Lemma C.15, (C.28) is in CC(H) and for t G H we have (g , L( = (g, Thus (g , L((j))(f)) = {g , 4> • /) as desired, and we have established (C.27). 1 B B0 It remains to verify that for b G Bo and / G Xo we have (x-b,x-b)<\\b\\2 (/,/). (C.30) * We've used (• , •) in place of (• , •) to emphasize that, a priori, L( We proceed exactly as above. Let X2 be the completion of Xo as a Hilbert CQ(G/H) xiT G-module. (Once we verify (C.30), then Xi and X2 will coincide with X by Proposition 3.11.) We define a right action of H on XQ by 1 1 f-ut(s) = f(st- )A(t- h(t). Note that t >—> / • ut is continuous for the inductive limit topology, and / • uth(s) = /(sh-H-^Aih-H-'hiht) 1 1 = (/ • «t)(^- )A(/i- )7(^) = (/ • «t) • «/,(«). Furthermore, for each p £ H we have l l l 1 Af">h,9' up)(r, s)= f f{stp' )g{r-^stp-^)6{p- )^{r- stp- ) du(t) = I f(st)g{r-1st)&(r-1st) dv(t) JH = E(f,9)(r,s). Thus if we view operators in £(X2) as operators on the right of X2 (so that x(TS) = (xT)S), p 1—>• Up is a strongly continuous homomorphism of H into C/£(X2). By Proposition C.17, R(b) := / b{t)utdv{t) -/JH. defines a homomorphism of C*{H) into £(X2). A computation similar to that in (C.29) shows that (fR(b) ,g) = Af-b,g) for all f,g G X0 and b e S0, E EQ and then (C.30) follows from ||ii(6)|| < ||b||. D In the course of proving Theorem C.23, we established the following result which will be useful in the next section. Corollary C.26. If H is a closed subgroup of a locally compact group G, then there is a covariant homomorphism (JC0(G/H)I3G) of (CQ(G/H),G) into C(X) charac terized by 1 JG(r)(f)(s)=Tr(f)(s) = f(r- s), and (C.31) Jco{G/H)(F)(f)(s) = F(s)f(s) (C.32) for f € CC{G), r,s€G, and F e C0{G/H). C.5 Induced Representations of Groups 293 C.5 Induced Representations of Groups Given a unitary representation V : H —> B(H,v), we want to build a unitary representation of G, using the bimodule X = E^C*(H) constructed in the previous section and the machinery of Section 2.4. By Corollary C.26, / >—> rs(f) defines a strongly continuous homomorphism JG •' G —> UC(X). The representation of G induced from V is the representation lndH V obtained via this action of G on X, through the natural left action of £(X) on the Hilbert space X ®c*(H) Wy of X-Ind7Ty. In other words, (Indg V)s:={ X-Ind^ nv) (jG(s)). (C.33) Thus the induced representation is given on elementary tensors / (g> h £ X0 0 Hy by (Ind% V)s(f®h)=rs(f)®h. Our first goal in this section is to characterize the representations of G which are induced from a fixed subgroup H. The following version of Mackey's imprimitivity theorem was first stated by Rieffel [147], building on ideas of Glimm [58]. Theorem C.27 (Mackey's Imprimitivity Theorem). Let H be a closed sub group of a locally compact group G. Suppose that U is a unitary representation of G on TLu- Then U is unitarily equivalent to a representation induced from H if and only if there is a representation M of CQ{G/H) on TLu such that (M, U) is a covariant representation of (CQ(G/H),G,T). Theorem C.27 will follow directly from Theorem C.23 and some basic results concerning representations of CQ{G/H) XT G. The first of these is the following. Lemma C.28. If H is a closed subgroup of a locally compact group G, there is a covariant homomorphism {ICQ{G/H)^G) °f (Co(G/H), G,T) into M(CQ{G/H) xr G) characterized by l l iG{r){(t))(t,s) = Proof In view of Theorem C.23, the left action of Co{G/H) xT G on X defines an isomorphism L of CQ{G/H) XT G into £(X) with range /C(X) (Proposition 3.8). Therefore L extends to an isomorphism of M{CQ(G/H) xr G) onto C(X) (Cor ollary 2.54). Thus the covariant homomorphism (JC0(G/H)IJG) °f Corollary C.26 gives us a covariant homomorphism {IC0(G/H)^G) into M{CQ{G/H) XT G) such that L(iG{r) L^CoiG/G/H){F)4>) =jCo(G/H)(F)L( Since L(){f) = <\> • / for 0 G CC(G x G/H) and / G CC(G), we have jG(r)L(0)(/) = rr(0./). Since JG = y,0(r-4,r-1-5)/(r-1.5)dM(t), (C.34) follows. A similar argument establishes (C.35). D The next result is the analogue of Corollary C.16 for CQ(G/H) xr G. Lemma C.29. Suppose that H is a closed subgroup of G and 0, ijj G CC(G x G/H). Then as elements ofCo(G/H) xr G, I ic0(G/H) ( s S Proof. On one hand, we can view s \-+ f(s) := ic0{G/H){(t>( , '))^G( )(^) as an element of CC(G, C0(G/H) xr G), and then 0:= [ f(s)d»{s) JG defines an element # of Co(G/H) xr G by Lemma C.3. On the other hand, we can also view / as taking values in CC(G x G/H), and define a function 9 G Cc(Gx G/H) by 9(t,r): = / iCo(G/i/)(0(s,-))iG(s)(^)(^r)d/i(s) = / 0(s,r)^(s~"4,s_1 • r)dfi(s) JG = cf>* ip(t,r). To prove the lemma, we need to show that 9 = 0 in CQ(G/H) XT G. TO do this, we use a variation of the proof of Lemma C.15. If F G CC(G x G x G/H), then (*,r)»- [ F(s,t,r)dn(s) JG is an element of GC(G x G/H). Composing with the inclusion of CC{G x G/H) into C0(G/H) xr G gives a linear map $ : GC(G x G x G/#) -> C0(G/H) xr G. Viewing 5 i—> F(s, •,•) as an element of CC(G,CQ(G/H) xr G), we get a linear map * : CC(G x G x G/ff) -• C0(G/H) xT G via Lemma C.3. The maps $ and ^ agree on functions of the form F(s,t,r) = g(s)k(t,r) for g G CC(G) and A: G GC(G x G/H). Just as in the proof of Lemma C.15, $ are \I> can be shown to be continuous with respect to the inductive limit topology on GC(G x G x G/H); thus $ and \I> agree on all of GC(G x G x G/H). The equality of 9 and 9 now follows by taking F(s, t, r) — (s, r)ip(s~1t, s"1 - r). D C.5 Induced Representations of Groups 295 We can now prove that every representation of CQ(G/H) XT G is the integrated form of a covariant representation. Proposition C.30. Suppose that L is a nondegenerate representation of L L Co(G/H) xT G on HL- Then there is a unique covariant representation (M , U ) L L of (C0(G/H),G,T) on HL such that L = M x U . Furthermore, L M (F):=L(iCo{G/H)(F)) and (C.37) U^:=L(iG(s)). (C.38) Proof Define ML and UL by (C.37) and (C.38), respectively. Using the strong L continuity of iG and the nondegeneracy of L, it is not hard to verify that U is L L strongly continuous. Then the covariance of {IC0{G/H)^G) implies that (M ,U ) is a covariant representation. Applying L to both sides of (C.36) shows that L L M x U ((f))L(ij) = L(0)Z#) for all 0,^ e CC(G x G/H). Since L is nondegenerate, it follows that L = ML x UL, as claimed. On the other hand, if L = M x U and 0 G CC(G x G/ff), then L(iG(r))L() = L(iG(r) = J UrM(${s,.))Usdii{s) JG = UrL( Since a similar computation shows that L(ic0(G/H){F))L{(l>) — M(F)L(0), the uniqueness assertion follows from the nondegeneracy of L. • Corollary C.31. Suppose that V is a unitary representation of a closed subgroup H of a locally compact group G. Let L = X-Ind7ry be the corresponding representa tion ofCo(G/H) xT G. Then L is the integrated form of a covariant representation (ML,UL) with UL = Indg V. Proof This follows from (C.33) and (C.38), because the identification of M{C0(G/H) xr G) with £(X) takes iG to jG. D Mackey's Imprimitivity Theorem now follows easily: Proof of Theorem C.27. If U is induced, then the existence of a representation M such that (M, U) is covariant follows from Corollary C.31. The "only if" part of the theorem follows. On the other hand, if (M, U) is covariant, then M x U is a nondegenerate representation of CQ(G/H) XT G. Since the latter is Morita equivalent to C*(H) 296 The Imprimitivity Theorem via X, M xi U is equivalent to X-Ind n for some nondegenerate representation TT of C*(H) (Theorem 3.29). But n = ny for some unitary representation V of H (Prop osition C.19), and U is equivalent to Ind^ V by Corollary C.31 and the uniqueness assertion in Proposition C.30. • It is sometimes convenient to have a more concrete description of lndH V and the space on which it acts. This is provided by Mackey's original construction of induced representations of second countable locally compact groups, which was extended to general locally compact groups by Blattner. (Details and references can be found in the Introduction and Chapter XI of [54].) We will give their description and prove that it is equivalent to ours. Suppose that if is a closed subgroup of a locally compact group G, and that V is a unitary representation of H on Hy. Let 1 Vi := {£ G Ch{G,Hv) : £{rt) = Vf (£(r)) for all r G G and teH}. Notice that if f e Vi, then \\£(rt)\\ = ||f (r)|| for t G H, and hence r »-> ||f (r)|| is a well-defined function on G/H. We set Vc := {£ G Vi : r h- ||£(r)|| belongs to CC(G/H) }. Let a be a quasi-invariant measure on G/H such as that constructed in Lemma C.2. Then (£ \V)~ [ (£(0 I V(r)) da(r) (C.39) JG/H is a well-defined positive-definite sesquilinear form on Vc. The completion V := Ly(G,a) is a Hilbert space. Hooptedoodle C.32. We use the notation V := Ly(G, a) because V can be realized as a space of (equivalence classes) of functions on G. Precisely, each vector in V is a class of functions £ : G —> Hv such that £(r£) = ^_1(£(r)) almost everywhere and r »—• ||£(r)|| belongs to L2(G/H,a). Theorem C.33. Suppose that H is a closed subgroup of a locally compact group G, and that V is a unitary representation of H on Hv • Let p : G —> (0, oo) be a continuous function on G such that p(rt) = (8(t)/A(i))p(r) for all r G G and teH. Let a be the quasi-invariant measure on G/H constructed from p as in Lemma C.2, and let V = Ly(G,a) be the Hilbert space constructed above. Then 1 (Ua€)(r) := (p^-MMr))*^" *-) for s,r G G and £ G Vc extends to a unitary representation of G on V, which is unitarily equivalent to Indgy. Proof. We can assume that a has been chosen so that (C.3) holds with respect to Haar measures [i and v on G and iJ, respectively. The left invariance of v allows us to define W : X0 0 Hy -> Vc by (W(f 0 h)(r) \k):= [ p{rt)~"*/\rt)(Vth \ k) du(t) for all keHy. JH C.5 Induced Representations of Groups 297 Using that p(rt) = j(t) 2p(r) for t G i7, we can compute the inner product in (f®h\g®k) = (irv{(9,f)BQ)h\k) = I (9,f)B (t)(Vth\k)du(t) JH B° = / /\(t)W)f(rt)(Vth\k)dv(r)dv(t) JH JG l = [ [ [ l{t)Ws)f{rst)p{rs)- {Vth\k)dv{s)da{r)dv{t) JH JG/H JH = I I I l{s-H)Ws)f{rt)p{rs)-\Vs-Hh\k)dv{s)dv{t)da(T) JG/H JH JH = [ [ I ^(s-H)W^f(rt)p(rs)-\Vth\Vsk)dv{s)dv(t)da(r) JG/H JH JH = [ [ I {p(rt)-^f(rt)Vth\p(rs)-ig{rs)Vsk)du(s)du(t)da(r) JG/H JH JH = [ (W(f Thus W is isometric, and extends to an isometry of X 0c* (H) ^V into V. To see that W is surjective, and therefore unitary, fix £ G Vc and e > 0. It will suffice to produce rj in the range of W such that ||£ — 771| < e. Notice that the range of W is closed under multiplication by elements of CC(G/H). Let C be a compact set in G such that the support of £ is contained in the interior of CH. We first claim that it will suffice to find for each r G supp(£) Pi C a function £r in the image of W such that Il6-(r)-g(r)||< € (C.40) a(q(C))* If (C.40) holds, then there is a neighbourhood Nr of r in G such that Nr C C and \\Us) - £(*)|| < , * 1 for all * G Nr; (C.41) since the norms of functions in Vc are constant on if-cosets, (C.41) holds for all 5 G NrH. By compactness there are 7*1,... ,rn in (7 such that C C UlLi ^i; and a partition of unity argument gives functions \X{ G C+{G/H) such that supp(/Xj) C for a11 5 Then q(Nr.), 0 < fjii < 1, EiMi(s) = 1 for s G Cif and X^(s) < 1 - 2^i /^£n belongs to the range of W, and II En 1 *(*)&•<(*) " CWII < , * ,1 for all 5 G G. (C.42) ^"^=1 a(g(C))2 Since (C.42) vanishes off Cif, we have || ^2i ^£r. — £|| < e. This proves the claim. 298 The Imprimitivity Theorem Now let r E CD (supp£). Since s i—> Vsh is continuous for all h G Wy, there is a neighbourhood N of e in G such that rN C C and ||V$(£(r)) -£(r)|| < ea(g(C))~i for tGiV. If z e C+(G) has supp(z) c rN and / p(rt)-h(rt)dv(t) = 1, then £r = W(z(g)£(r)) satisfies (C.40). This completes the proof that VF is unitary. To finish off, we compute {W(hid%V)a(f®h)(r) | fc) = / pirtyifis^rtXVth \ k) du(t) JH = (^y~) * ^ pis-'rty^fis-'rt^Vth | fc) di/(t) = (U3W(f®h)\k), and the Theorem follows. D C.6 The St one-von Neumann Theorem When H is trivial, Theorem C.23 is an abstract formulation of a classical result known as either the von Neumann uniqueness theorem [153, §10.4] or the Stone-von Neumann theorem. This theorem had its origins in reconciling the Heisenberg and Schrodinger formulations of quantum mechanics; further references and discussion can be found in [143]. Theorem C.34 (Stone-von Neumann Theorem). Suppose that G is a locally compact group. Let A be the left-regular representation of G on L2(G) given by -1 2 Xsh(r) = /i(s r), and let M be the representation ofCo(G) on L (G) by pointwise multiplication: M(f)h(r) = f(r)h(r). Then (M, A) is a covariant representation of (Co(G),G) and M x A is a faithful irreducible representation of CQ{G) XT G with range K(L2{G)). After giving the proof of this result, we'll use it to derive something more closely approximating the classical formulation (Theorem C.38). Proof of Theorem C.34- A quick calculation shows that (M, A) is covariant, and 2 hence there is a corresponding representation M x A of C0(G) xr G on L (G). Theorem C.23 implies that C$(G) xrG is Morita equivalent to C; thus Theorem 3.22 implies that CQ(G) XT G is simple, and M x A must be faithful. Therefore it suffices 2 to see that M xi A maps CC(G x G) onto /C(L (G)); irreducibility will then follow because JC(H) acts irreducibly on H. If K GCc(GxG), then fK(s, t) := K(t, s-H^s-H) (C.43) C.6 The Stone-von Neumann Theorem 299 defines an element of CC(G x G), and 1 M x X(fK)h(s) = / fK{r,s)h{r- s)dli{r) JG 1 = [ fK(sr-\s)A(r- )h(r)d^(r) JG = f K(s,r)h(r)dfjL(r). JG Thus M x X(fK) is the Hilbert-Schmidt operator with kernel K e CC(G x G) C L2(G x G) [31, Chap. XI §6]. In particular, M x X(fx) is compact, and since the Hilbert-Schmidt operators are dense in the compact operators, the range of M x A 2 contains /C(L (G)). On the other hand, given / G CC(G x G), ^(s,i):=/(si-1,s)A(i-1) satisfies (C.43) with / = //<-. Therefore the range of M x A is precisely /C(L2(G)). • Remark C.35. Since every nondegenerate representation of the compact op erators is equivalent to a direct sum of copies of the identity representation (Lemma B.34), it follows that every nondegenerate representation of Go(G) xT G is a equivalent to a direct sum of copies of M x A. Hooptedoodle C.36. Theorem C.23 can also be used to prove that CQ{G/H) XT G is iso 2 morphic to C*(H) ®a JC(L (G/H, a)), where a is any quasi-invariant measure on G/H [63, 85]. However, the isomorphism is not natural when H is nontrivial. When G is abelian, Theorem C.34 has a more classical form. To see this, we first want to remark that representations of Go(G) xa G are determined by two unitary representations satisfying a certain commutation relation (C.46). Let G be the Pontryagin dual of G. After identifying the dual of G with G via the Pontryagin Duality Theorem [55, Theorem 4.31], Co(G) can be identified with the the group G*-algebra C*{G) as in Example C.20: the Fourier transform F : CC(G) -> C0(G) defined by F(4>)(s)= f #7)7(s) TT(F(0)) = / #7)S?di/(7), and (C.44) JG S;=7t(F(is(7))), (C.45) where F is the extension of F to M(C*(G)). (It is not hard to check that the extension F of F to M(C*(G)) takes the generator 2^(7) to the function 7 G Cb{G) = M(C0(G)).) 300 The Imprimitivity Theorem A pair (i?, S) consisting of a unitary representation R of G on H and a unitary representation S of G on H is called a Heisenberg representation of G if S^Rs = 7(s)i?s57 for all s e G and 7 e G. (C.46) Example C.37. Define F : G -> *7L2(G) by Kyft(r) := -y(r)h(r). Then (A, V) is a Heisenberg representation of G on L2{G) called the Schrodinger representation. If 0 G CC(G) and s e G, define s • 0 by s • ^(7) := 7(5)^(7). Then F(s • 0)(r) = F(0)(5-V) = rs(F(0))(r). Thus if (R, S) is a Heisenberg representation of G on H and if ^5 is the integrated form of S, then it follows from (C.44) and Lemma C.ll that R3ir(F( = TT(TS(F^)))RS; in other words, (TTS,R) is a covariant representation of (Go(G),G, r). Conversely, suppose that (TT,R) is a covariant representation of (Go(G),G,r), and that TT is the integrated form of S = S^, so that S is given by (C.45). Then a computation shows that F(ig(7)(s.^))(r) =1(s)F(id(1) 57JRSTT(JF(0)) = 577r(rs(F(«/))))JRs = S7TT(F(S • 0))fls = 7r(F(ig(7)S • «/>))i?s = 7(s)7r(rs(F(id(1)4>)))Rs = 1(s)Rsn(F{iQ(7)4>)) = 1(s)RsS17r(F((j))). Since 7r is nondegenerate, this implies that (i?, S) is a Heisenberg representation. We deduce that (R,S) is a Heisenberg representation if and only if (TTS,R) is a covariant representation of (Go(G),G, r). Finally, notice that if /C C TL is an invariant subspace for n xi R, then /C is invariant for both TT and R. (This follows from Proposition C.30.) The space /C is also invariant for the unitary representation Sn corresponding to TT. Thus a direct- sum decomposition of TT XI R gives a direct-sum decomposition of the corresponding Heisenberg representation. Since any nondegenerate representation TT XI R is equiv alent to a direct sum of copies of M x A (Remark C.35), and since it is easy to see that SM = V, we have proved the classical St one-von Neumann uniqueness theorem. C.6 The St one-von Neumann Theorem 301 Theorem C.38 (von Neumann Uniqueness Theorem). Suppose that G is a locally compact abelian group. Then every Heisenberg representation of G is equiv alent to a direct sum of copies of the Schrodinger representation of G on L2(G). Appendix D Miscellany D.l Direct Limits We will need to use direct limits of groups to define sheaf cohomology groups. The definition itself is a bit messy, so we stress that a direct limit should be thought of as a generalized union. Indeed, the idea is to map all the groups to one large group, and then take the union of the images; the rub is that elements in different groups must be declared equal when their images coincide. More precisely, let A be a directed set. A directed system of groups on A is a family of groups { G\ }AGA together with homomorphisms T\@ : G\ —> Gp for each pair of indices such that A < /3, which satisfy T\\ = id for all A and r^ o r\p = r\1 whenever A < (3 < 7. A family of homomorphisms h\ : G\ —> H is said to be compatible if commutes for every pair satisfying A < /?. An inductive limit of the system { G\, r\/3 } is a group G together with a compatible family of homomorphisms rA : G\ —* G such that, whenever { h\ } is a compatible family of homomorphisms into H, there is a unique homomorphism h : G —+ H such that Gx >G h\ \ / h H commutes. Every directed system has a direct limit, and it is unique up to isomorphism. We denote it by lim G\. AGA 303 304 Miscellany The direct limit can be directly constructed as the quotient of a subgroup of Y\x Gx, as follows. Let G be the subgroup of those elements which eventually satisfy T\p(g\) =9(3 for A F := { x e G : xx = 0 for large A }. Then G/F is a direct limit for the system GA, with rA : GA —» G —• G/if defined by TA/3^) if A < /3, and TA( A < 7, P < 7 and TA7(#) = T^O')- A standard observation about direct limits is the following. Suppose that { Hx, px(3 } is another directed system of groups indexed by the same directed set A, and there are homomorphisms hx : GA —•> #A such that TA/3 GA G5 #A H, Pxp & commutes for every A < j3. Then we have a commutative diagram TX0 G, Gp H H. x PA/3 P p \ / P" limi^A whenever A < /?, so /ij := px ohi defines a compatible family of homomorphisms on { GA, TX(3 }, and there is a unique homomorphism h : HIUGA —• limi^x characterized byh(Tx(g))=px{hx(g)). ~* ~^ D.2 The Inductive Limit Topology Throughout this section, X will be a fixed locally compact HausdorfT space. For K C X, we let Cc(X;liO := {/ e CC(X) : supp/ C if}. D.2 The Inductive Limit Topology 305 equipped with the norm topology coming from the supremum norm. When K is compact, CC(X; K) is a Banach space. If JC(X) denotes the collection of compact subsets of X, then CC(X)= |J CC(X;K). KefC(x) Proposition D.l (see [54, Proposition II. 14.3]). There is a largest topology on CC(X) with respect to which (1) CC(X) is a locally convex linear space, and c (2) the inclusions LK : CC(X;K) -> CC(X) are continuous for all K G K,(X). This topology has the following universal property (P); a linear map F from CC(X) into a locally convex space M is continuous if and only if FOLK is continuous for every compact subset K of X. Remark D.2. Notice that it is not immediate that there is a unique topology which contains all the topologies satisfying (1) and (2): the existence of such a topology is part of the result. Definition D.3. If X is a locally compact Hausdorff space, then the topology on CC(X) given in Proposition D.l is called the inductive limit topology on CC(X). Remark D.4. The final topology for the family of inclusion maps LK is the unique largest topology making the inclusion maps continuous [125, §1.4.9]; in this topol ogy, A C CC(X) is open if and only if L^~{A) is open in CC(X; K) for all K G K(X). This is the "inductive limit topology" in the category of topological spaces and con tinuous maps. It is not clear under which circumstances this topology coincides with the inductive limit topology defined above. However, it will follow from Proposi tion D.7 that the final topology always contains the inductive limit topology. Property (P) implies that, if { f\ }AGA is a net which converges uniformly to /, and if the supports supp f\ are eventually contained in a fixed compact set K, then f\ —> / in the inductive limit topology. Regrettably, the converse fails for nets even when X = Z (see Example D.9), and only holds for sequences when X is cr-compact. (See the discussion in [41, §6.6], and note that a convergent sequence is always bounded.) The following observation is a direct consequence of property (P). Lemma D.5. Suppose that X is a locally compact Hausdorff space and that fi is any Borel measure on X assigning finite measure to all compact sets of X. Then 1 the inclusion of CC(X) in L (X,JA) is continuous with respect to the inductive limit topology. 1 Since the C*-norm on CC(G) is dominated by the L -norm, we have an imme diate corollary: Corollary D.6. The inclusion of CC(G) into C*(G) is continuous with respect to the inductive limit topology. 306 Miscellany The next proposition is based on Theorems 6.4 & 6.5 of [152], and requires some terminology. Recall that a subset S of a (complex) vector space V is convex if tS + (1 - i)S C S for all 0 < t < 1. We say that S is balanced if zS C S for all z G C with \z\ < 1. A collection of subsets (3 is a /oca/ 6ase for a topology on V if {v + W : v e V, W e /3} is a, base for the topology on V. A vector space V with a T\-topology is called a topological vector space if the operations of vector addition and scalar multiplication are continuous. Such a space V is an abelian topological group with respect to vector addition, so V is necessarily Hausdorff and any neighbourhood base at 0 is a local base for the topology. If V has a local base consisting of convex sets, then V is called locally convex. Thus normed vector spaces are are in particular locally convex topological vector spaces. Proposition D.l is an immediate corollary of the following. Proposition D.7. Suppose that X is a locally compact Hausdorff space. (a) If (3 is the collection of balanced convex sets W in CC(X) such that tJ^(W) is open in CC(X;K) for all compact subsets K of X, then /3 is a local base for a topology r on CC(X). (b) With respect to r, CC(X) is a locally convex topological vector space. (c) The inclusion LK '• CC(X;K) <^-> CC(X) is a homeomorphism onto its range for each compact set K in X. (d) (Property (P)) For any locally convex topological vector space M and any linear map F : CC(X) —> M, F is continuous if and only if FOLK is continuous for allK eJC(X). (e) Suppose a is a topology on CC(X) for which CC(X) is a locally convex topolog ical space and the inclusions LK are continuous for all compact subsets K of X. Then a is contained inr. In particular, r is the unique strongest topology on CC(X) such that (1) and (2) of Proposition D.l are satisfied. Proof. Let r denote the collection of sets in CC(X) which are unions of sets of the form / + W with / E CC(X) and W G /3. We proceed by first showing that r is a locally convex topology on CC(X) under which it is a topological vector space. We then show that (CC(X),T) satisfies (b)-(e). To see that r is a topology, it will suffice to see that if V\ and V2 are elements of r and if / G V\ fl V2, then there is a W G /3 such that f + w cv1nv2. (D.l) But for i = 1, 2, there must be fi G CC(X) and Wi G /3 such that fefi + WiCVi. Choose a compact set K containing the supports of /1, /2, and /. Then for i = 1, 2, we have / — fi G Wi D CC(X;K). Since the sets Wi are open, there exist Si > 0 such that / - fi G (1 - 6i)Wu D.2 The Inductive Limit Topology 307 and since each Wi is convex, / + SiWi = /» + (/- fi) + SiWi C fi + (1 - fiOWi + 6iWi Cfi + WiCVi. Therefore (D.l) holds with W := 81W1nS2W2. This establishes (a). The r-continuity of addition is an easy consequence of the convexity of the W's: l (f + 2W) + (g + ±W) C (f + g) + W. The r-continuity of scalar multiplication is a bit messy and follows because each W G (3 is balanced. To see this, fix z0 G C, /0 G CC{X) and W e (3. Since / G CC(X; if) for some K and since WnCc(X; K) is a neighbourhood of 0 in CC(X; if), there exists 6 > 0 such that <5/o G |VF. Choose c such that 2c(|zo| + S) = 1. Now if / - /o G W and |z - zo| < 5, then */ " *o/o = Z(f - /o) + (2 - *0)/0 G 2C^ + \W C c(« + \z0\)W + ±W C ±W + \W C TV. Thus scalar multiplication is continuous. If f,g G CC(X), then TV := {h G CC(X) : \\h\loo < \\f - g]^ } is an element of (3 and f ^ g + W. Thus the singleton { / } is r-closed, and r is a T\-topology. This proves (b). Let TK denote the norm topology on CC(X;K). Suppose that V G r and that / G V C\CC(X;K). The definition of r implies that there exists W G (3 such that f + W CV. Thus {f + W)nCc(X;K) c FnCc(X;K). Since WnCc(X;K) G rK, we deduce that V nCc(X;K) £TK- This proves that each LK is continuous. To see that LK is a homeomorphism, it suffices to see that the relative topology on CC(X;K) coincides with TK' if E G TK, we need to find V G r such that f E = 7nCc(I;K). But if / G £, there exists 6 > 0 such that B*/(/) := {^ € Cc(X;if) : \\g - /|U < ^ } C E. f Since W/ := {g G CC(X) : \\g\\oo < 8 } G /?, and since (/ + W^) H CC(X; if) = / + (W^ H CC(X; if)) C £7, it follows that V := Ufe£ / + Wf has the desired property. This proves (c). Next we show that r has property (P). Let M be a locally convex topological vector space and F : CC{X) —> M a linear map. Suppose that F o LK is continuous for all if G /C(X). Then F_1(V) G /3 for every convex neighbourhood V of 0 in M. Therefore F is continuous with respect to r. The converse is clear, and (d) follows. Finally, suppose that a is any other topology with the prescribed properties. Then property (P) implies that the identity map id : (CC{X),T) —> (Cc(X),a) is continuous. Thus a C r, as required. • 308 Miscellany Example D.8. Let $ := ZM denote the collection of all functions from Z to R+ ;= { x G E : x > 0 }. For each pej, let Wp:={f eCc(Z) : |/(n)| < p(n) for all n G Z}. Each Wp is a nonempty element of the local base /? for the inductive limit topology, and it is not hard to see that (30 := { Wp : p G $ } is itself a local base for the induc tive limit topology on CC(Z). In fact, if W G /?, the openness of W D Cc(Z; [—n, n]) allows us to define pG? inductively such that VFP C W. If { /A }IEA is a net in CC(X), then we say the net is eventually compactly sup ported if there is a compact set K and an index Ao such that supp f\ C K for all A > Ao- The next example shows that a convergent net in CC(X) need not be eventually compactly supported even if X = Z. Example D.9. Let 6n G CC(Z) be the indicator function of {n}. Note that A := # x Z+ is directed by (p, n) > (g, m) iff n > m and p(fc) > g(fc) for all k G Z. If -1 we define /(p,n) := p(n) <5n G CC(Z), then it is not hard to see that {f\ }AGA is eventually in every Wp belonging to the local base fio as defined in Example D.8. In other words, f\ —> 0 in the inductive limit topology on CC(Z). However, the net { fx }\eA is not eventually compactly supported. Suppose that Y and Z are topological spaces, and that F : X —» Y is a func tion. If { y\ }A Lemma D.10. Suppose that X is a locally compact Hausdorff space, that M is a locally convex linear space, and that F : CC(X) —> M is linear. Then F is continuous if and only if F maps eventually compactly supported convergent nets in CC(X) to convergent nets in M. Proof. Recall that F is continuous if and only if F o iK is continuous for all K G IC(X) (Property (P)). For the latter, it certainly suffices to check only nets in CC(X) which are eventually compactly supported. • Remark D.ll. One can still wonder if there is a topology on CC(X) for which the uniformly convergent eventually compactly supported nets are precisely the con vergent nets. In fact, there are criteria — given in Theorem 9 of [86, Chap. 2] — on a collection of nets with proscribed limits (a convergence class), which charac terize those which are the collection of convergent nets for some topology. It is not hard to see that the uniformly convergent eventually compactly supported nets fail condition (d) on page 74 of [86]. Index A, 210 C%t(U,S), 87-88 A 260 G£re([/,T), 75 ^A, 9, 11 C*(G), 15, 281 K(^0, 19 C*(G), 281 A1, 24 d", 74 AdU, 4 deg/, 86 AF, 120 0„ 89 i4(i/), 139 6{A), 127 A|j(t/), 138 6n, 184 A »„ B, 242 6n, 188 i4(t), 116 A, 79 AQB, 236 /*, 162 A®TB, 155 /M, 161 A(t/), 123 0, 69 Au, 120 £([/), 68 Aut A, 3, 141 G, 70 is Polish, 190 7(£), 287 AutCo(T) A, 141 T(X), 110 AutCo(T) G0(T, £(W)), 75 h(I), 217 4(17, i/), 125 ftx, 60 F a , 142 HA, 13, 65, 144, 150 a., 141 H\Z,g),^r 2 a®T/3, 159 H (G,T), 184 [21/j.], 240 #n(G,A), 185 21<8>CQ3, 238 ^"(G,i4),188 n G B(W)*.S, 29 H {T;Z) , 196 Bn{U,S), 74 Hn{U,S), 74 n C0(T,/C(W)),6, 215 # (X, 309 310 Index Ind7r, 35 X, 49 Inde 7T, 36 Xc*(if),269 Indg(i4,a), 163 XF, 120 k(F), 217 X-Ind^ K, 143 map on ideals, 54 K(H), 1 X-Ind^ Tr, 36 is a C*-algebra, 2 X-Ind7r, 36 /C(X), 18 Xj, 53, 55 role in Morita equivalence, 43 KX, 53 £(X,Y), 18 X^BHTT, 34 P(s,n), 32 X ®B Y, 48 L2(T,H;dn), 32 X • J, 53 £(X), 17 ZM(A), 224 nondegenerate homomorphism C(<*), 75 n into, 26 Z (*7, 7T(z,t)5 138 action on a space, 97 Prim A, 212 action on the spectrum, 174 Prim A/1, 216 adjoining an identity, 24 Prim/, 216 adjointable operators, 17 Q(A), 230 acting on X, 30 P0T, 241 almost Hausdorff, 231 pv,t/, 68, 69 a-continuous, 257 S", 262 automorphism, 141 5, 71 fixes Dixmier-Douady class, 162 a(a), 219 homeomorphism induced by, 141 supp(/), 84 Baer sum, 186 tr, 121 Baire property, 230 Z7(W), 3 balanced tensor product, 155 contractible in norm topology, Banach .A-module, 22 103 nondegenerate, 22 contractible strong topology, 102 Borel measure, 269 UM(A) boundary operator, 89 is Polish, 191 Brown-Green-Rieffel Theorem, 63, U, 102 150 Index 311 Bruhat approximate cross-section, conjugate Hilbert space, 19 271 conjugate vector space, 49 cut-down, 273 connecting homomorphism, 79, 80 bundle, see fibre bundle naturality of, 81 constant presheaf, see presheaf, C0(T)-algebras, 118, 195 constant Calkin algebra, 221 constant sheaf, see sheaf, constant categorical Morita equivalence, 59 continuous fields of C*-algebras, 114 Cauchy-Schwarz inequality, 9 continuous fields of Hilbert spaces, for states, 203 12, 114 CCR, 221 continuous-trace C*-algebra, 121 coboundary, 74 is CCR, 122 coboundary map, 74 local rank-one projections, 121 cochain, 73 product of, 155 alternating, 87 continuous-trace element, 121 cochain complex, 75 convex combination, 208 homomorphism of, 77 convex set, 208, 306 short exact sequence of, 80 corner, 50 associated long exact complementary, 50 sequence, 81 full, 50 cochain homotopy, 78 count ably generated, 144 cocycle, 74 covariance condition, 175 values in nonabelian group, 97 covariant representation, 47, 175, 285 Cohen Factorisation, 22 cross-norm, 249 cohomology cross-section, 270 abstract sheaf, 73 crossed homomorphism, 185 Cech, 73 principal, 185 cup product, 168 crossed product, 175 functorial, 77 C*-norm, 246 functorial C*-seminorm, 249 Cech vs. topological, 88 cyclic projection, 240 group, 184 maximal family of, 240 long exact sequence in, 81 cyclic representation, 280 Moore, 188 cyclic vector, 204 theories agree on nice spaces, 89 cohomology group, 74 Dauns-Hofmann Theorem, 224 cohomology ring, 168 de Rham Theorem, 89 commutant, 240, 262 deck transformation, 101 compact operators, 1, 143 deformation, 124 irreducible representations, 210 diagram chase, 81 compact operators on X, 18 direct limit, 303 compactification, 25 compatible homomorphisms, complex line bundle, 97 303 concrete C*-algebras, 237 direct sum of C*-algebras, 220 conjugate actions, 177 Dixmier-Douady class, 127 conjugate algebra, 157 classification theorem, 131 312 Index invariant under automorphism, G-module, 184 162 G-space, 98 of a bundle, 109 GCR, 221 of a G*-algebra, 131 Ga-set, 230 stable isomorphism invariant, Gelfand-Naimark Theorem, 201, 203, 151 207 Dixmier-Douady classification germ, 70 theorem, see Glimm algebra, 223 Dixmier-Douady class GNS, 203 double centralizers, 26 GNS-construction, 204-206 Double Commutant Theorem, 262 GNS map, 227 dual action, 176 group action, 97 dual coaction, 177 group G*-algebra, 15, 280, 281 dual group, 176, 285 group extension, 185 dual module, 49, 50 central, 185 dynamical system, 175 equivalent, 185 classical, 175 topological, 186, 191 effective action, 98 Haar functional, 270 elementary abelian group, 192 Haar measure, 14, 270 elementary tensors, 30 Hahn-Banach Separation Theorem, equivalence of categories, 59 228 essential ideal, 24 Heisenberg representation, 300 essential subspace, 203 hereditary subalgebra, 232, 233 exact G*-algebra, 253 Hermitian line bundle, 97 exponential map, 71 Hermitian structure, 94 exterior equivalent, 177 Hewitt-Cohen Factorisation, 22 external tensor product, 62 Hilbert ^-module, 9, 11 extreme point, 208 complemented submodule, 144 countably generated, 144 factor representation, 263 full, 11 has prime kernel, 265 example not —, 12 faithful, 203 Hilbert bundle, 12 Fell's condition, 121 n-homogeneous G*-algebras, 114 fibre bundle, 92, 93 homogeneous space, 270 isomorphism of, 93 homomorphisms of G*-algebras, xi section of, 110 Hooptedoodle, xiii fibre product, 94 hull, 217 fibres, 92 hull-kernel topology, 213 fine sheaf, 85 finite-rank operator, 1 ideals in G*-algebras, xi, 212 flip isomorphism, 243 imprimitivity algebra, 18, 269 forgetful homomorphism, 192 of tensor product, 62 Fourier transform, 285 imprimitivity bimodule, 41-43 free action, 98 completion of pre —, 45 fundamental group, 101 isomorphism of, 57 Index 313 over T, 118, 119 locally trivial bundle, 92 induced algebra, 163 locally unitary, 192 induced homomorphism on long exact sequence in cohomology, cohomology, 162 81 induced representation, 33-38 functorial, 37 Mackey obstruction, 192 notations for, 36 Mackey's Imprimitivity Theorem, of groups, 38-40, 293-301 41, 293 of tensor product, 64 matrix unit, 245 on ideals, 38 maximal C*-norm, 249 preserves containment, see weak maximal tensor product, 249 containment minimal projection, 5 space of, 34 minimal tensor product, 257 inductive limit, see direct limit Mobius band, 93 inductive limit topology, 281 modular function, 270 inner automorphism, 5, 6 Moore cohomology group, 188 inner product A-module, 8 Morita equivalence automatically nondegenerate, 9 C*(H) and Co (G/H) x G, completion of pre —, 15 46-47, 293 norm on, 10 categorical, 59 integrated form, 283 complementary full corners, 50 internal tensor product, 48 equivalence relation, 50 intertwines, 205 equivalent categories of invariant subspace, 202 representations, 59 reducing, 203 implies stable isomorphism, 150 irreducible closed set, 231 lattice isomorphism of ideals, 53 irreducible representation, 202 Morita equivalence of, 55 isomorphism of imprimitivity linking algebra, 52 bimodules, 57 local, 122 over T, 118 Jacobson topology, 213 characterization of, 119 relative topology from X(A), 61 stronger than Morita equivalence, 162 Kasparov Stabilization Theorem, 144 preserved under tensor product, kernel, 217 62 ring theory, 41 lattice, 53 same as strong —, 47 left-regular representation, 281 spectrum preserving, 118 liminal, 221 weaker than isomorphism, 47 linking algebra, 22, 51, 52 Morita equivalent, 47 local sections multiple, 254 continuous, 100 multiplier algebra, 26 example no continuous —, 101 multipliers, 26 local trivializations, 93 localization, 120 net, 2, 277 locally finite refinement, 73 n-homogeneous C*-algebras, 114 314 Index nondegenerate homomorphism, 26 product state, 239 nondegenerate representation, 203 projection, 11, 121 norm-decreasing, 280 proper action, 98 normalized 2-cocycle, 186 translation map, 100 nuclear C*-algebra, 258 pull-back C*-algebra, 161 pure state, 208 open map characterization of, 166 quasi-invariant measure, 272 opposite algebra, 157 quasi-state space, 230 orbit, 98 orbit map, 98 reduced C*-norm, 281 orbit space, 98 reduced group C*-algebra, 281 outer conjugate, 177 reducing subspace, 203 refinement, 78 paracompact, 73 locally finite, 73 shrinking lemma, 82 refining map, 78 partition of unity, 84 regular measure, 269 Phillips-Raeburn obstruction, 192 representation, 201 point-norm topology, 3 direct sum, 202 pointwise unitary, 192 faithful, 203 polarization identity, 20 invariant subspace for, 202 Polish abelian G-module, 187 irreducible, 202 Polish group, 187 means *-preserving, xi Polish space, 187, 229 nondegenerate, 203 Pontryagin dual, see dual group unitary equivalence, 202 Pontryagin Duality Theorem, 176 representation group, 190 positive element, 203 restriction map, 61, 69 positive functional, 203 rho-function, 272 Cauchy-Schwarz inequality for, Rieffel correspondence, 54, 60 241 Morita equivalence of postliminal, 221 corresponding ideals and pre-imprimitivity bimodule, 44 quotients, 55 completion of, 45 restriction to Prim A, 60 pre-inner product Ao-module, 15 self compatible, 60 completion of, 15 Rieffel homeomorphism, 60 presheaf, 68 rotation algebra, 176 constant, 70 sheafification of, 71 saturation, 271 prime ideal, 212 Schrodinger representation, 300 is primitive, 231 separable topological space, 229 primitive ideal, 211 sheaf, 69 is prime, 212 constant, 70 primitive ideal space, 211, 214 sheafification of constant and ideal structure, 219 presheaf, 71 principal G-bundle, 97, 98 fine, 85 classification of, 102 global sections, 69, 73 Index 315 homomorphism of, 71 strictly positive element, 145 of germs of G-valued functions, strong Morita equivalence, see 69 Morita equivalence, strong notation for, 69 strong operator topology, 3 restriction maps, 69 strongly continuous, 276 section, 69, 71 homomorphism, 277 short exact sequence of, 72 strongly continuous action, 163 associated long exact structure group, 92 sequence, 81 subcross, 249 soft, 102, 104 sum of ideals, 224 stalk, 70 Symmetric Imprimitivity Theorem, stalk picture, 70 194 sheaf cohomology group, 79 sheaf homomorphism, 71 Takesaki-Takai Duality Theorem, short exact sequence of C*-algebras, 176 252 tensor products shrinking lemma, 82 algebraic (0) vs. complete (0), cr-continuous, 242 32 cr-unital, 145 associativity, 243, 251 simple, 214 £?-balanced, 48 simplex, 89 balanced, 34 vertices of, 89 Hilbert space, 31 simplicial endomorphism, 90 modules, see external and spatial norm, 242 internal tensor products faithful representation of, 242 representations of, 247 formula for, 241 irreducible, 254, 255 product states cr-continuous, 242 semi-norms always subcross, 249 spatial tensor product, 242 vector spaces, 30 is minimal, 257 Toeplitz algebra, 220 spectrum, 210, 214 topological group, 4 surjection onto Prim, 213 trace, 121 spectrum preserving, 118 transformation group, 176 stable C*-algebra, 144 transformation group C*-algebra, stably isomorphic, 63, 144 176, 286 implies Morita equivalence, 150 transition functions, 95 *-strong topology, 276 trivial bundle, 93 and strict topology, 276 Type I, 263 state, 203 associated to 7r, 228 UHF-algebra, 223 extension to A1, 205 unimodular, 14, 270 Stone-Cech compactification, 226 unitary 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Lemma B.31. Suppose <\> \ A —* B and ip : C —> D are homomorphisms of C*- algebras. Then there is a unique homomorphism <\> 0max ip • A 0max C —> B 0max ^ 51/c/i tftat (j) ®max ^(a 0 c) = 0(a) ® ^(c). Proof. Because is and Z£> have commuting ranges, so do Ho-a B) . For general C*- algebras, the same proof shows that it is a homeomorphism of A x B into {A 0a B) , and we will show that it is still surjective if either A or B is GCR. When A is GCR, 7r H-» ker n is a homeomorphism of A onto Prim A, and it doesn't make any difference whether we work with spectra or primitive ideal spaces. For more general classes of algebras, it does make a difference, and it is often easier to describe Prim(A 0^ B). We will show that, provided A or B is nuclear, we can identify ker(7r 0 77) with the ideal (ker ir) a B + A 0^ (ker 77), and deduce from the previous results that (P, Q) ^ P ®a B + A 0a Q is a homeomorphism of Prim A x Prim P into Prim(.A 0^ 5). Under the same nuclearity hypothesis, this homeomorphism is surjective, whereas the map on spectra need not be. Without the nuclearity hypothesis, the map on primitive ideal spaces need not be surjective either [165]. a B, and it is contained in ker7r. Let w : A 0a B —» (v4 0a B)/7 be the quotient map. It will suffice to show that for each ^-=1 o^ <8> 6j G A©5,