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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.
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