I World Scientific An Introduction to the Classification of Amenable C-Algebras An Introduction to the Classification of Amenable C-Algebras Huaxin Lin University of Oregon, USA V fe World Scientific wB NNeeww J Jeerrsseeyy * LLoonnddoonn '•S Simin gapore • Hong Kong • Bangalore Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. AN INTRODUCTION TO THE CLASSIFICATION OF AMENABLE C*-ALGEBRAS Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 981-02-4680-3 Printed in Singapore by Mainland Press To whom I love Preface The theory and applications of C*-algebras are related to such diverse fields as operator theory, group representations, topology, quantum me chanics, non-commutative geometry and dynamical systems. In light of the Gelfand transformation, the theory of C*-algebras is also regarded as non- commutative topology. Despite the great influence of this subject to other fields, the understanding of C*-algebras itself was very limited. About a decade ago, George A. Elliott initiated the program of classification of C*- algebras (up to isomorphism) by their if-theoretical data. It started with the classification of AT -algebras with real rank zero. Since then, great efforts have been made to classify amenable C*-algebras, a class of C*- algebras that appears most naturally. Large classes of simple amenable C*-algebras were discovered to be classifiable. With these rapid develop ment, the theory of C*-algebras becomes increasely important to many other fields. For example, the applications of these results to dynamical systems have been well established. The purpose of this book is to introduce some of the recent develop ments of the theory of classification of amenable C*-algebras to a broad range of readers including non-experts and graduate students. It is an am bitious plan. However, the material presented here has been limited by the author's knowledge as well as the page limitation of this volume. For ex ample, the aspects of classification of purely infinite simple C*-algebras which is quite complete are not mentioned in this volume. The author's effort was concentrated to finite C*-algebras. Even in this case, only simple C*-algebras with tracial topological rank zero are treated in detail. The first three chapters contain the basics of the theory of C*-algebras vii Vlll Preface which are particularly important to the theory of the classification of amenable C*-algebras. References to these three chapters include (but not limited to) [147], [173], [143] and [48]. Chapter 4 offers the classification of the so-called j4T-algebras of real rank zero. The results in Chapter 6 cover the results in Chapter 4, however, the proofs given in Chapter 4 are much more elementary. It is the author's intention to present the classification of simple AT-algebras of real rank zero with limited tools so non-experts and graduate students may be able to read it without advanced knowledge of C*-algebras and if-theory. The first four chapters and first 6 sections of Chapter 5 are self-contained. This part could serve as a text book for a graduate course on C*-algebras. Indeed the author used it for a graduate course in University of Oregon and a lecture series in East China Normal University. The last two chapters contain more advanced topics. In partic ular, they contain the classification theorem for simple AH-algebras with real rank zero, the work of Elliott and Gong. To achieve these goals in such a limited volume, the author was often forced to give some new proofs (to avoid introducing too much new concepts and materials). Starting Chapter 4, at the end of each chapter, brief remarks are inserted. The intention is to give the reader some rough idea of the development related to the ma terial presented there. They are bound to contain errors. The author asks forgiveness from those experts whose works have not been mentioned. The majority of this book was written when the author was visiting East China Normal University during the summer of 2000, the Mathematical Sci ences Research Institute at Berkeley during the fall of 2000 and University of California at Santa Barbara in spring of 2001. It is his pleasure to express his gratitude for all hospitalities he received at these institutes. The exciting environment at the MSRI has left him with a great memories far beyond this book. Even though it is difficult for the author to express his exact appreciation to the people whom he met at MSRI, now is perhaps the only opportunity to record his sincere appreciation. The author thanks Professor N. C. Phillips and Shuang Zhang for their comments and suggestions. He would also like to take this opportunity to thank following persons who were exposed to earlier drafts and made corrections, comments and suggestions: Warren Akers, Shanwen Hu, Bobby Ilapogu, Benjamin Itza-Ortiz, Junping Liu, Shudong Liu, Nancy Livingston, Lisa Oberbroeckling, Michael Raney and Tadg Woods. Contents Preface vii Chapter 1 The Basics of C*-algebras 1 1.1 Banach algebras 1 1.2 C**-algebras 9 1.3 Commutative C*-algebras 12 1.4 Positive cones 16 1.5 Approximate identities, hereditary C*-subalgebras and quotients 20 1.6 Positive linear functionals and a Gelfand-Naimark theorem . . 25 1.7 Von Neumann algebras 32 1.8 Enveloping von Neumann algebras and the spectral theorem 38 1.9 Examples of C*-algebras 42 1.10 Inductive limits of C*-algebras 51 1.11 Exercises 60 1.12 Addenda 65 Chapter 2 Amenable C*-algebras and if-theory 67 2.1 Completely positive linear maps and the Stinespring represen tation 67 2.2 Examples of completely positive linear maps 72 2.3 Amenable C*-algebras 76 2.4 if-theory 82 2.5 Perturbations 89 2.6 Examples of if-groups 97 ix