Equivariant K-Theory and Freeness of Group Actions on C*-Algebras
Freeness of an action of a compact Lie group on a compact Hausdorff space is equivalent to a simple condition on the corresponding equivariant K-theory. This fact can be regarded as a theorem on actions on a commutative C*-algebra, namely the algebra of c
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1274 N. Christopher Phillips
Equivariant K-Theory and Freeness of Group Actions on C*-Algebras
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Author
N. Christopher Phillips Department of Mathematics, University of California Los Angeles, CA 90024-1555, USA
Mathematics Subject Classification (1985): Primary: 46L55, 46L80, 46M20 Secondary: 19K33, 19K99, 19L47 ISBN 3-540-18277-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-18277-2 Springer-Verlag New York Berlin Heidelberg
Library of Congress Cataloging-in-Publication Data, Phillips, N, Christopher (Norman Christopher), 1956-, Equivariant K-theory and freeness of group actions on C'-algebras, (Lecture notes in mathematics; 1274) Bibliography: p. Includes index, 1, K-theory, 2, Cr-alqebras. 3, Lie groups, I. Title, II. Series: Lecture notes in mathematics (Springer-Verlag); 1274, QA3,L28 no 1274 510s 87-23345 [QA612.33] [512',55] ISBN 0-387-18277-2 (U,S.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24,1985, and a copyright fee must always be paid, Violations fall under the prosecution act of the German Copyright Law, © Springer-Verlag Berlin Heidelberg 1987 Printed in Germany
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To
Wang Kai-Shyang
Acknowledgments I would like to thank a number of people for their help and encouragement during the preparation of this book. Graeme Segal first called my attention to the paper from which I learned about the relation between equivariant Ktheory and freeness of actions. I have had valuable discussions with, among others, Claude Schochet concerning Kunneth theorems, 1. M. Singer concerning pseudo differential operators, Nigel Higson concerning extendible prequasihomomorphisms, and Jonathan Rosenberg concerning projective unitary representations. David Handelman suggested the main result of section 9.2 and its proof. Most of all, however, I would like to thank Marc Rieffel. Many of the results in this book first appeared in my Ph.D. thesis written under his direction, and many others were first proved while I was working on my thesis. His patience, encouragement, and suggestions, both before and after I received my Ph.D., have been extremely valuable, and he has helped eliminate many obscurities in the exposition. He was also kind enough to show me some of his unpublished research notes, and to allow me to use some of the material from them in chapter 7. This book is based on the author's Ph.D. thesis written at the University of California at Berkeley. However, it incorporates substantial revisions and many additional results. The research reported here has been partia
