Homotopy Theory of C*-Algebras
Homotopy theory and C*-algebras are central topics in contemporary mathematics. This book introduces a modern homotopy theory for C*-algebras. One basic idea of the setup is to merge C*-algebras and spaces studied in algebraic topology into one category c
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Advisory Editorial Board Leonid Bunimovich (Georgia Institute of Technology, Atlanta) Benoît Perthame (Université Pierre et Marie Curie, Paris) Laurent Saloff-Coste (Cornell University, Ithaca) Igor Shparlinski (Macquarie University, New South Wales) Wolfgang Sprössig (TU Bergakademie Freiberg) Cédric Villani (Ecole Normale Supérieure, Lyon)
Paul Arne Østvær
Homotopy Theory of
C*-Algebras
Paul Arne Østvær Department of Mathematics University of Oslo P.O. Box 1053, Blindern 0316 Oslo Norway e-mail: [email protected]
2010 Mathematics Subject Classification: 46L99, 55P99 ISBN 978-3-0346-0564-9 DOI 10.1007/978-3-0346-0565-6
e-ISBN 978-3-0346-0565-6
Library of Congress Control Number: 2010933114 © Springer Basel AG 2010 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. For any kind of use permission of the copyright owner must be obtained. Cover design: deblik, Berlin Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media www.birkhauser-science.com
Contents 1
Introduction
2 Preliminaries 2.1 C∗ -spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 G-C∗ -spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Model categories . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 15 16
3 Unstable C∗ -homotopy theory 3.1 Pointwise model structures . . . . . . 3.2 Exact model structures . . . . . . . . 3.3 Matrix invariant model structures . . 3.4 Homotopy invariant model structures 3.5 Pointed model structures . . . . . . . 3.6 Base change . . . . . . . . . . . . . .
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25 33 43 47 59 65
4 Stable C∗ -homotopy theory 4.1 C∗ -spectra . . . . . . . 4.2 Bispectra . . . . . . . 4.3 Triangulated structure 4.4 Brown representability 4.5 C∗ -symmetric spectra 4.6 C∗ -functors . . . . . .
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69 78 81 86 86 99
5 Invariants 5.1 Cohomology and homology theories . . . . . . . . 5.2 KK-theory and the Eilenberg-MacLane spectrum 5.3 HL-theory and the Eilenberg-MacLane spectrum 5.4 The Chern-Connes-Karoubi character . . . . . . . 5.5 K-theory of C∗ -algebras . . . . . . . . . . . . . . 5.6 Zeta functions . . . . . . . . . . . . . . . . . . . .
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107 108 111 112 113 121
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6 The slice filtration .
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