Asymptotic Cyclic Cohomology
The aim of cyclic cohomology theories is the approximation of K-theory by cohomology theories defined by natural chain complexes. The basic example is the approximation of topological K-theory by de Rham cohomology via the classical Chern character. A cyc
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Michael Puschnigg
Asymptotic Cyclic Cohomology
Springer
Author Michael Puschnigg Mathematics Institute University of Heidelberg 1m Neuenheimer Feld 288 D-69120 Heidelberg, Germany e-mail: [email protected]
Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Poschnigg, Michael: Asymptotic cyclic cohomology / Michael Puschnigg. - Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris; Santa Clara; Singapore; Tokyo: Springer, 1996 (Lecture notes in mathematics; 1642) ISBN 3-540-61986-0 NE:GT Mathematics Subject Classification (1991): 19D55, l8G60, 19K35, 19K56 ISSN 0075-8434 ISBN 3-540-61986-0 Springer-Verlag Berlin Heidelberg New York 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 any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer- Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1996 Printed in Germany
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Introduction
This work is a contribution to the study of topological K-Theory and cyclic cohomology of complete normed algebras. The aim is the construction of a cohomology theory, defined by a natural chain complex, on the category of Banach algebras which a) is the target of a Chern character from topological K-theory (resp. bivariant K-theory). h) has nice functorial properties which faithfully reflect the properties of topological K-theory. c) is closely related to cyclic cohomology but avoids the usual pathologies of cyclic cohomology for operator algebras. d) is accessible to computation in sufficiently many cases. The final goal is to establish a Grothendieck-Riemann-Roch theorem for the constructed Chern character which for commutative C*-algebras reduces to the classical Crothendieck-Riemann-Roch formula. In his "Noncommutative Geometry" Alain Connes has developed the framework for a large number of far reaching generalisations of the index theorems of Atiyah and Singer. To motivate the problem addressed in this book and to put it in the right context we recall some basic principles of index theory and noncommutative geometry. The classical index theorem fo
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