Real versus complex K–theory using Kasparov’s bivariant KK–theory

Schick, Thomas

doi:10.2140/agt.2004.4.333

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Real versus complex K–theory using Kasparov's bivariant KK–theory

Thomas Schick

Algebraic & Geometric Topology 4 (2004) 333–346

DOI: 10.2140/agt.2004.4.333

arXiv: math.KT/0311295

Abstract

In this paper, we use the $K K$ –theory of Kasparov to prove exactness of sequences relating the $K$ –theory of a real $C^{*}$ –algebra and of its complexification (generalizing results of Boersema). We use this to relate the real version of the Baum-Connes conjecture for a discrete group to its complex counterpart. In particular, the complex Baum–Connes assembly map is an isomorphism if and only if the real one is, thus reproving a result of Baum and Karoubi. After inverting 2, the same is true for the injectivity or surjectivity part alone.

Keywords

real $K$–theory, complex $K$–theory, bivariant $K$–theory

Mathematical Subject Classification 2000

Primary: 19K35, 55N15

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Publication

Received: 24 November 2003

Accepted: 29 May 2004

Published: 29 May 2004

Authors

Thomas Schick
Fachbereich Mathematik Georg-August-Universität Göttingen Germany
http://www.uni-math.gwdg.de/schick/

Volume 4, issue 1 (2004)