#### Volume 4, issue 1 (2004)

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

### Thomas Schick

Algebraic & Geometric Topology 4 (2004) 333–346
 arXiv: math.KT/0311295
##### Abstract

In this paper, we use the $KK$–theory of Kasparov to prove exactness of sequences relating the $K$–theory of a real ${C}^{\ast }$–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