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

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.