In this paper, we use the –theory of Kasparov to prove exactness of sequences relating the –theory of a real –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.
"Real versus complex K–theory using Kasparov's bivariant KK–theory." Algebr. Geom. Topol. 4 (1) 333 - 346, 2004. https://doi.org/10.2140/agt.2004.4.333