  In this paper, we use the KK-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.
