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

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. We use this to relate the real version of the Baum-Connes conjecture for a discrete group to its complex counterpart. In particular, one implies the other, and, after inverting 2, the same is true for the injectivity or surjectivity part alone, thus reproving a result of Baum and Karoubi. 1 Motivation In the majority of available sources about the subject, complex C-algebras and Banach algebras and their K-theory is studied. However, for geometrical reasons, the real versions also play a prominent role. Before describing the results of this paper, we want to give the geometric motivation why both variants are necessary. (1) Real K-Theory (meaning K-theory of real C-algebra) is more powerful since it contains additional information. Most notably this can be seen at Hitchins Z/2-obstructions to positive scalar curvature in dimensions 8k + 1 and 8k + 2 [5]. They take values in KOj(R) for j = 1, 2. Related to this is the fact that there are 8 different groups, and not just 2, since real K-theory does not have the 2-periodicity of complex K-theory, but is 8-periodic. In particular, we mention the following result of Stephan Stolz: if the real Baum-Connes map μR,red : RKO Γ ∗ (EΓ) → KO∗(C ∗ R,redΓ) is injective, then the stable Gromov-Lawson-Rosenberg conjecture is true for Γ. This means that a spin manifold with fundamental group Γ stably admits a metric with positive scalar curvature if and only if the Mishchenko-Fomenko index of its Dirac operator vanishes. (2) Unfortunately, a real structure of some kind is needed to define indices in real K-theory. In particular, there is no good way to define a (higher) real index of the signature operator in dimension 4k + 2.