Real versus complex K-theory using Kasparov’s bivariant KK-theory. (English summary) Algebr. Geom. Topol. 4 (2004), 333–346. Let A be a separable real σ-unital C ∗-algebra and AC = A ⊗ C. Boersema established a long exact sequence in K-theory · · · → KOq−1(A) → KOq(A) → Kq(AC) → KOq−2(A) → · · ·. In this paper, the author re-proves this result by different methods. In fact, he obtains an analogue of the above exact sequence in Kasparov’s equivariant bivariant K-theory, and shows that it is compatible with the Baum-Connes assembly map. As a corollary, given a discrete group Γ, 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
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