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*-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.
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.