Bivariant KK-Theory and the Baum-Connes conjecure

2017/03/31 by Siegfried Echterhoff, Echterhoff, Siegfried · 2 citations
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT)

paper · pdf · doi:10.48550/arxiv.1703.10912

openalex publication_date 2017/03/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This is a survey on Kasparov's bivariant KK-theory in connection with the Baum-Connes conjecture on the K-theory of crossed products A\rtimes_rG by actions of a locally compact group G on a C*-algebra A. In particular we shall discuss Kasparov's Dirac dual-Dirac method as well as the permanence properties of the conjecture and the "Going-Down principle" for the left hand side of the conjecture, which often allows to reduce K-theory computations for A\rtimes_rG to computations for crossed products by compact subgroups of G. We give several applications for this principle including a discussion of a method developed by Cuntz, Li and the author for explicit computations of the K-theory groups of crossed products for certain group actions on totally disconnected spaces. This provides an important tool for the computation of K-theory groups of semi-group C*-algebras.

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