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arXiv · 1703.10912

Bivariant $KK$-Theory and the Baum-Connes conjecure

Siegfried Echterhoff
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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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Siegfried Echterhoff. 2017-03-31. Bivariant $KK$-Theory and the Baum-Connes conjecure. https://arxiv.org/abs/1703.10912

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