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

## TL;DR

This survey explores Kasparov's KK-theory and its role in the Baum-Connes conjecture, highlighting methods like Dirac dual-Dirac and the Going-Down principle for simplifying K-theory calculations.

## Contribution

It provides a comprehensive overview of KK-theory's application to the Baum-Connes conjecture, including new insights into the Going-Down principle and explicit K-theory computation techniques.

## Key findings

- The Going-Down principle reduces K-theory computations to compact subgroup actions.
- Applications include explicit K-theory calculations for crossed products on totally disconnected spaces.
- The survey discusses the Dirac dual-Dirac method and its significance in the conjecture.

## 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.