Structure and K-theory of crossed products by proper actions

TL;DR

This paper analyzes the structure and K-theory of crossed product C*-algebras arising from proper group actions on spaces, providing detailed descriptions of primitive ideal spaces and K-theoretic computations under certain conditions.

Contribution

It offers a detailed description of the primitive ideal space and K-theory of crossed products by proper actions, extending known results to broader classes of groups and actions.

Findings

Primitive ideal space described in terms of the action

K-theory computed for isolated orbits with finite stabilizers

K-theoretic proof of Baum-Connes type result for finite groups

Abstract

We study the C*-algebra crossed product $C_0(X)\rtimes G$ of a locally compact group $G$ acting properly on a locally compact Hausdorff space $X$. Under some mild extra conditions, which are automatic if $G$ is discrete or a Lie group, we describe in detail, and in terms of the action, the primitive ideal space of such crossed products as a topological space, in particular with respect to its fibring over the quotient space $G\backslash X$. We also give some results on the $\K$-theory of such C*-algebras. These more or less compute the $\K$-theory in the case of isolated orbits with non-trivial (finite) stabilizers. We also give a purely $\K$-theoretic proof of a result due to Paul Baum and Alain Connes on (\K)-theory with complex coefficients of crossed products by finite groups.