$K$-theory of crossed products via homotopy theory

Abstract

In this paper we analyse for a $G$-$C^{*}$-algebra $A$ to which extent one can calculate the $K$-theory of the reduced crossed product $K(A\rtimes_{r}G)$ from the $K$-theory spectrum $K(A)$ with the induced $G$-action. We also consider some cases where one allows to use the $K$-theories of crossed products for some proper subgroups of $G$. Our central goal is to demonstrate the usefulness of a homotopy theoretic approach. We mainly concentrate on finite groups.