Shapiro's lemma for topological K-theory of groups

Abstract.

Let $X\rtimes G$ be the crossed product groupoid of a locally compact group G acting on a locally compact space X. For any $X\rtimes G$-algebra A we show that a natural forgetful map from the topological K-theory $\operatorname K_*^{\top}(X\rtimes G;A)$ of the groupoid $X\rtimes G$ with coefficients in A to the topological K-theory $\operatorname K_*^{\top}(G;A)$ of G with coefficients in A is an isomorphism. We then discuss several interesting consequences of this result for the Baum-Connes conjecture.