Shapiro's lemma for topological K-theory of groups

Let X x G be the crossed product groupoid of a locally compact group G acting on a locally compact space X. For any X x G-algebra A we show that a natural forgetful map from the topological K-theory K-*(top)(X x G; A) of the groupoid X x G with coefficients in A to the topological K-theory 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.