Shapiro's lemma for topological \(K\)-theory of groups
Let \(G\) be a locally compact group, \(A\) a \(C^{\ast}\)-algebra equipped with a strongly continuous action of \(G\) on \(A\) and \[ K^{\text{top}}_{\ast}(G;A)=\lim_{Z\subset {\mathcal E}(G)} KK^{G}_{\ast}(C_0(Z),A), \] where \({\mathcal E}(G)\) denotes a universal example for the proper actions of \(G\), \(Z\) runs through the \(G\)-compact subsets of \({\mathcal E}(G)\) and \(KK^{G}_{\ast}(C_0(Z),A))\) denotes Kasparov's equivariant \(KK\)-theory. The topological \(K\)-theory forms the left hand side of the Baum-Connes assembly map \[ \mu_{G,A}: K^{\text{top}}_{\ast}(G;A)\to K_{\ast}(A\times G), \] where \(A\times G\) denotes the reduced crossed product of \(A\) and \(G\). The Baum-Connes conjecture asserts that \(\mu_{G,A}\) should be an isomorphism. Let \(X\times G\) be the crossed product groupoid of \(G\) acting on a locally compact space \(X\). The main purpose of the paper is to show that a natural forgetful map from the topological \(K\)-theory \(K^{\text{top}}_{\ast}(X\times G;A)\) of the groupoid \(X\times G\) with coefficients in \(A\) to \(K^{\text{top}}_{\ast}(G;A)\) is an isomorphism. There are some interesting consequences of this result for the Baum-Connes conjecture.
scientific article; zbMATH DE number 1990092