On the \(K\)-theory of crossed products by automorphic semigroup actions (Q2851200)

Let \(P\) be a semigroup that embeds into a group \(G\) and such that the embedding satisfies the Toeplitz condition introduced by \textit{X. Li} [``Nuclearity of semigroup \(C^*\)-algebras and the connection to amenability'', \url{arXiv:1203.0021}]. The paper deals with reduced crossed products by automorphic actions of \(P\) on a \(C^*\)-algebra \(A\) under the assumption that \(G\) satisfies the Baum-Connes conjecture with coefficients. A formula for \(K_*(A\rtimes_{\alpha,r}P)\) is obtained as a consequence of a result on the \(K\)-theory of crossed products for special actions of \(G\) on totally disconnected spaces. This result is applied to two classes of semigroups.NEWLINENEWLINEFirst, let \(P\) be the positive cone in a quasi-lattice ordered group \(G\) that satisfies the Baum-Connes conjecture with coefficients. Then the inclusion \(P\subset G\) satisfies the Toeplitz condition and it is shown that \(K_*(A\rtimes_{\alpha,r}P)\cong K_*(A)\) for any such \(P\) and for any of its automorphic actions \(\alpha\) on \(A\). This result can be viewed as a generalization of the Pimsner-Voiculescu six-term exact sequence for crossed products by \(\mathbb Z\).NEWLINENEWLINESecond, let \(R\) be the ring of algebraic integers in a number field, \(R^{\times}\) its multiplicative semigroup, and \(P=R\rtimes R^{\times}\) its \(ax+b\)-semigroup. Let \(C^*_\lambda(P)\) (resp. \(C^*_\rho(P)\)) be the left (resp. right) regular \(C^*\)-algebra of \(P\). As opposed to group \(C^*\)-algebras, left and right semigroup algebras may be different, e.g., \(C^*_\rho(P)\) admits non-trivial one-dimensional representations, while \(C^*_\lambda(P)\) does not admit any finite-dimensional representation. Nevertheless, it is shown that \(C^*_\lambda(P)\) and \(C^*_\rho(P)\) are \(KK\)-equivalent.