\(K\)-theory of analytic crossed products (Q1802714)

The author proves a theorem which is simultaneously a non-selfadjoint analogue of Connes' Thom isomorphism and a generalization of a result of J. Peters. Let \(G\) be a locally compact, compactly generated, abelian group and that \(\Sigma\) is a subsemigroup of \(G\) which is the closure of its interior, meets its image under inversion at the identity element of \(G\), and generates \(G\). Let \(A\) be a \(C^*\)-algebra and let \(\alpha\) be a continuous homomorphism from \(G\) into \(\Aut(A)\). Let \(G\times_ \alpha A\) be the enveloping \(C^*\)-algebra of \(L^ 1(G,A)\) and call it the \(C^*\)-crossed product determined by the \(C^*\)-dynamical system \((A,G,\alpha)\). Then, for \(i=0,1\) \(K_ i(\Sigma\times_ \alpha A)\) is isomorphic to \(K_ i(A)\) if \(G\) is discrete and 0 otherwise.