Traces, unitary characters and crossed products by \(\mathbb{Z}\) (Q1918528)

Summary: We determine the character group of the infinite unitary group of a unital exact \(C^*\)-algebra in terms of K-theory and traces and obtain a description of the infinite unitary group modulo the closure of its commutator subgroup by the same means. The methods are then used to decide when the state space \(SK_0(A\times_\alpha \mathbb{Z})\) of the \(K_0\) group of a crossed product by \(\mathbb{Z}\) is homeomorphic to \(SK_0(A)_{\alpha_*}\) or \(T(A)_\alpha\). We also consider the crossed product \(A\times_\alpha G\) by a discrete countable Abelian group \(G\) and give necessary and sufficient conditions for the equality \(T(A\times_\alpha G)= T(A)_\alpha\) to hold.