Traces, Unitary Characters and Crossed Products by $Z$

Abstract

We determine the character group of the infinite unitary group of a unital exact $C^{\ast }$-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 $S{K}_0(A{\times }_{\alpha }Z)$ of the $K_0$ group of a crossed product by $Z$ is homeomorphic to $S{K}_0(A{)}_{{\alpha }_{\ast }}$ or $T(A{)}_{{\alpha }^{\ast }}$. 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.