Traces, Unitary Characters and Crossed Products by \mathbb{Z}

Abstract

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 SK₀(A×_α\mathbb{Z}) of the K₀ group of a crossed product by \mathbb{Z} is homeomorphic to SK₀(A)_α. or T(A)_α. We also consider the crossed product A×_αG discrete countable abelian group G and give necessary and sufficient conditions for the equality T(A×_αG)=T(A)_α to hold.