Traces, Unitary Characters and Crossed Products by 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 _SK_0(A x ℤ) of the K_0 group of a crossed product by ℤ is homeomorphic to SK_0(A)α* or T(A)α*. We also consider the crossed product A x α_G by a discrete countable abelian group G and give necessary and sufficient conditions for the equality T(A x α_G) = T(A)α to hold.