Traces, Unitary Characters and Crossed Products by $ℤ$

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 SKQ(A X fl Z) of the KQ group of a crossed product by Z is homeomorphic to SK0(A\t or TG4) a. We also consider the crossed product A X aG by a discrete countable abelian group G and give necessary and sufficient conditions for the equality T(A xaG) = T(A\ to hold.