Asymptotic unitary equivalence in C*-algebras

Abstract

Let C = C(X) be the unital C*-algebra of all continuous functions on a finite CW complex X and let A be a unital simple C*-algebra with tracial rank at most one. We show that two unital monomorphisms φ,ψ: C → A are asymptotically unitarily equivalent, i.e., there exists a continuous path of unitaries {u _t : t ∈ [0, 1)} ⊂ A such that lim _t→1 u* _t φ(f)u _t = ψ(f) for all f ∈ C(X) if and only if [φ] = [ψ] in KK(C,A), τ ◦φ = τ ◦ψ for all τ ∈ T(A), and φ ^† = ψ ^†, where T(A) is the simplex of tracial states of A and φ ^†, ψ ^†: U _∞(C)/DU _∞(C) → U _∞(A)/DU _∞(A) are the induced homomorphisms and where U _∞(A) = ∪ ^∞ _k=1 U(M _k (A)) and U _∞(_C) = ∪ ^∞ _k=1 U(M _k (C)) are usual infinite unitary groups, respectively, and DU _∞(A) and DU _∞(C) are the commutator subgroups of U _∞(A) and U _∞(C), respectively. We actually prove a more general result for the case in which C is any general unital AH-algebra.