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 {ut: t ∈ [0, 1)} ⊂ A such that limt→1u*tφ(f)ut = 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(Mk(A)) and U∞(C) = ∪k=1∞U(Mk(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.