The Range of Approximate Unitary Equivalence Classes of Homomorphisms from AH-algebras

Let C be a unital AH-algebra and A be a unital simple C∗-algebra with tracial rank zero. It has been shown that two unital monomorphisms φ, ψ : C → A are approximately unitarily equivalent if and only if [φ] = [ψ] in KL(C,A) and τ ◦ φ = τ ◦ ψ for all τ ∈ T (A), where T (A) is the tracial state space of A. In this paper we prove the following: Given κ ∈ KL(C,A) with κ(K0(C)+\{0}) ⊂ K0(A)+\{0} and with κ([1C ]) = [1A] and a continuous affine map λ : T (A) → Tf(C) which is compatible with κ, where Tf(C) is the convex set of all faithful tracial states, there exists a unital monomorphism φ : C → A such that [φ] = κ and τ ◦ φ(c) = λ(τ)(c) for all c ∈ Cs.a. and τ ∈ T (A). Denote by Moneau(C,A) the set of approximate unitary equivalence classes of unital monomorphisms. We provide a bijective map Λ : Moneau(C,A) → KLT (C,A), whereKLT (C,A) is the set of compatible pairs of elements inKL(C,A) and continuous affine maps from T (A) to Tf(C). Moreover, we realized that there are compact metric spaces X , unital simple AF-algebras A and κ ∈ KL(C(X), A) with κ(K0(C(X))+ \ {0}) ⊂ K0(A)+ \ {0} for which there is no homomorphism h : C(X) → A so that [h] = κ.