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

Abstract

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 $ϕ, ψ: C\to A$ are approximately unitarily equivalent if and only if $$ [ϕ]=[ψ] {\rm in} KL(C,A) and τ\circ ϕ=τ\circ ψ\tforal τ\in T(A), $$ where $T(A)$ is the tracial state space of $A.$ In this paper we prove the following: Given $κ\in KL(C,A)$ with $κ(K_0(C)_+\setminus \{0\})\subset K_0(A)_+\setminus \{0\}$ and with $κ([1_C])=[1_A]$ and a continuous affine map $λ: T(A)\to T_{\mathtt{f}}(C)$ which is compatible with $κ,$ where $T_{\mathtt{f}}(C)$ is the convex set of all faithful tracial states, there exists a unital monomorphism $ϕ: C\to A$ such that $$ [ϕ]=κ\andeqn τ\circ ϕ(c)=λ(τ)(c) $$ for all $c\in C_{s.a.}$ and $τ\in T(A).$ Denote by ${\rm Mon}_{au}^e(C,A)$ the set of approximate unitary equivalence classes of unital monomorphisms. We provide a bijective map $$ Λ: {\rm Mon}_{au}^e (C,A)\to KLT(C,A)^{++}, $$ where $KLT(C,A)^{++}$ is the set of compatible pairs of elements in $KL(C,A)^{++}$ and continuous affine maps from $T(A)$ to $T_{\mathtt{f}}(C).$ Moreover, we realized that there are compact metric spaces $X$, unital simple AF-algebras $A$ and $κ\in KL(C(X), A)$ with $κ(K_0(C(X))_+\setminus\{0\})\subset K_0(A)_+\setminus \{0\}$ for which there is no \hm $h: C(X)\to A$ so that $[h]=κ.$