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 phi, psi : C -> A are approximately unitarily equivalent if and only if [phi] = [psi] in KL(C, A) and tau omicron phi = tau omicron psi for all tau is an element of T(A), where T(A) is the tracial state space of A. In this paper we prove the following: Given kappa is an element of KL(C, A) with kappa(K-0(C)(+)\{0}) subset of K-0(A)(+)\{0} and with kappa([1(C)]) = [1(A)] and a continuous affine map lambda : T (A) -> T-f(C) which is compatible with., where T-f(C) is the convex set of all faithful tracial states, there exists a unital monomorphism phi : C -> A such that [phi] = kappa and tau omicron phi(c) = lambda(tau)(c) for all c. C-s.a. and tau is an element of T(A). Denote by Mon(au)(e)(C, A) the set of approximate unitary equivalence classes of unital monomorphisms. We provide a bijective map Lambda : Mon(au)(e)(C, A) -> 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-f(C). Moreover, we found that there are compact metric spaces X, unital simple AF-algebras A and kappa is an element of KL(C(X), A) with kappa(K0(C(X))+\{0}) subset of K-0(A)+\{0} for which there is no homomorphism h : C(X) -> A so that [h] = kappa.