Homomorphisms from AH-algebras

Let $C$ be a general unital AH-algebra and let $A$ be a unital simple $C^*$-algebra with tracial rank at most one. Suppose that $\phi, \psi: C\to A$ are two unital monomorphisms. We show that $\phi$ and $\psi$ are approximately unitarily equivalent if and only if \beq[\phi]&=&[\psi] {\rm in} KL(C,A), \phi_{\sharp}&=&\psi_{\sharp}\tand \phi^{\dag}&=&\psi^{\dag}, \eneq where $\phi_{\sharp}$ and $\psi_{\sharp}$ are continuous affine maps from tracial state space $T(A)$ of $A$ to faithful tracial state space $T_{\rm f}(C)$ of $C$ induced by $\phi$ and $\psi,$ respectively, and $\phi^{\ddag}$ and $\psi^{\ddag}$ are induced homomorphisms from $K_1(C)$ into $\Aff(T(A))/\bar{\rho_A(K_0(A))},$ where $\Aff(T(A))$ is the space of all real affine continuous functions on $T(A)$ and $\bar{\rho_A(K_0(A))}$ is the closure of the image of $K_0(A)$ in the affine space $\Aff(T(A)).$ In particular, the above holds for $C=C(X),$ the algebra of continuous functions on a compact metric space. An approximate version of this is also obtained. We also show that, given a triple of compatible elements $\kappa\in KL_e(C,A)^{++},$ an affine map $\gamma: T(C)\to T_{\rm f}(C)$ and a \hm $\af: K_1(C)\to \Aff(T(A))/\bar{\rho_A(K_0(A))},$ there exists a unital monomorphism $\phi: C\to A$ such that $[h]=\kappa,$ $h_{\sharp}=\gamma$ and $\phi^{\dag}=\af.$