Homomorphisms from AH-algebras

Let $C$ be a general unital AH-algebra and let $A$ be a unital simple $C^{\ast }$-algebra with tracial rank at most one. Suppose that $\varphi ,\psi :C\to A$ are two unital monomorphisms. We show that $\varphi$ 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 ${\varphi }_{\mathrm{♯}}$ and ${\psi }_{\mathrm{♯}}$ are continuous affine maps from tracial state space $T(A)$ of $A$ to faithful tracial state space $T_{\mathrm{f}}(C)$ of $C$ induced by $\varphi$ and $\psi ,$ respectively, and ${\varphi }^{\mathrm{‡}}$ and ${\psi }^{\mathrm{‡}}$ are induced homomorphisms from $K_1(C)$ into $\text{Aff}(T(A))\mathrm{/}\overline{{\rho }_A(K_0(A))},$ where $\text{Aff}(T(A))$ is the space of all real affine continuous functions on $T(A)$ and $\overline{{\rho }_A(K_0(A))}$ is the closure of the image of $K_0(A)$ in the affine space $\text{Aff}(T(A))\mathrm{.}$ 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 $\kappa \in K{L}_e(C,A{)}^{++},$ an affine map $\gamma :T(C)\to T_{\mathrm{f}}(C)$ and a \hm $\af: K_1©\to \Aff(T(A))/\bar{\rho_A(K_0(A))},$thereexistsaunitalmonomorphism$\phi: C\to A$suchthat$[h]=\kappa,$ $h_{\mathrm{♯}}=\gamma$ and ${\varphi }^{\mathrm{†}}=\text{af}\mathrm{.}$