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 $phi, psi: Cto A$ 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 $kappain KL_e(C,A)^{++},$ 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 $phi: Cto A$ such that $[h]=kappa,$ $h_{sharp}=gamma$ and $phi^{dag}=af.$
تحميل البحث