Homomorphisms into a simple Z-stable C*-Algebras

Abstract

Let $A$ and $B$ be unital separable simple amenable \CA s which satisfy the Universal Coefficient Theorem. Suppose {that} $A$ and $B$ are $\mathcal Z$-stable and are of rationally tracial rank no more than one. We prove the following: Suppose that $\phi, \psi: A\to B$ are unital {monomorphisms}. There exists a sequence of unitaries $\{u_n\}\subset B$ such that $$ \lim_{n\to\infty} u_n^*\phi(a) u_n=\psi(a)\tforal a\in A, $$ if and only if $$ [\phi]=[\psi]\,\,\,\text{in}\,\,\, KL(A,B), \phi_{\sharp}=\psi_{\sharp}\andeqn\phi^{\ddag}=\psi^{\ddag}, $$ where $\phi_{\sharp}, \psi_{\sharp}: \aff(T(A))\to \aff(T(B))$ and $\phi^{\ddag}, \psi^{\ddag}: U(A)/CU(A)\to U(B)/CU(B)$ are {the} induced maps and where $T(A)$ and $T(B)$ are tracial state spaces of $A$ and $B,$ and $CU(A)$ and $CU(B)$ are closure of {commutator} subgroups of unitary groups of $A$ and $B,$ respectively. We also show that this holds for some AH-algebras $A.$ {Moreover, if $\kappa\in KL(A,B)$ preserves the order and the identity, $\lambda: \aff(\tr(A))\to \aff(\tr(B))$ is a continuous affine map and $\gamma: U(A)/CU(A)\to U(B)/CU(B)$ is a \hm\, which are compatible, we also show that there is a unital \hm\, $\phi: A\to B$ so that $([\phi],\phi_{\sharp},\phi^{\ddag})=(\kappa, \lambda, \gamma),$ at least in the case that $K_1(A)$ is a free group,