Homomorphisms into a simple Z-stable C*-Algebras

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 $\varphi ,\psi :A\to B$ are unital {monomorphisms}. There exists a sequence of unitaries $\{u_n\}\subset B$ such that $$\underset{n\to \mathrm{\infty }}{\lim }{u}_n^{\ast }\varphi (a)u_n=\psi (a)\text{tforal}a\in A,$$ if and only if $$[\varphi ]=[\psi ]\text{in}KL(A,B),{\varphi }_{\mathrm{♯}}={\psi }_{\mathrm{♯}}\text{andeqn}{\varphi }^{\mathrm{‡}}={\psi }^{\mathrm{‡}},$$ where ${\varphi }_{\mathrm{♯}},{\psi }_{\mathrm{♯}}:\text{aff}(T(A))\to \text{aff}(T(B))$ and $\phi^{\ddag}, \psi^{\ddag}: U(A)/CU(A)\to U(B)/CU(B)$aretheinducedmapsandwhere$T(A)$and$T(B)$aretracialstatespacesof$A$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 $A\mathrm{.}$ {Moreover, if $\kappa \in KL(A,B)$ preserves the order and the identity, $\lambda: \aff(\tr(A))\to \aff(\tr(B))$isacontinuousaffinemapand$\gamma: U(A)/CU(A)\to U(B)/CU(B)$ is a \hm, which are compatible, we also show that there is a unital \hm, $\varphi :A\to B$ so that $([\varphi ],{\varphi }_{\mathrm{♯}},{\varphi }^{\mathrm{‡}})=(\kappa ,\lambda ,\gamma ),$ at least in the case that $K_1(A)$ is a free group,