# Journal of Operator Theory

Volume 71, Issue 2, Spring 2014 pp. 517-569.

Homomorphisms into simple ${\cal Z}$-stable $C^*$-algebras**Authors**: Huaxin Lin (1) and Zhuang Niu (2)

**Author institution:**(1) Department of Mathematics, East China Normal University, Shanghai, China; current address: Department of Mathematics, University of Oregon, Eugene, OR 97403, U.S.A.

(2) Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL A1C5S7, Canada; current address: Department of Mathematics, University of Wyoming, Laramie, WY 82071, U.S.A.

**Summary:**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\limits_{n\to\infty} u_n^*\phi(a) u_n=\psi(a)\mbox{ for all } a\in A, $ if and only if $ [\phi]=[\psi]\ \text{in } KL(A,B),\ \phi_{\sharp}=\psi_{\sharp}\mbox{ and }\phi^{\ddag}=\psi^{\ddag}, $ where $\phi_{\sharp}, \psi_{\sharp}: \aff(\tr(A))\to \aff(\tr(B))$ and $\phi^{\ddag}, \psi^{\ddag}: U(A)/CU(A)\to U(B)/CU(B)$ are {the} induced maps (where $\tr(A)$ and $\tr(B)$ are {the} tracial state spaces of $A$ and $B,$ and $CU(A)$ and $CU(B)$ are the closures of the commutator subgroups of the unitary groups of $A$ and $B,$ respectively). We also show that this holds if $A$ is a rationally AH-algebra which is not necessarily simple. Moreover, for any {strictly positive unit-preserving} $\kappa\in KL(A,B)$, %preserving the order and the identity, any continuous affine map $\lambda: \aff(\tr(A))\to \aff(\tr(B))$ and any continuous group \hm\ $\gamma: U(A)/CU(A)\to U(B)/CU(B)$ 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.

**DOI:**http://dx.doi.org/10.7900/jot.2012jul10.1975

**Keywords:**classification of $C^*$-algebras, AH-algebras, $\mathcal Z$-stable $C^*$-algebras, homotopy lemma, uniqueness theorems

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