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 $\varphi ,\psi :A\to B$ are unital {monomorphisms}. There exists a sequence of unitaries $\{u_n\}\subset B$ such that $$ \lim_{n\to\infty} u_n^*ϕ(a) u_n=ψ(a)\tforal a\in A, $$ if and only if $$ [ϕ]=[ψ]\,\,\,\text{in}\,\,\, KL(A,B), ϕ_{\sharp}=ψ_{\sharp}\andeqnϕ^‡=ψ^‡, $$ where $ϕ_{\sharp}, ψ_{\sharp}: \aff(T(A))\to \aff(T(B))$ and ${\varphi }^{‡},{\psi }^{‡}:U(A)/CU(A)\to U(B)/CU(B)$ 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, $λ: \aff(\tr(A))\to \aff(\tr(B))$ 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\, $\varphi :A\to B$ so that $([\varphi ],{\varphi }_{♯},{\varphi }^{‡})=(\kappa ,\lambda ,\gamma ),$ at least in the case that $K_1(A)$ is a free group,

The revision improves the original result. It is now 47 pages

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• Homomorphisms from AH-algebras
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Cited by in corpus (1)

• The corona algebra of stablized Jiang-Su algebra