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29 March 2024
 
  » arxiv » 1003.1760

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Homomorphisms into a simple Z-stable C*-Algebras
Huaxin Lin ; Zhuang Niu ;
Date 9 Mar 2010
AbstractLet $A$ 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 $phi, psi: A o B$ are unital {monomorphisms}. There exists a sequence of unitaries ${u_n}subset B$ such that $$ lim_{n oinfty} u_n^*phi(a) u_n=psi(a) foral ain A, $$ if and only if $$ [phi]=[psi],,, ext{in},,, KL(A,B), phi_{sharp}=psi_{sharp}andeqnphi^{ddag}=psi^{ddag}, $$ where $phi_{sharp}, psi_{sharp}: aff(T(A)) o aff(T(B))$ and $phi^{ddag}, psi^{ddag}: U(A)/CU(A) o U(B)/CU(B)$ 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 $kappain KL(A,B)$ preserves the order and the identity, $lambda: aff( r(A)) o aff( r(B))$ is a continuous affine map and $gamma: U(A)/CU(A) o U(B)/CU(B)$ is a hm, which are compatible, we also show that there is a unital hm, $phi: A o B$ so that $([phi],phi_{sharp},phi^{ddag})=(kappa, lambda, gamma),$ at least in the case that $K_1(A)$ is a free group,
Source arXiv, 1003.1760
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