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تسجيل مستخدم جديدThe Range of Approximate Unitary Equivalence Classes of Homomorphisms from AH-algebras
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Huaxin Lin
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Let $C$ be a unital AH-algebra and $A$ be a unital simple C*-algebra with tracial rank zero. It has been shown that two unital monomorphisms $phi, psi: Cto A$ are approximately unitarily equivalent if and only if $$ [phi]=[psi] {rm in} KL(C,A) and taucirc phi=taucirc psi tforal tauin T(A), $$ where $T(A)$ is the tracial state space of $A.$ In this paper we prove the following: Given $kappain KL(C,A)$ with $kappa(K_0(C)_+setminus {0})subset K_0(A)_+setminus {0}$ and with $kappa([1_C])=[1_A]$ and a continuous affine map $lambda: T(A)to T_{mathtt{f}}(C)$ which is compatible with $kappa,$ where $T_{mathtt{f}}(C)$ is the convex set of all faithful tracial states, there exists a unital monomorphism $phi: Cto A$ such that $$ [phi]=kappaandeqn taucirc phi(c)=lambda(tau)(c) $$ for all $cin C_{s.a.}$ and $tauin T(A).$ Denote by ${rm Mon}_{au}^e(C,A)$ the set of approximate unitary equivalence classes of unital monomorphisms. We provide a bijective map $$ Lambda: {rm Mon}_{au}^e (C,A)to KLT(C,A)^{++}, $$ where $KLT(C,A)^{++}$ is the set of compatible pairs of elements in $KL(C,A)^{++}$ and continuous affine maps from $T(A)$ to $T_{mathtt{f}}(C).$ Moreover, we realized that there are compact metric spaces $X$, unital simple AF-algebras $A$ and $kappain KL(C(X), A)$ with $kappa(K_0(C(X))_+setminus{0})subset K_0(A)_+setminus {0}$ for which there is no hm $h: C(X)to A$ so that $[h]=kappa.$
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Let $C$ be a general unital AH-algebra and let $A$ be a unital simple $C^*$-algebra with tracial rank at most one. Suppose that $phi, psi: Cto A$ are two unital monomorphisms. We show that $phi$ and $psi$ are approximately unitarily equivalent if and
only if beq[phi]&=&[psi] {rm in} KL(C,A), phi_{sharp}&=&psi_{sharp}tand phi^{dag}&=&psi^{dag}, eneq where $phi_{sharp}$ and $psi_{sharp}$ are continuous affine maps from tracial state space $T(A)$ of $A$ to faithful tracial state space $T_{rm f}(C)$ of $C$ induced by $phi$ and $psi,$ respectively, and $phi^{ddag}$ and $psi^{ddag}$ are induced homomorphisms from $K_1(C)$ into $Aff(T(A))/bar{rho_A(K_0(A))},$ where $Aff(T(A))$ is the space of all real affine continuous functions on $T(A)$ and $bar{rho_A(K_0(A))}$ is the closure of the image of $K_0(A)$ in the affine space $Aff(T(A)).$ In particular, the above holds for $C=C(X),$ the algebra of continuous functions on a compact metric space. An approximate version of this is also obtained. We also show that, given a triple of compatible elements $kappain KL_e(C,A)^{++},$ an affine map $gamma: T(C)to T_{rm f}(C)$ and a hm $af: K_1(C)to Aff(T(A))/bar{rho_A(K_0(A))},$ there exists a unital monomorphism $phi: Cto A$ such that $[h]=kappa,$ $h_{sharp}=gamma$ and $phi^{dag}=af.$
Let $A$ and $C$ be two unital simple C*-algebas with tracial rank zero. Suppose that $C$ is amenable and satisfies the Universal Coefficient Theorem. Denote by ${{KK}}_e(C,A)^{++}$ the set of those $kappa$ for which $kappa(K_0(C)_+setminus{0})subset
K_0(A)_+setminus{0}$ and $kappa([1_C])=[1_A]$. Suppose that $kappain {KK}_e(C,A)^{++}.$ We show that there is a unital monomorphism $phi: Cto A$ such that $[phi]=kappa.$ Suppose that $C$ is a unital AH-algebra and $lambda: mathrm{T}(A)to mathrm{T}_{mathtt{f}}(C)$ is a continuous affine map for which $tau(kappa([p]))=lambda(tau)(p)$ for all projections $p$ in all matrix algebras of $C$ and any $tauin mathrm{T}(A),$ where $mathrm{T}(A)$ is the simplex of tracial states of $A$ and $mathrm{T}_{mathtt{f}}(C)$ is the convex set of faithful tracial states of $C.$ We prove that there is a unital monomorphism $phi: Cto A$ such that $phi$ induces both $kappa$ and $lambda.$ Suppose that $h: Cto A$ is a unital monomorphism and $gamma in mathrm{Hom}(Kone(C), aff(A)).$ We show that there exists a unital monomorphism $phi: Cto A$ such that $[phi]=[h]$ in ${KK}(C,A),$ $taucirc phi=taucirc h$ for all tracial states $tau$ and the associated rotation map can be given by $gamma.$ Applications to classification of simple C*-algebras are also given.
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