Homomorphisms from AH-algebras

Abstract

Let $C$ be a general unital AH-algebra and let $A$ be a unital simple $C^{\ast }$-algebra with tracial rank at most one. Suppose that $\varphi ,\psi :C\to A$ are two unital monomorphisms. We show that $\varphi$ and $\psi$ are approximately unitarily equivalent if and only if \beq[ϕ]&=&[ψ] {\rm in} KL(C,A), ϕ_{\sharp}&=&ψ_{\sharp}\tand ϕ^†&=&ψ^†, \eneq where ${\varphi }_{♯}$ and ${\psi }_{♯}$ are continuous affine maps from tracial state space $T(A)$ of $A$ to faithful tracial state space $T_{\mathrm{f}}(C)$ of $C$ induced by $\varphi$ and $\psi ,$ respectively, and ${\varphi }^{‡}$ and ${\psi }^{‡}$ are induced homomorphisms from $K_1(C)$ into $\Aff(T(A))/\bar{ρ_A(K_0(A))},$ where $\Aff(T(A))$ is the space of all real affine continuous functions on $T(A)$ and $\overline{{\rho }_A(K_0(A))}$ 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 $\kappa \in K{L}_e(C,A{)}^{++},$ an affine map $\gamma :T(C)\to T_{\mathrm{f}}(C)$ and a \hm $\af: K_1(C)\to \Aff(T(A))/\bar{ρ_A(K_0(A))},$ there exists a unital monomorphism $\varphi :C\to A$ such that $[h]=\kappa ,$ $h_{♯}=\gamma$ and $ϕ^†=\af.$

References in corpus (1)

• Approximate Homotopy of Homomorphisms from $C(X)$ into a Simple $C^{\ast }$-algebra

Cited by in corpus (9)

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• On Local AH algebras
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• The kernel of the determinant map on certain simple C*-algebras
• The corona algebra of stablized Jiang-Su algebra
• C*-algebras of minimal dynamical systems of the product of a Cantor set and an odd dimensional sphere
• Distance between unitary orbits of normal elements in simple C*-algebras of real rank zero