Asymptotic unitary equivalence in $C^{\ast }$-algebras

Abstract

Let $C=C(X)$ be the unital $C^{\ast }$-algebra of all continuous functions on a finite CW complex $X$ and let $A$ be a unital simple $C^{\ast }$-algebra with tracial rank at most one. We show that two unital monomorphisms $\varphi ,\psi :C\to A$ are asymptotically unitarily equivalent, i.e., there exists a continuous path of unitaries $\{u_t:t\in [0,1)\}\subset A$ such that $$\underset{t\to 1}{\lim }{u}_t^{\ast }\varphi (f)u_t=\psi (f)\mathrm{forall}\in C(X),$$ if and only if \beq [ϕ]&=&[ψ] {\rm in} KK(C, A), τ\circ ϕ&=&τ\circ ψ{\rm for all} τ\in T(A), and ϕ^†&=&ψ^†, \eneq where $T(A)$ is the simplex of tracial states of $A$ and ${\varphi }^{†},{\psi }^{†}:U(M_{\infty }(C))/DU(M_{\infty }(C))\to$ $U(M_{\infty }(A))/DU(M_{\infty }(A))$ are induced homomorphisms and where $U(M_{\infty }(A))$ and $U(M_{\infty }(C))$ are groups of union of unitary groups of $M_k(A)$ and $M_k(C)$ for all integer $k\ge 1,$ $DU(M_{\infty }(A))$ and $DU(M_{\infty }(C))$ are commutator subgroups of $U(M_{\infty }(A))$ and $U(M_{\infty }(C)),$ respectively. We actually prove a more general result for the case that $C$ is any general unital AH-algebra.

References in corpus (4)

• Approximate Homotopy of Homomorphisms from $C(X)$ into a Simple $C^{\ast }$-algebra
• Localizing the Elliott conjecture at strongly self-absorbing C*-algebras
• Homomorphisms from AH-algebras
• On Local AH algebras

Cited by in corpus (1)

• Kishimoto's Conjugacy Theorems in simple $C^{\ast }$-algebras of tracial rank one