Asymptotic unitary equivalence in $C^*$-algebras

Abstract

Let $C=C(X)$ be the unital $C^*$-algebra of all continuous functions on a finite CW complex $X$ and let $A$ be a unital simple $C^*$-algebra with tracial rank at most one. We show that two unital monomorphisms $\phi, \psi: C\to A$ are asymptotically unitarily equivalent, i.e., there exists a continuous path of unitaries $\{u_t: t\in [0,1)\}\subset A$ such that $$ \lim_{t\to 1} u_t^*\phi(f)u_t=\psi(f) {\rm for all} \in C(X), $$ if and only if \beq [\phi]&=&[\psi] {\rm in} KK(C, A), \tau\circ \phi&=&\tau\circ \psi {\rm for all} \tau\in T(A), and \phi^{\dag}&=&\psi^{\dag}, \eneq where $T(A)$ is the simplex of tracial states of $A$ and $\phi^{\dag}, \psi^{\dag}: U(M_{\infty}(C))/DU(M_{\infty}(C))\to$ $U(M_{\infty}(A))/DU(M_{\infty}(A))$ 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.