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Asymptotic unitary equivalence in $C^*$-algebras

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 $ϕ, ψ: C\to A$ are asymptotically unitarily equivalent, i.e., there exists a continuous path of unitaries $\{u_t: t\in [0,1)\}\subset A$ such that $$ \lim_{t\to 1} u_t^*ϕ(f)u_t=ψ(f) {\rm for all} \in C(X), $$ 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 $ϕ^†, ψ^†: U(M_{\infty}(C))/DU(M_{\infty}(C))\to$ $U(M_{\infty}(A))/DU(M_{\infty}(A))$ 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.