# Journal of Operator Theory

Volume 47, Issue 2, Spring 2002 pp. 343-378.

Stable approximate unitary equivalence of homomorphisms**Authors**: Huaxin Lin

**Author institution:**Department of Mathematics, East China Normal University, Shanghai, China

**Summary:**Let $A$ be a separable unital nuclear \CAl and let $B$ be a unital \CA. Suppose that $A$ satisfies the Universal Coefficient Theorem and $\alpha,\beta: A\to B$ are \hm s. We show that $\alpha$ and $\beta$ are stably approximately unitarily equivalent if they induce the same element in $\KK(A,B)$ and either $A$ or $B$ is simple. In particular, an automorphism $\alpha$ on $A$ is stably approximately inner if $[\alpha]=[{\rm id}_E]$ in $\KK(A,A).$ If $B$ is simple and $A$ is ``{\rm K}-theoretically locally finite" then $\alpha$ and $\beta$ are stably approximately unitarily equivalent if and only if they induce the same element in ${\rm KL}(A,B).$ In the case that $A$ and $B$ are separable purely infinite simple \CA s and $A$ is nuclear and satisfies the UCT, then $\phi$ and $\psi$ are approximately unitarily equivalent if and only if $[\alpha]=[\psi]$ in ${\rm KL}(A,B).$

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