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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