Asymptotically unitary equivalence and asymptotically inner automorphisms

Abstract

Let $C$ be a unital AH-algebra and let $A$ be a unital separable simple \CA with tracial rank zero. Suppose that ${\varphi }_1,{\varphi }_2:C\to A$ are two unital monomorphisms. We show that there is a continuous path of unitaries $\{u_t:t\in [0,\infty )\}$ of $A$ such that $$ \lim_{t\to\infty}u_t^*ϕ_1(a)u_t=ϕ_2(a)\tforal a\in C $$ if and only if $[{\varphi }_1]=[{\varphi }_2]$ in $KK(C,A),$ $\tau \circ {\varphi }_1=\tau \circ {\varphi }_2$ for all $\tau \in T(A)$ and the rotation map ${\tilde{\eta }}_{{\varphi }_1,{\varphi }_2}$ associated with ${\varphi }_1$ and ${\varphi }_2$ is zero. In particular, an automorphism $\af$ on a unital separable simple \CA $A$ in $\mathcal{N}$ with tracial rank zero is asymptotically inner if and only if $$ [\af]=[{\rm id}_A] \text{in} KK(A,A) $$ and the rotation map ${\tilde{\eta }}_{{\varphi }_1,{\varphi }_2}$ is zero. Let $A$ be a unital AH-algebra (not necessarily simple) and let $\af\in Aut(A)$ be an automorphism. As an application, we show that the associated crossed product $A\rtimes_{\af}\Z$ can be embedded into a unital simple AF-algebra if and only if $A$ admits a strictly positive $\af$-invariant tracial state.

This is a revision of 04/07

Cited by in corpus (1)

• Lifting KK-elements, asymptotical unitary equivalence and classification of simple C*-algebras