Asymptotically unitary equivalence and asymptotically inner automorphisms

Abstract

Let $C$ be a unital AH-algebra and let $A$ be a unital separable simple $C^*$-algebra with tracial rank zero. Suppose that $\phi_1, \phi_2\colon \ C\to A$ are two unital monomorphisms. We show that there is a continuous path of unitaries $\{u_t\colon \ t\in [0, \infty)\}$ of $A$ such that $$ \lim_{t\to\infty}u_t^*\phi_1(a)u_t=\phi_2(a)\quad\hbox{\rm for all}\quad a\in C $$ if and only if $[\phi_1]=[\phi_2]$ in $KK(C,A),$ $\tau\circ \phi_1=\tau\circ \phi_2$ for all $\tau\in T(A)$ and the rotation map ${\tilde\eta}_{\phi_1,\phi_2}$ associated with $\phi_1$ and $\phi_2$ is zero. In particular, an automorphism $\alpha$ on a unital separable simple $C^*$-algebra $A$ in ${\cal N}$ with tracial rank zero is asymptotically inner if and only if $$ [\alpha]=[{\rm id}_A]\quad{\rm in}\quad KK(A,A) $$ and the rotation map ${\tilde\eta}_{\phi_1, \phi_2}$ is zero. Let $A$ be a unital AH-algebra (not necessarily simple) and let $\alpha\in Aut(A)$ be an automorphism. As an application, we show that the associated crossed product $A\rtimes_{\alpha}{\Bbb Z}$ can be embedded into a unital simple AF-algebra if and only if $A$ admits a strictly positive $\alpha$-invariant tracial state.