Kishimoto's Conjugacy Theorems in simple $C^*$-algebras of tracial rank one

Abstract

Let $A$ be a unital separable simple amenable $C^*$-algebra with finite tracial rank which satisfies the Universal Coefficient Theorem (UCT). Suppose $\af$ and $\bt$ are two automorphisms with the Rokhlin property that {induce the same action on the $K$-theoretical data of $A$.} We show that $\af$ and $\bt$ are strongly cocycle conjugate and uniformly approximately conjugate, that is, there exists a sequence of unitaries $\{u_n\}\subset A$ and a sequence of strongly asymptotically inner automorphisms $σ_n$ such that $$ \af={\rm Ad}\, u_n\circ σ_n\circ \bt\circ σ_n^{-1}\andeqn \lim_{n\to\infty}\|u_n-1\|=0, $$ and that the converse holds. {We then give a $K$-theoretic description as to exactly when $\af$ and $\bt$ are cocycle conjugate, at least under a mild restriction. Moreover, we show that given any $K$-theoretical data, there exists an automorphism $\af$ with the Rokhlin property which has the same $K$-theoretical data.