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. Suppose α and β are two automorphisms with the Rokhlin property that induce the same action on the K-theoretical data of A. We show that α and β are strongly outer 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 $${\sigma_n}$$ such that $$\alpha = {\rm Ad} \, u_n \circ \sigma_n \circ \beta \circ \sigma_n^{-1} {\rm and} \lim_{n \rightarrow \infty}\|u_n - 1\| = 0,$$ and that the converse holds. We then give a K-theoretic description as to exactly when α and β are outer conjugate, at least under a mild restriction. Moreover, we show that given any K-theoretical data, there exists an automorphism α with the Rokhlin property which has the same K-theoretical data.