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

TL;DR

This paper proves Kishimoto's conjugacy theorems for automorphisms with the Rokhlin property on simple unital $C^*$-algebras of tracial rank one, establishing conditions for strong cocycle conjugacy and approximate conjugacy based on $K$-theory.

Contribution

It extends Kishimoto's conjugacy results to a broader class of $C^*$-algebras with finite tracial rank, providing $K$-theoretic criteria for cocycle conjugacy.

Findings

Automorphisms with Rokhlin property are strongly cocycle conjugate if they induce the same $K$-theoretical data.

Existence of automorphisms with Rokhlin property matching given $K$-theoretical data.

Characterization of cocycle conjugacy via $K$-theory under mild restrictions.

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 $\sigma_n$ such that $$ \af={\rm Ad}\, u_n\circ \sigma_n\circ \bt\circ \sigma_n^{-1}\andeqn \lim_{n\to\infty}\|u_n-1\|=0, $$ 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.