Certain aperiodic automorphisms of unital simple projectionless C*-algebras

TL;DR

This paper studies aperiodic automorphisms of certain unital simple projectionless C*-algebras, showing conjugacy results and the existence of Rohlin automorphisms asymptotically equivalent to any given automorphism.

Contribution

It proves conjugacy of aperiodic automorphisms modulo weak inner automorphisms and constructs Rohlin automorphisms asymptotically unitarily equivalent to any automorphism in a broad class.

Findings

Aperiodic automorphisms are conjugate modulo weak inner automorphisms.

Existence of Rohlin automorphisms asymptotically unitarily equivalent to any automorphism.

Extension of results to a class of unital simple C*-algebras including AT-algebras of real rank zero.

Abstract

Let $G$ be an inductive limit of finite cyclic groups and let $A$ be a unital simple projectionless C*-algebra with $K_1(A) \cong G$ and with a unique tracial state, as constructed based on dimension drop algebras by Jiang and Su. First, we show that any two aperiodic elements in $\Aut(A)/\WInn(A)$ are conjugate, where $\WInn(A)$ means the subgroup of $\Aut(A)$ consisting of automorphisms which are inner in the tracial representation. In the second part of this paper, we consider a class of unital simple C*-algebras with a unique tracial state which contains the class of unital simple AT-algebras of real rank zero with a unique tracial state. This class is closed under inductive limits and under crossed products by actions of $\Z$ with the Rohlin property. Let $A$ be a TAF-algebra in this class. We show that for any automorphism $\alpha$ of $A$ there exists an automorphism $\widetilde{\alpha}$ of $A$ with the Rohlin property such that $\widetilde{\alpha}$ and $\alpha$ are asymptotically unitarily equivalent. In its proof we use an aperiodic automorphism of the Jiang-Su algebra.