Crossed Products by Automorphisms with the Tracial Quasi-Rokhlin Property

TL;DR

This paper introduces the tracial quasi-Rokhlin property for automorphisms of unital C*-algebras, establishing conditions for simplicity of the crossed product and a correspondence of tracial states, with applications to minimal dynamical systems.

Contribution

It defines the tracial quasi-Rokhlin property for automorphisms and proves its implications for the structure and tracial state space of the associated crossed product C*-algebras.

Findings

Crossed product C*-algebras are simple under the property.

Bijection between tracial states on the crossed product and invariant tracial states on A.

Application to automorphisms extending minimal dynamical systems.

Abstract

We introduce the tracial quasi-Rokhlin property for an automorphism alpha of a unital C*-algebra A, which is not assumed to be simple. We show that under suitable hypotheses, the associated crossed product C*-algebra C*(Z,A,alpha) is simple, and there is a bijection between the space of tracial states on C*(Z,A,alpha)$ and the alpha-invariant tracial states on A. We show that, for a minimal dynamical system (X,h) and a simple, separable, unital C*-algebra A, the automorphism beta which extends the action of h on C(X) has the tracial quasi-Rokhlin property, and hence that C*(Z,C(X,A),beta) has the structural properties described above.