Existence of the Matui-Sato tracial Rokhlin property

TL;DR

This paper constructs actions of elementary amenable groups on certain nuclear C*-algebras that possess the Matui-Sato tracial Rokhlin property, ensuring the crossed products retain simplicity and low tracial rank.

Contribution

It demonstrates the existence of group actions with the Matui-Sato tracial Rokhlin property on a broad class of nuclear C*-algebras, expanding understanding of their structure and crossed products.

Findings

Existence of actions with the tracial Rokhlin property for elementary amenable groups.

Crossed products remain simple with low tracial rank.

Applicable to unital simple separable nuclear C*-algebras with tracial rank at most one.

Abstract

We show by construction that when $G$ is an elementary amenable group and $A$ is a unital simple nuclear and tracially approximately divisible $C^*$-algebra, there exists an action $\omega$ of $G$ on $A$ with the tracial Rokhlin property in the sense of Matui and Sato. In particular, group actions with this Matui-Sato tracial Rokhlin property always exist for unital simple separable nuclear $C^*$-algebras with tracial rank at most one. If $A$ is simple with rational tracial rank at most one, then the crossed product $A\rtimes_{\omega}G$ is also simple with rational tracial rank at most one.