Rohlin actions of finite groups on the Razak-Jacelon algebra

TL;DR

This paper proves that strongly outer finite group actions on a certain class of C*-algebras extend to have the Rohlin property when tensoring with the Razak-Jacelon algebra, leading to classification up to conjugacy.

Contribution

It demonstrates that such group actions become Rohlin actions on the tensor product with the Razak-Jacelon algebra, establishing their uniqueness up to conjugacy.

Findings

Actions have Rohlin property after tensoring with al.

Such actions are unique up to conjugacy.

Extension of classification results to these actions.

Abstract

Let $A$ be a simple separable nuclear C$^*$-algebra with a unique tracial state and no unbounded traces, and let $\alpha$ be a strongly outer action of a finite group $G$ on $A$. In this paper, we show that $\alpha\otimes \mathrm{id}$ on $A\otimes\mathcal{W}$ has the Rohlin property, where $\mathcal{W}$ is the Razak-Jacelon algebra. Combing this result with the recent classification results and our previous result, we see that such actions are unique up to conjugacy.