Strongly outer actions of certain torsion-free amenable groups on the Razak-Jacelon algebra

TL;DR

This paper proves that for a certain class of torsion-free amenable groups, all strongly outer actions on the Razak-Jacelon algebra are essentially the same, extending known results for strongly self-absorbing C*-algebras.

Contribution

It establishes that all strongly outer actions of groups in a specific class on the Razak-Jacelon algebra are cocycle conjugate, generalizing Szabó's results.

Findings

All strongly outer actions of groups in class $rak{C}$ on $ w$ are cocycle conjugate.

The class $rak{C}$ includes torsion-free abelian and poly-$bz$ groups.

Extension of classification results to a broader class of amenable groups.

Abstract

Let $\mathfrak{C}$ be the smallest class of countable discrete groups with the following properties: (i) $\mathfrak{C}$ contains the trivial group, (ii) $\mathfrak{C}$ is closed under isomorphisms, countable increasing unions and extensions by $\mathbb{Z}$. Note that $\mathfrak{C}$ contains all countable discrete torsion-free abelian groups and poly-$\mathbb{Z}$ groups. Also, $\mathfrak{C}$ is a subclass of the class of countable discrete torsion-free elementary amenable groups. In this paper, we show that if $\Gamma\in \mathfrak{C}$, then all strongly outer actions of $\Gamma$ on the Razak-Jacelon algebra $\mathcal{W}$ are cocycle conjugate to each other. This can be regarded as an analogous result of Szab\'o's result for strongly self-absorbing C$^*$-algebras.