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

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 Szabó's result for strongly self-absorbing C${}^{\ast }$-algebras.

8 pages, I have changed Definition 2.2, to appear in Proc. Roy. Soc. Edinburgh Sect. A