Actions of certain torsion-free elementary amenable groups on strongly self-absorbing C*-algebras

TL;DR

This paper studies actions of a broad class of torsion-free elementary amenable groups on strongly self-absorbing C*-algebras, establishing uniqueness and absorption properties of such actions.

Contribution

It proves the uniqueness and absorption of strongly outer group actions on strongly self-absorbing C*-algebras for a wide class of amenable groups, extending prior results.

Findings

Unique strongly outer actions up to cocycle conjugacy.

Actions absorb all other actions on the same algebra up to cocycle conjugacy.

Application of equivariant property (SI) in the monotracial case.

Abstract

In this paper we consider a bootstrap class $\mathfrak C$ of countable discrete groups, which is closed under countable unions and extensions by the integers, and we study actions of such groups on C*-algebras. This class includes all torsion-free abelian groups, poly-$\mathbb Z$-groups, as well as other examples. Using the interplay between relative Rokhlin dimension and semi-strongly self-absorbing actions established in prior work, we obtain the following two main results for any group $\Gamma\in\mathfrak C$ and any strongly self-absorbing C*-algebra $\mathcal D$: (1) There is a unique strongly outer $\Gamma$-action on $\mathcal D$ up to (very strong) cocycle conjugacy. (2) If $\alpha: \Gamma\curvearrowright A$ is a strongly outer action on a separable, unital, nuclear, simple, $\mathcal D$-stable C*-algebra with at most one trace, then it absorbs every $\Gamma$-action on $\mathcal D$ up to (very strong) cocycle conjugacy. In fact we establish more general relative versions of these two results for actions of amenable groups that have a predetermined quotient in the class $\mathfrak C$. For the monotracial case, the proof comprises an application of Matui--Sato's equivariant property (SI) as a key method.