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

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.