Actions of certain torsion-free elementary amenable groups on strongly self-absorbing $$\mathrm {C}^*$$ 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 $$\mathrm {C}^*$$ -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 $$\mathrm {C}^*$$ -algebra $$\mathcal {D}$$ : 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.