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 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-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 Γ ∈ C and any strongly self-absorbing C * -algebra D:(1) There is a unique strongly outer Γ-action on D up to (very strong) cocycle conjugacy.(2) If α : ΓA is a strongly outer action on a separable, unital, nuclear, simple, D-stable C * -algebra with at most one trace, then it absorbs every Γ-action on 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 C. For the monotracial case, the proof comprises an application of Matui-Sato's equivariant property (SI) as a key method.Contents 4 2. Relative Rokhlin-type conditions 7 3. Actions on strongly self-absorbing C * -algebras 14 References 16

Rohlin Actions of Finite Groups on the Razak–Jacelon Algebra

“…(Note that every strongly self-absorbing C * -algebra is unital by definition.) In this paper, we study group actions on W and show an analogous result of Szabó's result in [40] for group actions on strongly self-absorbing C * -algebras (see also [11], [12], [13], [21], [22], [23], [25] and [36] for pioneering works). We refer the reader to [10] for the importance and some difficulties of studying group actions on C * -algebras.…”

“…Let C be the smallest class of countable discrete groups with the following properties: (i) C contains the trivial group, (ii) C is closed under isomorphisms, countable increasing unions and extensions by Z. Note that C is the same class as in [40,Definition B]. It is easy to see that C contains all countable discrete torsion-free abelian groups and poly-Z groups, and C is a subclass of the class of countable discrete torsion-free elementary amenable groups.…”

“…It is easy to see that C contains all countable discrete torsion-free abelian groups and poly-Z groups, and C is a subclass of the class of countable discrete torsion-free elementary amenable groups. Szabó showed that if Γ ∈ C and D is a strongly self-absorbing C * -algebra, then there exists a unique strongly outer action of Γ on D up to cocycle conjugacy ( [40,Corollary 3.4]). In this paper, we show an analogous result of this result.…”