Rokhlin dimension: absorption of model actions

Abstract

In this paper, we establish a connection between Rokhlin dimension and the absorption of certain model actions on strongly selfabsorbing C * -algebras. Namely, as to be made precise in the paper, let G be a well-behaved locally compact group. If D is a strongly selfabsorbing C * -algebra, and α : G A is an action on a separable, Dabsorbing C * -algebra that has finite Rokhlin dimension with commuting towers, then α tensorially absorbs every semi-strongly self-absorbing Gactions on D. In particular, this is the case when α satisfies any version of what is called the Rokhlin property, such as for G = R or G = Z k . This contains several existing results of similar nature as special cases. We will in fact prove a more general version of this theorem, which is intended for use in subsequent work. We will then discuss some nontrivial applications. Most notably it is shown that for any k ≥ 1 and on any strongly self-absorbing Kirchberg algebra, there exists a unique R k -action having finite Rokhlin dimension with commuting towers up to (very strong) cocycle conjugacy.

Actions of certain torsion-free elementary amenable groups on strongly self-absorbing $$\mathrm {C}^*$$ C ∗ -algebras

“…We note that in a similar but slightly different context, the type of approximately central embedding technique utilized here has also been independently discovered by Gardella-Hirshberg [4] and in ongoing work of Gardella-Phillips-Wang [7]. Compared to similar Rokhlintype theorems such as [26,27], the key difference is that we can arrange the resulting Rokhlin towers to commute with each other, which is both necessary for the methods of [44] to be applicable and is in a sense a special feature of Z as a (relative) acting group.…”

“…The most important tool in the proof of our main results is given by the interplay between Rokhlin dimension relative to subgroups and the absorption of semi-strongly self-absorbing actions, as established in [44]. In essence, our proof goes by showing that, if viewed as a property of amenable groups Γ, the statements in the two theorems above are closed under extensions by Z.…”

“…The present work is a continuation of the work initiated in [44]. As such, our aim is to study C * -dynamical systems and to classify them up to cocycle conjugacy.…”

“…In order to handle extensions by Z, one needs to show that all strongly outer actions as in Theorem D satisfy certain Rokhlin-type conditions relative to any normal subgroup H such that Γ/H ∼ = Z, which allows one to apply the main result from [44]. This Rokhlin-type theorem is the only ingredient where the proof has to handle the case of finite and infinite C *algebras separately.…”

“…As such, our aim is to study C * -dynamical systems and to classify them up to cocycle conjugacy. We refer to the introduction of [44], [43], or in particular [12] and the references therein for a proper motivation and historical overview of the classification theory of (discrete) group actions on von Neumann algebras and C * -algebras.…”