Rokhlin dimension: absorption of model actions

Szabó, Gábor

doi:10.2140/apde.2019.12.1357

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Rokhlin dimension: absorption of model actions

Gábor Szabó

Vol. 12 (2019), No. 5, 1357–1396

DOI: 10.2140/apde.2019.12.1357

Abstract

We establish a connection between Rokhlin dimension and the absorption of certain model actions on strongly self-absorbing $C^{*}$ -algebras. Namely, as to be made precise in the paper, let $G$ be a well-behaved locally compact group. If $D$ is a strongly self-absorbing $C^{*}$ -algebra and $α : G ↷ A$ is an action on a separable, $D$ -absorbing $C^{*}$ -algebra that has finite Rokhlin dimension with commuting towers, then $α$ tensorially absorbs every semi-strongly self-absorbing $G$ -action 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 $k \geq 1$ 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.

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Keywords

Rokhlin dimension, C*-dynamical system, strongly self-absorbing C*-algebra

Mathematical Subject Classification 2010

Primary: 46L55

Milestones

Received: 16 April 2018

Revised: 12 September 2018

Accepted: 18 October 2018

Published: 15 December 2018

Authors

Gábor Szabó
Department of Mathematics KU Leuven Leuven Belgium

Vol. 12, No. 5, 2019