Rokhlin dimension: absorption of model actions

Abstract

We establish a connection between Rokhlin dimension and the absorption of certain model actions on strongly self-absorbing $C^{\ast }$-algebras. Namely, as to be made precise in the paper, let $G$ be a well-behaved locally compact group. If $\mathcal{D}$ is a strongly self-absorbing $C^{\ast }$-algebra and $\alpha :G\curvearrowright A$ is an action on a separable, $\mathcal{D}$-absorbing $C^{\ast }$-algebra that has finite Rokhlin dimension with commuting towers, then $\alpha$ tensorially absorbs every semi-strongly self-absorbing $G$-action on $\mathcal{D}$. In particular, this is the case when $\alpha$ satisfies any version of what is called the Rokhlin property, such as for $G=ℝ$ or $G={\mathbb{Z}}^k$. 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 $k\ge 1$ 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.