Rokhlin dimension: absorption of model actions

TL;DR

This paper links Rokhlin dimension to the absorption of model actions on strongly self-absorbing C*-algebras, extending existing results and providing applications like classifying certain group actions.

Contribution

It establishes a general theorem connecting Rokhlin dimension with absorption of semi-strongly self-absorbing actions, broadening the scope of prior results.

Findings

Finite Rokhlin dimension implies absorption of certain model actions.

Uniqueness of
2^kb1-actions with finite Rokhlin dimension on Kirchberg algebras.

Extension of known results to more general group actions.

Abstract

In this paper, 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 $\mathcal D$ is a strongly self-absorbing C*-algebra, and $\alpha: G\curvearrowright A$ is an action on a separable, $\mathcal D$-absorbing C*-algebra that has finite Rokhlin dimension with commuting towers, then $\alpha$ tensorially absorbs every semi-strongly self-absorbing $G$-actions on $\mathcal D$. 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 non-trivial applications. Most notably it is shown that for any $k\geq 1$ and on any strongly self-absorbing Kirchberg algebra, there exists a unique $\mathbb R^k$-action having finite Rokhlin dimension with commuting towers up to (very strong) cocycle conjugacy.