Strongly self-absorbing C*-dynamical systems, III

TL;DR

This paper advances the theory of strongly self-absorbing C*-dynamical systems by strengthening equivariant absorption results and establishing new reduction techniques, leading to uniqueness results for certain group actions on key C*-algebras.

Contribution

It extends the theory of semi-strongly self-absorbing actions, improves equivariant McDuff-type theorems, and introduces new reduction methods for equivariant absorption, culminating in new uniqueness results.

Findings

Strengthened equivariant McDuff-type theorem.

Reduction of equivariant Z-stability to UHF-stability.

Uniqueness of strongly outer G-actions on certain C*-algebras.

Abstract

In this paper, we accomplish two objectives. Firstly, we extend and improve some results in the theory of (semi-)strongly self-absorbing C*-dynamical systems, which was introduced and studied in previous work. In particular, this concerns the theory when restricted to the case where all the semi-strongly self-absorbing actions are assumed to be unitarily regular, which is a mild technical condition. The central result in the first part is a strengthened version of the equivariant McDuff-type theorem, where equivariant tensorial absorption can be achieved with respect to so-called very strong cocycle conjugacy. Secondly, we establish completely new results within the theory. This mainly concerns how equivariantly $\cal Z$-stable absorption can be reduced to equivariantly UHF-stable absorption with respect to a given semi-strongly self-absorbing action. Combining these abstract results with known uniqueness theorems due to Matui and Izumi-Matui, we obtain the following main result. If $G$ is a torsion-free abelian group and $\cal D$ is one of the known strongly self-absorbing C*-algebras, then strongly outer $G$-actions on $\cal D$ are unique up to (very strong) cocycle conjugacy. This is new even for $\mathbb{Z}^3$-actions on the Jiang-Su algebra.