Publication date:
20 August 2017
Source:Advances in Mathematics, Volume 316
Author(s): Gábor Szabó
In this paper, we accomplish two objectives. Firstly, we extend and improve some results in the theory of (semi-)strongly self-absorbing ${\mathrm{C}}^{⁎}$-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 $\mathcal{Z}$-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 $\mathcal{D}$ is one of the known strongly self-absorbing ${\mathrm{C}}^{⁎}$-algebras, then strongly outer G-actions on $\mathcal{D}$ are unique up to (very strong) cocycle conjugacy. This is new even for ${\mathbb{Z}}^3$-actions on the Jiang–Su algebra.