TY - JOUR

T1 - Strongly self-absorbing C*-dynamical systems, III

AU - Szabo, Gabor

N1 - The work presented in this paper has benefited from a visit to the Department of Mathematics at the University of Kyoto in January 2016, and I would like to express my gratitude to Masaki Izumi for the hospitality and support.
Open Access funded by Engineering and Physical Sciences Research Council

PY - 2017/8/20

Y1 - 2017/8/20

N2 - 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 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 D is one of the known strongly self-absorbing ⁎C⁎-algebras, then strongly outer G-actions on D are unique up to (very strong) cocycle conjugacy. This is new even for Z3-actions on the Jiang–Su algebra.

AB - 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 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 D is one of the known strongly self-absorbing ⁎C⁎-algebras, then strongly outer G-actions on D are unique up to (very strong) cocycle conjugacy. This is new even for Z3-actions on the Jiang–Su algebra.

KW - Strongly self-absorbing C-algebra

KW - Noncommutative dynamical systems

U2 - 10.1016/j.aim.2017.06.008

DO - 10.1016/j.aim.2017.06.008

M3 - Article

VL - 316

SP - 356

EP - 380

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -