Strongly self-absorbing C*-dynamical systems, III

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 $\mathcal{Z}$-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 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.

22 pages; v3 some added remarks and simplified argument in section 5

References in corpus (11)

• Quasidiagonality of nuclear C*-algebras
• Completely positive maps of order zero
• Classification of finite simple amenable $\mathcal{Z}$-stable $C^{\ast }$-algebras
• Strongly self-absorbing C*-algebras are Z-stable
• Strongly self-absorbing C*-dynamical systems
• Sequentially split $\ast$-homomorphisms between ${\mathrm{C}}^{\ast }$-algebras
• Strongly self-absorbing C*-dynamical systems, II
• The spatial Rokhlin property for actions of compact quantum groups
• Actions of amenable groups and crossed products of Z-absorbing C*-algebras
• Classification of outer actions of Z^N on O_2
• Z^N-actions on UHF algebras of infinite type

Cited by in corpus (16)

• Equivariant Kirchberg-Phillips-type absorption for amenable group actions
• Equivariant property (SI) revisited
• On a categorical framework for classifying C*-dynamics up to cocycle conjugacy
• Actions of certain torsion-free elementary amenable groups on strongly self-absorbing C*-algebras
• Cartan subalgebras and the UCT problem, II
• The classification of Rokhlin flows on C*-algebras
• Rokhlin dimension: absorption of model actions
• Strongly outer actions of amenable groups on $\mathcal{Z}$-stable nuclear $C^{\ast }$-algebras
• Equivariant higher twisted K-theory of SU(n) for exponential functor twists
• Equivariant bundles and absorption
• Equivariant Kirchberg-Phillips type absorption for the Razak-Jacelon algebra
• A classification of anomalous actions through model action absorption
• Quasi-invariant lifts of completely positive maps for groupoid actions
• Actions of rigid groups on UHF-algebras
• Rohlin actions of finite groups on the Razak-Jacelon algebra
• The stable uniqueness theorem for unitary tensor category equivariant KK-theory