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 selfabsorbing 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 semistrongly 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 Z 3 -actions on the Jiang-Su algebra.

Equivariant property (SI) revisited

“…This certainly includes for example the induced action on traces, or Γ-equivariant Kasparov theory [39]. Like in the ordinary classification program, this idea leads one to the concept of equivariant Jiang-Su stability: Definition (see [85,77]). Let Γ be a countable group and α : Γ…”

“…This provides a sufficient criterion for A ∼ = A ⊗ Z; see [43,85]. Using instead the equivariant version of property (SI), an entirely analogous method can be applied to maps going into the fixed point algebra F ω (A) Γ , yielding equivariant Jiang-Su stability via [77].…”

On characterizations of amenable $$\hbox {C}^*$$-dynamical systems and new examples

“…) is a G-C * -algebra and B ⊂ A ∞ is invariant under the diagonal G-action, then we denote by F ∞,α (B, A) the set of all G-continuous elements in F ∞ (B, A) [49]. Note that F ∞,α (B, A), equipped with the restriction action, is a G-C * -algebra.…”

An Invitation to Model Theory and C*-Algebras

“…An instance of this phenomenon is the general theory of strongly self-absorbing C*-algebras, which admits a natural model-theoretic treatment; see Section 6. The equivariant analog strongly self-absorbing C*algebras has been recently introduced and studied by Szabó in a series of papers [75][76][77], where the theory is developed in close parallel to the nonequivariant setting.…”

“…The model-theoretic treatment of strongly self-absorbing C*-algebras is the subject of [37]. The modeltheoretic proof of the characterization of D-absorption presented here is in some respects original, although heavily inspired by the proofs from the literature [47,75,83]. Finally, the technique of model-theoretic forcing in the metric setting has been first considered in [12] and then further developed in [36,42,53].…”