Strongly self-absorbing C*-dynamical systems, II

Abstract

This is a continuation of the study of strongly self-absorbing actions of locally compact groups on C*-algebras. Given a strongly self-absorbing action $\gamma :G\curvearrowright \mathcal{D}$, we establish permanence properties for the class of separable C*-dynamical systems absorbing $\gamma$ tensorially up to cocycle conjugacy. Generalizing results of both Toms-Winter and Dadarlat-Winter, it is proved that the desirable equivariant analogues of the classical permanence properties hold in this context. For the permanence with regard to equivariant extensions, we need to require a mild extra condition on $\gamma$, which replaces $K_1$-injectivity assumptions in the classical theory. This condition turns out to be automatic for equivariantly Jiang-Su absorbing C*-dynamical systems, yielding a large class of examples. It is left open whether this condition is redundant for all strongly self-absorbing actions, and we do consider examples that satisfy this condition but are not equivariantly Jiang-Su absorbing.

32 pages; this is a late upload of the version that has been accepted for publication in J. Noncomm. Geom

References in corpus (3)

• 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

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• Equivariant ${\mathcal{O}}_2$-absorption theorem for exact groups
• The classification of Rokhlin flows on C*-algebras
• Rokhlin dimension: absorption of model actions
• Equivariant higher twisted K-theory of SU(n) for exponential functor twists
• Equivariant higher Dixmier-Douady Theory for circle actions on UHF-algebras
• Quasi-invariant lifts of completely positive maps for groupoid actions
• Equivariant property Gamma and the tracial local-to-global principle for C*-dynamics
• A classification of anomalous actions through model action absorption