Strongly self-absorbing C*-dynamical systems, II

TL;DR

This paper extends the theory of strongly self-absorbing actions of groups on C*-algebras, establishing permanence properties and conditions under which these properties hold, generalizing previous results and providing new examples.

Contribution

It generalizes classical permanence properties to the equivariant setting for strongly self-absorbing actions, introducing a new mild condition and analyzing its implications.

Findings

Permanence properties hold for strongly self-absorbing actions under tensorial absorption.

A mild extra condition replaces $K_1$-injectivity in the equivariant extension context.

Examples satisfying the condition include equivariantly Jiang-Su absorbing systems.

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.