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.