Strongly self-absorbing \(C^*\)-dynamical systems. II (Q1747040)

Summary: This is a continuation of our 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 \textit{A. S. Toms} and \textit{W. Winter} [Trans. Am. Math. Soc. 359, No. 8, 3999--4029 (2007; Zbl 1120.46046)] and \textit{M. Dadarlat} and \textit{W. Winter} [Math. Scand. 104, No. 1, 95--107 (2009; Zbl 1170.46065)], 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 require a mild extra condition on \(\gamma\), which replaces the \( K_1\)-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. For Part I, see [Trans. Am. Math. Soc. 370, No. 1, 99--130 (2018; Zbl 1392.46054)].