Corona algebras and strongly self-absorbing $\mathrm{C}^{\ast}$-dynamics

Abstract

This article concerns the structure of $\mathrm{C}^{\ast}$-algebraic group actions induced on corona algebras from a given $\sigma$-unital $\mathrm{C}^{\ast}$-dynamical system over a locally compact group $G$. We prove that such actions satisfy the so-called dynamical folding property, which generalizes a fundamental property observed for corona algebras in works of Manuilov--Thomsen and Phillips--Weaver. We then focus on corona actions induced from $G$-$\mathrm{C}^{\ast}$-dynamics that are assumed to absorb a given strongly self-absorbing and unitarily regular $G$-action $\gamma$. It is proved that these corona actions are $\gamma$-saturated, which is a stronger property than being separably $\gamma$-stable. Conversely, if one assumes that the underlying $\mathrm{C}^{\ast}$-dynamics absorbs the trivial action on the compact operators, then $\gamma$-saturation of the corona action is equivalent to the original action being $\gamma$-absorbing. These results are a dynamical version of recent work by Farah and the third-named author.