Continuous actions on primitive ideal spaces lift to ${\mathrm{C}}^{\ast }$-actions

Abstract

We prove that for any second-countable, locally compact group $G$, any continuous $G$-action on the primitive ideal space of a separable, nuclear ${\mathrm{C}}^{\ast }$-algebra $B$ such that $B\cong B\otimes \mathcal{K}\otimes {\mathcal{O}}_2$ is induced by an action on $B$. As a direct consequence, we establish that every continuous action on the primitive ideal space of a separable, nuclear ${\mathrm{C}}^{\ast }$-algebra is induced by an action on a ${\mathrm{C}}^{\ast }$-algebra with the same primitive ideal space. Moreover, we discuss an application to the classification of equivariantly ${\mathcal{O}}_2$-stable actions.

19 pages; v2 minor changes; this version is accepted for publication in Bull. London Math. Soc