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

TL;DR

This paper proves that continuous actions on the primitive ideal space of certain nuclear C*-algebras can be lifted to actions on the algebras themselves, aiding classification of equivariant actions.

Contribution

It demonstrates that all continuous primitive ideal space actions of nuclear C*-algebras are induced by algebra actions, extending previous understanding in the field.

Findings

Actions on primitive ideal spaces lift to algebra actions for nuclear C*-algebras.

Every primitive ideal space action is induced by a C*-algebra action with the same primitive ideal space.

Application to classifying equivariantly O_2-stable 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.