Crossed module actions on continuous trace \(C^*\)-algebras

Abstract

We lift an action of a torus \(\mathbb{T}^n\) on the spectrum of a continuous trace algebra to an action of a certain crossed module of Lie groups that is an extension of \(\mathbb{R}^n\). We compute equivariant Brauer and Picard groups for this crossed module and describe the obstruction to the existence of an action of \(\mathbb{R}^n\) in our framework.