Crossed module actions on continuous trace $C^*$-algebras

TL;DR

This paper explores lifting torus actions on continuous trace $C^*$-algebras to crossed modules, computing related algebraic invariants and identifying obstructions to extending these actions to $ eal^n$.

Contribution

It introduces a framework for lifting torus actions to crossed modules and computes associated equivariant Brauer and Picard groups, revealing obstructions to $ eal^n$ actions.

Findings

Computed equivariant Brauer groups for the crossed module

Computed equivariant Picard groups for the crossed module

Identified obstructions to extending $ orus^n$ actions to $ eal^n$

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.