Groupoid Crossed Products of Continuous-Trace C*-Algebras

TL;DR

This paper demonstrates that for a groupoid dynamical system with a continuous trace algebra, the crossed product is Morita equivalent to a twisted groupoid C*-algebra, extending known results from group actions.

Contribution

It generalizes the classical result for group actions to the setting of groupoid dynamical systems with continuous trace algebras.

Findings

Crossed product is Morita equivalent to a twisted groupoid C*-algebra.

Establishes a groupoid analogue of a classical crossed product result.

Provides a framework for analyzing continuous-trace C*-algebras under groupoid actions.

Abstract

We show that if $(A,G,\alpha)$ is a groupoid dynamical system with $A$ continuous trace, then the crossed product $A\rtimes_{\alpha}G$ is Morita equivalent to the C*-algebra $C*(\underline G,\underline E)$ of a twist $\underline E$ over a groupoid $\underline G$ equivalent to $G$. This is a groupoid analogue of the well known result for the crossed product of a group acting on an elementary C*-algebra.