The Mackey Machine for Crossed Products by Regular Groupoids. II

TL;DR

This paper extends the Mackey machine framework to analyze the spectra of crossed products by regular groupoids, establishing a homeomorphism with orbit spaces and strengthening results for groupoid algebras.

Contribution

It introduces a new method to relate the spectrum of crossed products by regular groupoids to orbit spaces, generalizing previous Mackey theory results.

Findings

Spectra of crossed products are homeomorphic to orbit spaces under certain conditions.

Established an action of the groupoid on the spectrum of the crossed product.

Provided strengthened results for the case of groupoid algebra crossed products.

Abstract

We prove that given a regular groupoid $G$ whose isotropy subgroupoid $S$ has a Haar system, along with a dynamical system $(A,G,\alpha)$, there is an action of $G$ on the spectrum of $A\rtimes S$ such that the spectrum of $A\rtimes G$ is homeomorphic to the orbit space of this action via induction. In addition, we give a strengthening of these results in the case where the crossed product is a groupoid algebra.