Groupoid Semidirect Product Fell Bundles II- Principal Actions and Stabilization

TL;DR

This paper extends the theory of Fell bundles over groupoids by establishing an equivalence between semidirect-product and fixed-point bundles under groupoid actions, and relates stabilization to crossed-product duality.

Contribution

It generalizes previous results from groups to groupoids, providing a new perspective on Fell bundle actions and stabilization in the context of groupoid theory.

Findings

Established an equivalence between semidirect-product and fixed-point Fell bundles for groupoid actions.

Connected the Stabilization Theorem for Fell bundles to crossed-product duality.

Extended prior group-based results to the more general setting of groupoids.

Abstract

Given a free and proper action of a groupoid on a Fell bundle (over another groupoid), we give an equivalence between the semidirect-product and the generalized-fixed-point Fell bundles, generalizing an earlier result where the action was by a group. As an application, we show that the Stabilization Theorem for Fell bundles over groupoids is essentially another form of crossed-product duality.