The Fell compactification and non-Hausdorff groupoids

TL;DR

This paper introduces a Fell compactification method for locally compact non-Hausdorff groupoids, transforming them into Hausdorff groupoids and offering a geometric perspective on certain representations.

Contribution

It applies Fell compactification to non-Hausdorff groupoids, producing Hausdorff groupoids and clarifying the structure of left-regular representations in the etale case.

Findings

Fell compactification yields Hausdorff groupoids from non-Hausdorff ones.

Provides a geometric interpretation of left-regular representations.

Enhances understanding of groupoid structures in operator algebras.

Abstract

A compactification of Fell is applied to locally compact non-Hausdorff groupoids and yields locally compact Hausdorff groupoids. In the etale case, this construction provides a geometric picture for the left-regular representations introduced by Khoshkam and Skandalis.