Equivalence and Exact Groupoids

TL;DR

This paper demonstrates that the linking groupoid construction can be used to prove that groupoid exactness is preserved under equivalence, paralleling the role of linking algebras in $C^*$-algebra theory.

Contribution

It provides a new proof that groupoid exactness is preserved under equivalence using the linking groupoid construction.

Findings

Linking groupoid serves as a tool for analyzing groupoid exactness.

The proof parallels the linking algebra approach in $C^*$-algebra Morita theory.

Groupoid equivalence preserves exactness via linking groupoid methods.

Abstract

Given two locally compact Hausdorff groupoids $G$ and $H$ and a $(G,H)$-equivalence $Z$, one can construct the associated linking groupoid $L$. This is reminiscent of the linking algebra for Morita equivalent $C^*$-algebras. Indeed, Sims and Williams reestablished Renault's equivalence theorem by realizing $C^*(L)$ as the linking algebra for $C^*(G)$ and $C^*(H)$. Since the proof that Morita equivalence preserves exactness for $C^*$-algebras depends on the linking algebra, the linking groupoid should serve the same purpose for groupoid exactness and equivalence. We exhibit such a proof here.