Quasi-locality for \'{e}tale groupoids

TL;DR

This paper introduces the concept of quasi-locality for operators on the Hilbert module of an étale groupoid, providing a new geometric criterion to identify elements of the reduced groupoid C*-algebra, extending Roe's metric space results.

Contribution

It generalizes Roe's quasi-locality concept to étale groupoids and establishes an equivalence between quasi-locality and membership in the reduced groupoid C*-algebra for amenable, -compact groupoids.

Findings

Quasi-local operators characterize elements of the reduced groupoid C*-algebra.

The approach recovers known results for metric spaces.

New characterizations for reduced crossed products and uniform Roe algebras for groupoids.

Abstract

Let $\mathcal{G}$ be a locally compact \'{e}tale groupoid and $\mathscr{L}(L^2(\mathcal{G}))$ be the $C^*$-algebra of adjointable operators on the Hilbert $C^*$-module $L^2(\mathcal{G})$. In this paper, we discover a notion called quasi-locality for operators in $\mathscr{L}(L^2(\mathcal{G}))$, generalising the metric space case introduced by Roe. Our main result shows that when $\mathcal{G}$ is additionally $\sigma$-compact and amenable, an equivariant operator in $\mathscr{L}(L^2(\mathcal{G}))$ belongs to the reduced groupoid $C^*$-algebra $C^*_r(\mathcal{G})$ if and only if it is quasi-local. This provides a practical approach to describe elements in $C^*_r(\mathcal{G})$ using coarse geometry. Our main tool is a description for operators in $\mathscr{L}(L^2(\mathcal{G}))$ via their slices with the same philosophy to the computer tomography. As applications, we recover a result by \v{S}pakula and the second-named author in the metric space case, and deduce new characterisations for reduced crossed products and uniform Roe algebras for groupoids.