Quasi-locality for étale groupoids

Abstract

Let $\mathcal{G}$ be a locally compact étale groupoid and $\mathcal{L}(L^2(\mathcal{G}))$ be the $C^{\ast }$-algebra of adjointable operators on the Hilbert $C^{\ast }$-module $L^2(\mathcal{G})$. In this paper, we discover a notion called quasi-locality for operators in $\mathcal{L}(L^2(\mathcal{G}))$, generalising the metric space case introduced by Roe. Our main result shows that when $\mathcal{G}$ is additionally $\ddot{I}ƒ$-compact and amenable, an equivariant operator in $\mathcal{L}(L^2(\mathcal{G}))$ belongs to the reduced groupoid $C^{\ast }$-algebra $C_r^{\ast }(\mathcal{G})$ if and only if it is quasi-local. This provides a practical approach to describe elements in $C_r^{\ast }(\mathcal{G})$ using coarse geometry. Our main tool is a description for operators in $\mathcal{L}(L^2(\mathcal{G}))$ via their slices with the same philosophy to the computer tomography. As applications, we recover a result by Å 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.

Published in Comm. Math. Phy