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 $\sigma$-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 Špakula and the second-named author in the metric space case, and deduce new characterisations for reduced crossed products and uniform Roe algebras for groupoids.

arXiv commentPublished in Comm. Math. Phy

References in corpus (4)

• Amenable actions and exactness for discrete groups
• Relative commutant pictures of Roe algebras
• Cartan subalgebras in uniform Roe algebras
• A quasi-local characterisation of $L^p$-Roe algebras