Quasi-Locality for étale Groupoids

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