Quasi-Locality for étale Groupoids.

Let G be a locally compact étale groupoid and L (L 2 (G)) be the C ∗ -algebra of adjointable operators on the Hilbert C ∗ -module L 2 (G) . In this paper, we discover a notion called quasi-locality for operators in L (L 2 (G)) , generalising the metric space case introduced by Roe. Our main result shows that when G is additionally σ -compact and amenable, an equivariant operator in L (L 2 (G)) belongs to the reduced groupoid C ∗ -algebra C r ∗ (G) if and only if it is quasi-local. This provides a practical approach to describe elements in C r ∗ (G) using coarse geometry. Our main tool is a description for operators in L (L 2 (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. [ABSTRACT FROM AUTHOR]