Limit operator theory for groupoids

We extend the symbol calculus and study the limit operator theory for σ-compact, étale, and amenable groupoids, in the Hilbert space case. This approach not only unifies various existing results which include the cases of exact groups and discrete metric spaces with Property A, but also establish new limit operator theories for group/groupoid actions and uniform Roe algebras of groupoids. In the process, we extend a monumental result by Exel, Nistor, and Prudhon, showing that the invertibility of an element in the groupoid C*- algebra of a σ-compact amenable groupoid with a Haar system is equivalent to the invertibility of its images under regular representations.