Limit operator theory for groupoids

TL;DR

This paper develops a unified limit operator theory for -tale, amenable groupoids, extending symbol calculus and invertibility results, thereby connecting and broadening existing theories for groups, spaces, and groupoid actions.

Contribution

It introduces a comprehensive limit operator framework for groupoids, extending previous results and establishing new invertibility criteria in groupoid $C^*$-algebras.

Findings

Unified limit operator theory for -tale, amenable groupoids

Extension of invertibility equivalence in groupoid $C^*$-algebras

New results for group/groupoid actions and uniform Roe algebras

Abstract

We extend the symbol calculus and study the limit operator theory for $\sigma$-compact, \'{e}tale 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 $\sigma$-compact amenable groupoid with a Haar system is equivalent to the invertibility of its images under regular representations.