A Steinberg algebra approach to \'etale groupoid C*-algebras

TL;DR

This paper develops a new approach to constructing C*-algebras of ample and étale groupoids using Steinberg algebras, establishing equivalence with standard methods and extending results to non-Hausdorff cases.

Contribution

It introduces a Steinberg algebra framework for étale groupoid C*-algebras and extends boundedness results to non-Hausdorff groupoids.

Findings

Constructs full and reduced C*-algebras from Steinberg algebras.

Shows the equivalence of this construction with standard methods.

Extends boundedness of *-homomorphisms to non-Hausdorff groupoids.

Abstract

We construct the full and reduced C*-algebras of an ample groupoid from its complex Steinberg algebra. We also show that our construction gives the same C*-algebras as the standard constructions. In the last section, we consider an arbitrary locally compact, second-countable, \'etale groupoid, possibly non-Hausdorff. Using the techniques developed for Steinberg algebras, we show that every $*$-homomorphism from Connes' space of functions to $B(\mathcal{H})$ is automatically I-norm bounded. Previously, this was only known for Hausdorff groupoids.