Simplicity of algebras associated to \'etale groupoids

TL;DR

This paper characterizes when C*-algebras associated with second-countable, étale, amenable groupoids are simple, linking algebraic simplicity to topological properties of the groupoid, and extends results to Steinberg's algebra for totally disconnected units.

Contribution

It provides necessary and sufficient conditions for simplicity of C*-algebras and Steinberg algebras associated with étale groupoids, connecting algebraic simplicity to topological groupoid properties.

Findings

C*-algebra simplicity characterized by topologically principal and minimal groupoids

Steinberg algebra simplicity linked to isotropy subgroupoid and minimality

Conditions are both necessary and sufficient for algebraic simplicity

Abstract

We prove that the C*-algebra of a second-countable, \'etale, amenable groupoid is simple if and only if the groupoid is topologically principal and minimal. We also show that if G has totally disconnected unit space, then the associated complex *-algebra introduced by Steinberg is simple if and only if the interior of the isotropy subgroupoid of G is equal to the unit space and G is minimal.