Simplicity of inverse semigroup and \'etale groupoid algebras

TL;DR

This paper characterizes when the algebra of an étale groupoid with totally disconnected units is simple, extending previous results by removing the topologically principal condition and providing new examples, with applications to inverse semigroup algebras.

Contribution

It establishes new criteria for simplicity of étale groupoid and inverse semigroup algebras, including the first examples of certain non-topologically principal groupoids, and analyzes characteristic dependence via Galois descent.

Findings

Simplicity characterized by minimality and effectiveness of the groupoid.

First examples of minimal, effective but not topologically principal étale groupoids.

Simplicity depends only on the field's characteristic, with liftability from positive characteristic.

Abstract

In this paper, we prove that the algebra of an \'etale groupoid with totally disconnected unit space has a simple algebra over a field if and only if the groupoid is minimal and effective and the only function of the algebra that vanishes on every open subset is the null function. Previous work on the subject required the groupoid to be also topologically principal in the non-Hausdorff case, but we do not. Furthermore, we provide the first examples of minimal and effective but not topologically principal \'etale groupoids with totally disconnected unit spaces. Our examples come from self-similar group actions of uncountable groups. More generally, we show that the essential algebra of an \'etale groupoid (the quotient by the ideal of functions vanishing on every open set), is simple if and only if the groupoid is minimal and topologically free, generalizing to the algebraic setting a recent result for essential $C^*$-algebras. The main application of our work is to provide a description of the simple contracted inverse semigroup algebras, thereby answering a question of Munn from the seventies. Using Galois descent, we show that simplicity of \'etale groupoid and inverse semigroup algebras depends only on the characteristic of the field and can be lifted from positive characteristic to characteristic $0$. We also provide examples of inverse semigroups and \'etale groupoids with simple algebras outside of a prescribed set of prime characteristics.