Prime \'etale groupoid algebras with applications to inverse semigroup and Leavitt path algebras

TL;DR

This paper establishes conditions for prime and semiprime properties of étale groupoid algebras, linking algebraic primeness to the dynamical property of topological transitivity, and applies these to inverse semigroup and Leavitt path algebras.

Contribution

It provides new necessary and sufficient conditions for primeness in étale groupoid algebras, connecting algebraic properties with dynamical groupoid features, and recovers known results for specific algebra classes.

Findings

Primeness is characterized by topological transitivity of the groupoid.

Necessary and sufficient conditions for semiprimeness are established.

Applications include known primeness results for inverse semigroup and Leavitt path algebras.

Abstract

In this paper we give some sufficient and some necessary conditions for an \'etale groupoid algebra to be a prime ring. As an application we recover the known primeness results for inverse semigroup algebras and Leavitt path algebras. It turns out that primeness of the algebra is connected with the dynamical property of topological transitivity of the groupoid. We obtain analogous results for semiprimeness.