Ideals of etale groupoid algebras and Exel's Effros-Hahn conjecture

TL;DR

This paper generalizes a recent result about ideals in ample groupoid algebras over fields to arbitrary commutative rings, providing a shorter proof and exploring the structure of primitive ideals via induced representations.

Contribution

It extends Demeneghi's result to broader base rings, introduces a new proof using the Disintegration Theorem, and analyzes conditions under which primitive ideals are kernels of irreducible induced representations.

Findings

Every ideal is an intersection of kernels of induced representations from isotropy groups.

Every primitive ideal is the kernel of an induced representation from an isotropy group.

Under certain conditions, primitive ideals are kernels of irreducible induced representations.

Abstract

We extend to arbitrary commutative base rings a recent result of Demeneghi that every ideal of an ample groupoid algebra over a field is an intersection of kernels of induced representations from isotropy groups, with a much shorter proof, by using the author's Disintegration Theorem for groupoid representations. We also prove that every primitive ideal is the kernel of an induced representation from an isotropy group; however, we are unable to show, in general, that it is the kernel of an irreducible induced representation. If each isotropy group is finite (e.g., if the groupoid is principal) and if the base ring is Artinian (e.g., a field), then we can show that every primitive ideal is the kernel of an irreducible representation induced from isotropy.