Irreducible Representations of Groupoid $C^*$-algebras

TL;DR

This paper proves that for a certain class of groupoids, all representations induced from irreducible stability group representations remain irreducible, advancing the understanding of their representation theory.

Contribution

It establishes that induced representations from irreducible stability group representations are always irreducible in second countable locally compact Hausdorff groupoids with Haar systems.

Findings

Induced representations from irreducible stability group representations are irreducible.

Provides a characterization of irreducible representations in groupoid $C^*$-algebras.

Enhances the theoretical framework for analyzing groupoid $C^*$-algebras.

Abstract

If $G$ is a second countable locally compact Hausdorff groupoid with Haar system, we show that every representation induced from an irreducible representation of a stability group is irreducible.