Simplicity criteria for etale groupoid $C^*$-algebras

TL;DR

This paper establishes criteria for simplicity of reduced C*-algebras of Hausdorff etale groupoids by analyzing specific subalgebras and their state extensions, with applications to crossed products of abelian C*-algebras.

Contribution

It introduces a new framework for simplicity criteria based on non-degenerate subalgebras and state extension control, applicable to groupoid and crossed product C*-algebras.

Findings

Provided simplicity criteria for reduced C*-algebras of Hausdorff etale groupoids.

Applied the framework to crossed products of abelian C*-algebras by discrete groups.

Developed methods to analyze state extensions in the context of groupoid C*-algebras.

Abstract

We develop a framework suitable for obtaining simplicity criteria for reduced $C^*$-algebras of Hausdorff etale groupoids. This is based on the study of certain non-degenerate $C^*$-subalgebras (in the case of groupoids, the $C^*$-algebra of the interior isotropy bundle), for which one can control (non-unique) state extensions to the ambient C*-algebra. As an application, we give simplicity criteria for reduced crossed products of abelian $C^*$-algebras by discrete groups.