On dense subalgebras of the singular ideal in groupoid C*-algebras

TL;DR

This paper analyzes the structure of ideals in amenable second-countable étale groupoid $C^*$-algebras, proving that such ideals are completely determined by their isotropy fibres. It then reduces the question of density of Connes' algebra within the singular ideal to density questions inside isotropy group $C^*$-algebras, and provides concrete criteria (via density properties of isotropy groups and quasi-regular representations) for when density holds. The authors establish a general density framework, prove density for broad classes (notably abelian isotropy and contracting self-similar group groupoids), and show that in ample groupoids the algebraic singular ideal $J_{bC}$ is dense in the analytic singular ideal, enabling a tractable, algebraic approach to the density problem. These results advance understanding of simplicity, essentiality, and the fine structure of singular ideals in non-Hausdorff groupoid $C^*$-algebras with implications for dynamics and foliation-related constructions.

Abstract

We prove that ideals in amenable second-countable non-Hausdorff étale groupoid $C^*$-algebras are determined by their isotropy fibres. As an application, we characterise when the singular functions in Connes' algebra are dense in the singular ideal in terms of a property of explicit ideals in the isotropy group $C^*$-algebras.

Theorems & Definitions (52)

• Definition 1.1: CEPSS, KM21, EP22
• Definition 1.2: Tim
• Lemma 1.3
• proof
• Remark 2.1
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Proposition 2.4