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

TL;DR

This paper characterizes the structure of ideals in certain groupoid C*-algebras, linking their properties to isotropy fibers, and identifies conditions under which singular functions are dense in the singular ideal.

Contribution

It provides a new characterization of ideals in amenable second-countable non-Hausdorff étale groupoid C*-algebras based on isotropy fibers and explores density conditions of singular functions.

Findings

Ideals are determined by isotropy fibers in the specified groupoid C*-algebras.

Density of singular functions in the singular ideal is characterized by properties of isotropy group C*-algebras.

Density holds for groupoids with finite-by-nilpotent isotropy groups.

Abstract

We prove that ideals in amenable second-countable non-Hausdorff \'etale 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. We then show this density property holds for all $C^*$-algebras of groupoids with finite-by-nilpotent isotropy groups.