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

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