Characterizations of zero singular ideal in \'etale groupoid C*-algebras via compressible maps

Abstract

We show the singular ideal in a non-Hausdorff \'etale groupoid C*-algebra is zero if and only if every unit is contained, at the level of group representation theory, in the collection of subgroups of the unit's isotropy group obtained as limit sets of nets in the "Hausdorff part" of the unit space. This is achieved through a study of the interplay between the Hausdorff cover and restriction maps on C*-algebras of groupoids to reductions by closed locally invariant subsets, which we show are compressible to *-homomorphisms and therefore have many of the same properties. We also prove a simpler algebraic characterization of zero singular ideal that holds whenever the isotropy group C*-algebras satisfy a certain ideal intersection property. We prove this property holds for all direct limits of virtually torsion free solvable groups.