Stability of Lie groupoid C*-algebras

TL;DR

This paper proves that the C*-algebra of a Lie groupoid is stable if the groupoid has no zero-dimensional orbits, extending previous results to singular foliations and their holonomy groupoids.

Contribution

It generalizes the stability theorem for foliation C*-algebras to Lie groupoids without zero-dimensional orbits and to certain singular foliations.

Findings

C*-algebra stability for Lie groupoids with no zero-dimensional orbits

Extension of stability results to singular foliations

Application to holonomy groupoids of singular foliations

Abstract

In this paper we generalize a theorem of M. Hilsum and G. Skandalis stating that the $C^*$- algebra of any foliation of non zero dimension is stable. Precisely, we show that the C*-algebra of a Lie groupoid is stable whenever the groupoid has no orbit of dimension zero. We also prove an analogous theorem for singular foliations for which the holonomy groupoid as defined by I. Androulidakis and G. Skandalis is not Lie in general.