Stability of Lie groupoid C∗C∗-algebras

In this paper we generalize a theorem of M. Hilsum and G. Skandalis stating that the C∗C∗-algebra of any foliation of nonzero dimension is stable. Precisely, we show that the C∗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.