Smooth Cartan triples and Lie twists over Hausdorff \'etale Lie groupoids

Abstract

We characterise when a smooth structure on the unit space of a Hausdorff \'etale groupoid can be extended to a Lie-groupoid structure on the whole groupoid. We introduce Lie twists over Hausdorff Lie groupoids, building on Kumjian's notion of a twist over a topological groupoid. We establish necessary and sufficient conditions on a family of sections of a twist over a Lie groupoid under which the twist can be made into a Lie twist so that all the specified sections are smooth. We obtain conditions on a twist over an \'etale groupoid whose unit space is a smooth manifold and a family of sections of the twist that characterise when the pair can be made into a Lie twist for which the given sections are smooth. We use these results to describe conditions on a Cartan pair of C*-algebras and a family of normalisers of the subalgebra, under which Renault's Weyl twist for the pair can be made into a Lie twist for which the given normalisers correspond to smooth sections.Comment: 32 page