2023/09/17 by Duwenig, Anna, Sims, Aidan · 1 citation
#46L87 (primary) #58B34 (secondary) #58H05 #Differential Geometry (math.DG) #FOS: Mathematics #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2309.09177
We describe how to recover a Lie structure on a twist over a Hausdorff étale groupoid from functional-analytic data in the spirit of Connes' reconstruction theorem for manifolds. We first characterise when a smooth structure on the unit space of a Hausdorff étale 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 use these results in the setting of twists over étale groupoids 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.