Kumjian–Renault theory allows us to reconstruct an étale groupoid twist from a Cartan pair of C*-algebras. I will explain what additional functional analytic data is needed to reconstruct a smooth structure on said twist. Our result gives rise to a correspondence between so-called "Lie groupoid twists" and "smooth Cartan triples", and it ties in Kumjian and Renault's theorems with Connes' reconstruction of a smooth manifold structure from certain spectral triples over commutative C*-algebras.

on joint work with Aidan Sims (University of Wollongong)