Operator Algebras and Poisson Manifolds¶Associated to Groupoids

Abstract

It is well known that a measured groupoid G defines a von Neumann algebra W *(G), and that a Lie groupoid G canonically defines both a C *-algebra C *(G) and a Poisson manifold A *(G). We construct suitable categories of measured groupoids, Lie groupoids, von Neumann algebras, C *-algebras, and Poisson manifolds, with the feature that in each case Morita equivalence comes down to isomorphism of objects. Subsequently, we show that the maps G↦W *(G), G↦C *(G), and G↦A *(G) are functorial between the categories in question. It follows that these maps preserve Morita equivalence.