Duality for actions and coactions of finite groupoids on von Neumann algebras (Q1320213)

The author generalizes to finite groupoids the notion of group actions and coactions on von Neumann algebras. By analogy with the crossed product algebra construction from a given covariant system, he shows how to associate a dual coaction (resp-action) to a given action (resp- coaction). Then a duality type theorem holds as in the group case. On the other hand, given a coaction it is possible to construct a new von Neumann algebra by integrating sections of the groupoid. In the case when the groupoid is principal, there is a close relationship between this von Neumann algebra and the equivalent one obtained by using the bidual coaction.