Inclusions of von Neumann algebras and quantum groupoïds III

In a former article, in collaboration with Jean-Michel Vallin, we have constructed two “quantum groupoı¨ds” dual to each other, from a depth 2 inclusion of von Neumann algebras M0⊂M1, in such a way that the canonical Jones'tower associated to the inclusion can be described as a tower of successive crossed-products by these two structures. We are now investigating in greater details these structures in the presence of an appropriate modular theory on the basis M0′∩M1, and we show how these examples fit with Lesieur's “measured quantum groupoı¨ds”.