Inclusions of von Neumann Algebras and Quantum groupoids III

In a former article, in collaboration with Jean-Michel Vallin, we have constructed two “quantum groupo"{\i}ds” dual to each other, from a depth 2 inclusion of von Neumann algebras $M_0\subset M_1$, 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 $M_0^{\mathrm{\prime }}\cap M_1$, and we show how these examples fit with Lesieur’s “measured quantum groupo"{\i}ds”.