Inclusions of Von Neumann Algebras and Quantum Groupoı̈ds II

In a previous article, in collaboration with Jean-Michel Vallin, we constructed two quantum groupoı̈ds, dual to each other, from a depth 2 inclusion of von Neumann algebras M0⊂M1. In this paper we investigate this structure in greater detail. In the previous article, we constructed the analog of a co-product, while in this paper we define a co-inverse, by making the polar decomposition of the analog of the antipode, and left and right invariant Haar operator-valued weights. These two structures of quantum groupoı̈ds, dual to each other, can be placed on the relative commutants M′0∩M2 and M′1∩M3 in such a way that the canonical Jonesʹ tower associated to the inclusion can be described as a tower of successive crossed-products by these two structures.