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Inclusions of von Neumann algebras, and quantum groupoïds. (English) Zbl 0974.46055
Authors’ abstract: From a depth 2 inclusion of von Neumann algebras $$M_0\subset M_1$$, which an operator-valued weight verifying a regularity condition, we construct a pseudo-multiplicative unitary, which leads to two structures of Hopf bimodules, dual to each other. Moreover, we construct an action of one of these structures on the algebra $$M_1$$ such that $$M_0$$ is the fixed point subalgebra, the algebra $$M_2$$ given by the basic construction being then isomorphic to the crossed-product. We construct on $$M_2$$ an action of the other structure, which can be considered as the dual action. If the inclusion $$M_0\subset M_1$$ is irreducible, we recover quantum groups, as proved in former papers. This construction generalizes the situation which occurs for actions (or co-actions) of groupoïds. Other examples of “quantum groupoïds” are given.

##### MSC:
 46L89 Other “noncommutative” mathematics based on $$C^*$$-algebra theory 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 46N50 Applications of functional analysis in quantum physics 46L60 Applications of selfadjoint operator algebras to physics 46L10 General theory of von Neumann algebras 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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