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Inclusions of von Neumann algebras and quantum groupoïds. III. (English) Zbl 1088.46036
Summary: In Part I, in collaboration with J.-M. Vallin [J. Funct. Anal. 172, No. 2, 249–300 (2000; Zbl 0974.46055)], we have constructed two “quantum groupoïds” 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 with the inclusion can be described as a tower of successive crossed products by these two structures. We are now investigating in greater detail these structures in the presence of an appropriate modular theory on the basis \(M_0^{\prime}\cap M_1\), and we show how these examples fit with Lesieur’s “measured quantum groupoïds”.
[For Part II, see ibid. 178, No. 1, 156–225 (2000; Zbl 0982.46046).]

MSC:
46L37 Subfactors and their classification
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
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