Inclusions of von Neumann algebras and quantum groupoïds III

Abstract

In a former article, in collaboration with Jean-Michel Vallin, we have constructed two "quantum groupoïds" dual to each other, from a depth 2 inclusion of von Neumann algebras M 0 ⊂ 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 ∩ M 1 , and we show how these examples fit with Lesieur's "measured quantum groupoïds".

Measured quantum groupoids associated with matched pairs of locally compact groupoids

“…In an other article ( [10]), starting with any depth 2 inclusion of von Neumann algebras M 0 ⊂ M 1 , with an operator-valued weight T 1 verifying a regularity condition, Michel Enock and the author have given a pseudomultiplicative unitary generating two Hopf bimodules in duality; one of them acts on M 1 in such a way that M 0 is isomorphic to the fixed point algebra and the von Neumann algebra M 2 , given by the basic construction, is isomorphic to the crossed product.…”

Comonoidal W*-Morita equivalence for von Neumann bialgebras

“…If the right M 2 -module generated by 12 .N 1 / is -weakly dense in N 2 , then .e/ D f , and hence unital.…”

“…Then we denote Q 1 Q 2 for the corner of Q 1 x Q 2 by the projection e˝f C .1 e/˝.1 f /. The reason for this notation is that this can (easily) be shown to be a special case of a fibred product of von Neumann algebras (i.e., fibred over C 2 ); see [12], Sections 2.3 and 2.4.…”

Pseudo-multiplicative unitaries on C*-modules and Hopf C*-families I

“…In the setting of von Neumann algebras, the theory of quantum groups was extended to a theory of measured quantum groupoids by Lesieur [10], building on work of Vallin and Enock [5], [6], [21]. Central concepts in this theory are Hopf-von Neumann bimodules and pseudo-multiplicative unitaries on Hilbert spaces, which generalize Hopf C*-algebras and multiplicative unitaries, respectively.…”

“…Pseudo-multiplicative unitaries on C*-modules generalize the multiplicative unitaries of Baaj and Skandalis [1], and are analogues of the pseudo-multiplicative unitaries on Hilbert spaces studied by Enock, Lesieur, Vallin [5], [10], [21]. We introduce Hopf C*-families on C*-bimodules and associate to special classes of pseudo-multiplicative unitaries two Hopf C*-families.…”