The unitary implementation of a measured quantum groupoid action

Frank Lesieur a introduit une notion de groupoïde quantique mesuré, dans le cadre des algèbres de von Neumann, en s’inspirant des groupes quantiques localement compacts de Kustermans et Vaes (dans la version de cette construction faite dans le cadre des algèbres de von Neumann). Dans un article précédent, l’auteur a introduit les notions d’action, de produit croisé, d’action duale d’un groupoïde quantique mesuré ; un théorème de bidulaité des actions a éte démontré. Cet article continue ce programme : nous démontrons l’existence d’une implémentation standard d’une action, et un théorème de bidulaité pour les poids. Sont ainsi généralisés des résultats qui avaient été démontrés par S. Vaes pour les groupes quantiques localement compacts, et par T. Yamanouchi pour les groupoïdes mesurés.

Mimicking the von Neumann version of Kustermans and Vaes’ locally compact quantum groups, Franck Lesieur had introduced a notion of measured quantum groupoid, in the setting of von Neumann algebras. In a former article, the author had introduced the notions of actions, crossed-product, dual actions of a measured quantum groupoid; a biduality theorem for actions has been proved. This article continues that program: we prove the existence of a standard implementation for an action, and a biduality theorem for weights. We generalize this way results which were proved, for locally compact quantum groups by S. Vaes, and for measured groupoids by T. Yamanouchi.

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