Measured Quantum Groupoids in action

TL;DR

This paper develops the theory of measured quantum groupoids, including actions, crossed-products, and biduality, extending concepts from locally compact quantum groups within the framework of von Neumann algebras.

Contribution

It introduces a simplified axiomatic framework for measured quantum groupoids and establishes new duality and depth 2 inclusion results, generalizing prior work on quantum groups.

Findings

Established a biduality theorem for measured quantum groupoids.

Proved the depth 2 property of the inclusion into crossed-products.

Constructed a measured quantum groupoid from any given action, generalizing locally compact quantum groups.

Abstract

Franck Lesieur had introduced in his thesis (now published in an expended and revised version in the {\it M\'emoires de la SMF} (2007)) a notion of measured quantum groupoid, in the setting of von Neumann algebras and a simplification of Lesieur's axioms is presented in an appendix of this article. We here develop the notions of actions, crossed-product, and obtain a biduality theorem, following what had been done by Stefaan Vaes for locally compact quantum groups. Moreover, we prove that the inclusion of the initial algebra into its crossed-product is depth 2, which gives a converse of a result proved by Jean-Michel Vallin and the author. More precisely, to any action of a measured quantum groupoid, we associate another measured quantum groupoid. In particular, starting from an action of a locally compact quantum group, we obtain a measured quantum groupoid canonically associated to this action; when the action is outer, this measured quantum groupoid is the initial locally compact quantum group