[Paper Review] Measured Quantum Groupoids in action

Michel Enock|ArXiv.org|Oct 29, 2007

Advanced Operator Algebra Research13 references4 citations

TL;DR

This paper develops the theory of actions, crossed-products, and duality for measured quantum groupoids in von Neumann algebras, extending Vaes' biduality theorem for locally compact quantum groups. It proves that the inclusion of a von Neumann algebra into its crossed-product under such an action is depth 2, providing a converse to a result by Vallin and Enock, and constructs a canonical measured quantum groupoid from any outer action of a locally compact quantum group.

ABSTRACT

Franck Lesieur had introduced in his thesis (now published in an expended and revised version in the {\it Mémoires 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

Motivation & Objective

• To extend the theory of locally compact quantum groups to the more general framework of measured quantum groupoids in von Neumann algebras.
• To develop the notion of groupoid actions and crossed-products in this setting, generalizing Vaes' work on quantum groups.
• To establish a biduality theorem for measured quantum groupoids, mirroring the duality theorems in quantum group theory.
• To prove that the inclusion of the initial algebra into its crossed-product is depth 2, offering a converse to a known result on depth 2 inclusions.
• To construct a canonical measured quantum groupoid from any outer action of a locally compact quantum group, recovering the original group when the action is outer.

Proposed method

• Utilizes the framework of pseudo-multiplicative unitaries and Hopf-bimodules to define and study measured quantum groupoids.
• Applies operator-valued weights and Jones' basic construction to analyze depth 2 inclusions of von Neumann algebras.
• Employs slice maps and Radon-Nikodym theorems (Vaes-type) to handle modular structures and implement duality.
• Introduces a standard implementation of an action using a δ-invariant weight to construct the crossed-product algebra.
• Uses the theory of corepresentations and dual actions to define and analyze the biduality structure.
• Applies the theory of co-inverse and scaling groups to characterize modular properties and ensure consistency of duality.

Experimental results

Research questions

• RQ1 How can the notion of action and crossed-product be generalized from locally compact quantum groups to measured quantum groupoids?
• RQ2 Does a biduality theorem hold for measured quantum groupoids, analogous to the one for locally compact quantum groups?
• RQ3 Can the depth 2 property of the inclusion M₀ ⊂ M₁ be characterized as a converse to the known result that depth 2 inclusions give rise to measured quantum groupoids?
• RQ4 What is the structure of the measured quantum groupoid canonically associated to an outer action of a locally compact quantum group?
• RQ5 How do the modular automorphisms and scaling groups behave under duality in this generalized framework?

Key findings

• The inclusion of the initial von Neumann algebra into its crossed-product under a measured quantum groupoid action is proven to be depth 2, providing a converse to a result by Vallin and Enock.
• A biduality theorem is established for measured quantum groupoids, extending Vaes' result for locally compact quantum groups to the groupoid setting.
• For any outer action of a locally compact quantum group, the associated measured quantum groupoid is isomorphic to the original quantum group, demonstrating a canonical duality.
• The co-inverse and scaling group of a measured quantum groupoid are shown to preserve the structure under duality, ensuring consistency of the modular objects.
• The modular automorphisms σΦ and σΦ∘R commute, and the modular parameter λ is shown to be affiliated to β(N) and fixed by the co-inverse R.
• The standard implementation of an action, using a δ-invariant weight, leads to a well-defined crossed-product with a dual action, confirming the duality structure.