2007/10/29 by Michel Enock, Enock, Michel
Mathematics · Physics and Astronomy · #46L55 #46L89 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Noncommutative and Quantum Gravity Theories #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.0710.5364
openalex publication_date 2007/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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