Journal of Operator Theory

Volume 66, Issue 1, Summer 2011 pp. 3-58.

Measured quantum groupoids with a central basis

Authors: Michel Enock
Author institution: Institut de Mathematiques de Jussieu, Unite Mixte Paris 6 / Paris 7, France; and CNRS de Recherche 7586, 175, rue du Chevaleret, Plateau 7E, F-75013 Paris, France

Summary: Mimicking the von Neumann version of Kustermans and Vaes' locally compact quantum groups, Franck Lesieur has introduced a notion of measured quantum groupoid, in the setting of von Neumann algebras. In this article, we suppose that the basis of the measured quantum groupoid is central; in that case, we prove that a specific sub-$C^*$-algebra is invariant under all the data of the measured quantum groupoid; moreover, this sub-$C^*$-algebra is a continuous field of $C^*$-algebras; when the basis is central in both the measured quantum groupoid and its dual, we get that the measured quantum groupoid is a continuous field of locally compact quantum groups. On the other hand, using this sub-$C^*$-algebra, we prove that any abelian measured quantum groupoid comes from a locally compact groupoid.

Keywords: Measured quantum groupoids, continuous fields of $C^*$-algebras, locally compact quantum groups

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