Measured quantum groupoids with a central basis

Abstract

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 this article, we suppose that the basis of the measured quantum groupoid is central; in that case, we prove that a specific sub-${\mathbf{C}}^{\ast }$ algebra is invariant under all the data of the measured quantum groupoid; moreover, this sub-${\mathbf{C}}^{\ast }$-algebra is a continuous field of ${\mathbf{C}}^{\ast }$-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-${\mathbf{C}}^{\ast }$-algebra, we prove that any abelian measured quantum groupoid comes from a locally compact groupoid.

to be published in Journal of Operator Theory

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Cited by in corpus (1)

• Locally compact quantum groupoid