Measured quantum groupoids with a central basis

Details Measured quantum groupoids with a central basis Measured quantum groupoids with a central basis

2008-08-29

arxiv.org/abs/0808.4052v2

Mathematics

Operator Algebras

to be published in Journal of Operator Theory

Scientific paper

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-${\bf C}^*$ algebra is invariant under all the data of the measured quantum groupoid; moreover, this sub-${\bf C}^*$-algebra is a continuous field of ${\bf C}^*$-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-${\bf C}^*$-algebra, we prove that any abelian measured quantum groupoid comes from a locally compact groupoid.

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