Measured Quantum Groupoids with a Central Basis

In ([L2]), Franck Lesieur had introduced a notion of measured quantum groupoid, in the setting of von Neumann algebras. In an annex of [E6], Lesieur’s axioms have been simplified. In this article, we suppose that the basis is central; in that case, we prove that a specific sub-C∗ algebra is, in a sense, invariant under all the data which define the measured quantum group, which allow us to prove that any abelian measured quantum groupoid comes from a locally compact groupoid; moreover, this sub-C∗-algebra is a continuous field of C∗-algebras, which allow us to characterize the measured quantum groupoid whose basis is central, and central also in the dual, as continuous fields of locally compact quantum groups. This allows us to re-use as quantum groupoids some examples introduced by Blanchard in [B2].