Compact quantum groupoids in the setting of C -algebras

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 prov...

C-pseudo-multiplicative unitaries and Hopf C-bimodules

We introduce C∗-pseudo-multiplicative unitaries and concrete Hopf C∗-bimodules for the study of quantum groupoids in the setting of C∗-algebras. These unitaries and Hopf C∗-bimodules generalize multiplicative unitaries and Hopf C∗-algebras and are analogues of the pseudo-multiplicative unitaries and Hopf–von Neumann-bimodules studied by Enock, Lesieur and Vallin. To each C∗-pseudo-multiplicativ...

Reiter’s Properties for the Actions of Locally Compact Quantum Goups on von Neumann Algebras

ar X iv : m at h - ph / 9 91 20 06 v 1 7 D ec 1 99 9 Compact Quantum Groupoids

Quantum groupoids are a joint generalization of groupoids and quantum groups. We propose a definition of a compact quantum groupoid that is based on the theory of C *-algebras and Hilbert bimodules. The essential point is that whenever one has a tensor product over C in the theory of quantum groups, one now uses a certain tensor product over the base algebra of the quantum groupoid.

C*-algebras on r-discrete Abelian Groupoids

We study certain function algebras and their operator algebra completions on r-discrete abelian groupoids, the corresponding conditional expectations, maximal abelian subalgebras (masa) and eigen-functionals. We give a semidirect product decomposition for an abelian groupoid. This is done through a matched pair and leads to a C*-diagonal (for a special case). We use this decomposition to study ...