| mots-clés : Algèbres d'opérateurs – groupes quantiques – groupoïdes – unitaires pseudo-multiplicatifs – poids quasi-invariants – poids opératoriels de Haar – antipode – coinvolution – groupe d'échelle – module – opérateur d'échelle – théorème d'unicité – dualité – bidualité |
| autres localisations : http://www.univ-orleans.fr/SCIENCES/MAPMO/publications/lesieur/these.php http://www.math.unicaen.fr/~lesieur/recherche.html |
|
| Measured quantum groupoids: axiomatic, study, duality, examples |
| In this thesis, we propose a definition for measured quantum groupoid. The aim is the construction of objects with duality including both quantum groups and groupoids. We base ourselves on J. Kustermans and S. Vaes' works about locally compact quantum groups that we generalize thanks to formalism introduced by M. Enock and J.M. Vallin in the case of inclusion of von Neumann algebras. From a structure of Hopf-bimodule with left and right invariant operator-valued weights, we define a fondamental pseudo-multiplicative unitary. We introduce the notion of quasi-invariant weight on the basis and, then, we construct an antipode with polar decomposition, a coinvolution, a scaling group, a module and a scaling operator. The construction of the dual structure needs an extra condition which is satisfied in a lot of examples. We prove a biduality theorem when the basis is semifinite. This theory is illustrated with different examples. |
| mots-clés en anglais : Operator algebras – quantum groups – groupoids – pseudo-multiplicative unitaries – quasi-invariant weights – Haar operator-valued weights – antipode – co-involution – scalar group – module – scalar operator – unicity theorem – duality – biduality |
