Mémoires de la Société Mathématique de France 2008; 122 pp; softcover Number: 109 ISBN10: 285629233X ISBN13: 9782856292334 List Price: US$40 Individual Members: US$36 Order Code: SMFMEM/109
 In this volume, the author gives a definition for measured quantum groupoids. He aims to construct objects with duality including both quantum groups and groupoids. J. Kustermans and S. Vaes' works about locally compact quantum groups the author generalizes thanks to formalism introduced by M. Enock and J. M. Vallin in the case of inclusion of von Neumann algebras. From a structure of Hopfbimodule with left and right invariant operatorvalued weights, the author defines a fundamental pseudomultiplicative unitary. To get a satisfying duality in the general case, he assumes the existence of an antipode given by its polar decomposition. This theory is illustrated with many examples, among them the inclusion of von Neumann algebras (M. Enock) and a sub family of measured quantum groupoids with easier axiomatic. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Readership Graduate students and research mathematicians interested in analysis. Table of Contents  Introduction
 Recalls
 Fundamental pseudomultiplicative unitary
Part I. Measured quantum groupoids  Definition
 Uniqueness, modulus and scaling operator
 A density theorem
 Manageability of the fundamental unitary
 Duality
Part II. Examples  Adapted measured quantum groupoids
 Groupoids
 Finite quantum groupoids
 Quantum groups
 Compact case
 Quantum space quantum groupoid
 Pairs quantum groupoid
 Inclusions of von Neumann algebras
 Operations on adapted measured quantum groupoids
 Bibliography
