Measured quantum groupoids

Abstract

In this article, part of the author's 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 fundamental 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 modulus and a scaling operator. This theory is illustrated with different examples. Duality of measured quantum groupoids will be discussed in a forthcoming article.