On Lesieur's Measured Quantum Groupoids

TL;DR

This paper simplifies the axioms of measured quantum groupoids within von Neumann algebras, building on Lesieur's prior complex framework and the concept of pseudo-multiplicative units.

Contribution

The paper provides a simplified set of axioms for measured quantum groupoids, making the theory more accessible and easier to work with.

Findings

Simplified the axioms of measured quantum groupoids

Clarified the role of pseudo-multiplicative units

Enhanced understanding of quantum groupoid structure

Abstract

In his thesis ([L1]), which is published in an expended and revised version ([L2]), Franck Lesieur had introduced a notion of measured quantum groupoid, in the setting of von Neumann algebras, using intensively the notion of pseudo-multiplicative unitary, which had been introduced in a previous article of the author, in collaboration with Jean-Michel Vallin [EV]. In [L2], the axioms given are very complicated and are here simplified.