The Unitary Implementation of a Measured Quantum Groupoid action

TL;DR

This paper extends the theory of measured quantum groupoids by establishing a standard implementation for actions and a biduality theorem for weights, generalizing prior results from quantum groups and measured groupoids.

Contribution

It proves the existence of a standard implementation for actions and a biduality theorem for weights in the setting of measured quantum groupoids, advancing the theoretical framework.

Findings

Established a standard implementation for measured quantum groupoid actions.

Proved a biduality theorem for weights in this setting.

Generalized results from locally compact quantum groups and measured groupoids.

Abstract

Mimicking the von Neumann version of Kustermans and Vaes' locally compact quantum groups, Franck Lesieur had introduced a notion of measured quantum groupoid, in the setting of von Neumann algebras. In a former article, the author had introduced the notion of actions, crossed-product, dual actions of a measured quantum groupoid: a biduality theorem for actions had been proved. This article continues that program : we prove the existence of a standard implementation for an action, and a bidulaity theorem for weights. We generalize this way results which were proved, for locally compact quantum groups by S. Vaes, and for measured groupoids by T. Yamanouchi.