Outer actions of measured quantum groupoids

TL;DR

This paper extends the concept of outer actions from locally compact quantum groups to measured quantum groupoids, demonstrating their existence and linking them to depth 2 von Neumann algebra inclusions.

Contribution

It introduces an appropriate definition of outer actions for measured quantum groupoids and proves their existence, generalizing previous results for quantum groups.

Findings

Every measured quantum groupoid admits an outer action.

Outer actions are used to realize quantum groupoids from depth 2 von Neumann algebra inclusions.

The work generalizes the concept of outer actions from quantum groups to quantum groupoids.

Abstract

Mimicking a recent article of Stefaan Vaes, in which was proved that every locally compact quantum group can act outerly, we prove that we get the same result for measured quantum groupoids, with an appropriate definition of outer actions of measured quantum groupoids. This result is used to show that every measured quantum groupoid can be found from some depth 2 inclusion of von Neumann algebras.