SAT actions of discrete quantum groups and minimal injective extensions of their von Neumann algebras

TL;DR

This paper generalizes SAT states for quantum group actions, showing that for discrete Kac-type quantum groups, unique stationary SAT states imply rigidity in their von Neumann algebra extensions.

Contribution

It introduces a generalized SAT state concept for quantum groups and establishes new rigidity results for their von Neumann algebra extensions in the Kac type case.

Findings

Unique stationary SAT states imply rigidity of injective extensions

Generalization of SAT states to quantum group actions

Results specific to discrete Kac-type quantum groups

Abstract

We introduce a natural generalization of the notion of strongly approximately transitive (SAT) states for actions of locally compact quantum groups. In the case of discrete quantum groups of Kac type, we show that the existence of unique stationary SAT states entails rigidity results concerning injective extensions of quantum group von Neumann algebras.