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

Mehrdad Kalantar, Fatemeh Khosravi, Mohammad S. M. Moakhar

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.

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

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.
Paper Structure (8 sections, 11 theorems, 29 equations)

This paper contains 8 sections, 11 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Relative Yetter--Drinfeld algebras
  4. Strongly Approximately Transitive (SAT) State
  5. Unique stationary SAT states
  6. Examples and applications
  7. Orthogonal free quantum groups
  8. $\widehat{\mathbb G}$-injectivity

Key Result

Lemma 2.1

Let $\mathbb G$ be a discrete quantum group, and let $N$ be a $\mathbb G$-von Neumann algebra. The canonical conditional expectation $E:\mathbb G\ltimes_{\alpha}N\to \alpha(N)$ is $\mathbb G$-equivariant if and only if $\mathbb G$ is of Kac type.

Theorems & Definitions (27)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Remark 3.5
  • ...and 17 more