Classification of actions of compact abelian groups on subfactors with index less than 4

TL;DR

This paper classifies actions of abelian groups on certain von Neumann algebra inclusions, providing a comprehensive understanding of their structure and equivalence classes, especially for compact abelian groups on AFD factors of type II_1 with index less than 4.

Contribution

It introduces a classification framework for abelian group actions on von Neumann algebra inclusions, extending to compact abelian groups on AFD factors with specific index constraints.

Findings

Actions of discrete abelian groups are classified up to cocycle conjugacy.

Actions of compact abelian groups on AFD factors of type II_1 with index less than 4 are classified up to stable conjugacy.

The classification results provide a complete understanding of these group actions in the specified setting.

Abstract

We calssify actions of discrete abelian groups on some inclusions of von Neumann algebras, up to cocycle conjugacy. As an application, we classify actions of compact abelian groups on the inclusions of AFD factors of type II_1 with index less than 4, up to stable conjugacy.