A Comment on Jones Inclusions with infinite Index

TL;DR

This paper demonstrates how certain infinite von Neumann algebra inclusions with depth 2 can be explicitly realized as fixed point algebras under outer actions of compact Kac-algebras, providing an alternative proof of a known result.

Contribution

It offers an explicit construction of the fixed point algebra description for depth 2 inclusions of infinite von Neumann algebras, extending previous results.

Findings

Explicit realization of $ abla$ as fixed point algebra under Kac-algebra action

Alternative proof of a general result by Enock and Nest

Clarification of the structure of depth 2 inclusions in infinite von Neumann algebras

Abstract

Given an irreducible inclusion of infinite von-Neumann-algebras $\cn \subset \cm$ together with a conditional expectation $ E : \cm \rightarrow \cm $ such that the inclusion has depth 2, we show quite explicitely how $\cn $ can be viewed as the fixed point algebra of $\cm$ w.r.t. an outer action of a compact Kac-algebra acting on $\cm$. This gives an alternative proof, under this special setting of a more general result of M. Enock and R. Nest, [E-N], see also S. Yamagami, [Ya2].