Normal intermediate subfactors

TL;DR

This paper introduces the concept of normal intermediate subfactors in type II$_1$ factors, characterizing their normality in terms of subgroup normality when the factors are crossed products by finite groups.

Contribution

It defines normality for intermediate subfactors and characterizes it in the depth 2 case, linking it to subgroup normality in crossed product scenarios.

Findings

Normal intermediate subfactors are characterized by depth conditions.

In depth 2 cases, normality corresponds to subgroup normality.

Normality is preserved under certain factorization conditions.

Abstract

Let $N \subset M$ be an irreducible inclusion of type type II$_1$ factors with finite Jones index. We shall introduce the notion of normality for intermediate subfactors of the inclusion $N \subset M$. If the depth of $N \subset M$ is 2, then an intermediate subfactor $K$ for $N \subset M$ is normal in $ N \subset M$ if and only if the depths of $N \subset K$ and $K \subset M$ are both 2. In particular, if $M$ is the crossed product $N \rtimes G$ of a finite group $G$, then $K = N \rtimes H$ is normal in $N \subset M$ if and only if $H$ is a normal subgroup of $G$.