Commuting-Square Subfactors and Central Sequences

TL;DR

This paper investigates the properties of subfactors constructed from commuting squares, showing that their central sequence inclusions have infinite index, with applications to subfactors derived from groups.

Contribution

It establishes that subfactors from commuting squares have infinite Pimnser-Popa index in their central sequence inclusion, extending to certain group-related subfactors.

Findings

Central sequence inclusion has infinite index for commuting square subfactors.

Results apply to specific infinite-depth hyperfinite subfactors from groups.

Provides new insights into the structure of subfactors and their indices.

Abstract

Let $M_0 \subset M_1$ be a finite-index infinite-depth hyperfinite $II_1$ subfactor and $\omega$ a free ultrafilter of the natural numbers. We show that if this subfactor is constructed from a commuting square then the central sequence inclusion $M_0^{\omega} \cap M_1' \subset (M_0)_{\omega}$ has infinite Pimnser-Popa index. We will also demonstrate this result for certain infinite-depth hyperfinite subfactors coming from groups.