Commuting squares and planar subalgebras

TL;DR

This paper explores the deep connection between commuting squares of II_1 factors and subfactor planar algebras, establishing a bidirectional construction linking these structures.

Contribution

It demonstrates that non-degenerate smooth commuting squares of II_1 factors correspond to inclusions of subfactor planar algebras, using the Guionnet-Jones-Shlyakhtenko construction.

Findings

Establishes a correspondence between commuting squares and planar algebra inclusions

Provides a construction method for translating between the two structures

Enhances understanding of subfactor theory and planar algebra relationships

Abstract

We show a close relationship between non-degenerate smooth commuting squares of $II_1$-factors with all inclusions of finite index and inclusions of subfactor planar algebras by showing that each leads to a construction of the other. One direction of this uses the Guionnet-Jones-Shlyakhtenko construction.