Random matrices, free probability, planar algebras and subfactors

TL;DR

This paper constructs a tower of non-commutative probability spaces from planar algebras using random matrix theory, providing an alternative proof that every planar algebra can be realized by a subfactor.

Contribution

It introduces a new approach linking planar algebras, free probability, and subfactors via graded algebra structures and random matrices.

Findings

Constructs II_1 factors from planar algebras

Realizes planar algebras as systems of higher relative commutants

Provides an alternative proof of Popa's realization theorem

Abstract

Using a family of graded algebra structures on a planar algebra and a family of traces coming from random matrix theory, we obtain a tower of non-commutative probability spaces, naturally associated to a given planar algebra. The associated von Neumann algebras are II$_{1}$ factors whose inclusions realize the given planar algebra as a system of higher relative commutants. We thus give an alternative proof to a result of Popa that every planar algebra can be realized by a subfactor.