Free product von Neumann algebras associated to graphs and Guionnet, Jones, Shlyakhtenko subfactors in infinite Depth

TL;DR

This paper extends the understanding of free product von Neumann algebras associated with subfactor planar algebras, showing that infinite-depth cases produce factors isomorphic to L(F∞), generalizing previous finite-depth results.

Contribution

It proves that for infinite-depth subfactor planar algebras, the associated von Neumann factors are isomorphic to L(F∞), broadening the classification of these algebras.

Findings

Finite-depth planar algebras yield interpolated free group factors.

Infinite-depth planar algebras produce factors isomorphic to L(F∞).

Generalization of subfactor algebra classification to infinite-depth cases.

Abstract

Given a subfactor planar algebra P, Guionnet, Jones and Shlyakhtenko give a diagrammatic construction of a II_{1} subfactor whose planar algebra is P. They showed if P is finite-depth, then the factors are interpolated free group factors, and they identified the parameters. We prove if P is infinite-depth, then the factors are isomorphic to L(F\infty}$.