On the Guionnet-Jones-Shlyakhtenko construction for graphs

TL;DR

This paper demonstrates that applying the Guionnet-Jones-Shlyakhtenko construction to finite depth subfactor planar algebras results in inclusions of interpolated free group factors, reaffirming their universality.

Contribution

It extends the Guionnet-Jones-Shlyakhtenko construction to graphs, providing a new proof of universality for finite depth planar algebras.

Findings

Construction yields interpolated free group factors with finite parameters

Provides alternative proof of universality for finite depth planar algebras

Connects subfactor planar algebras to free probability structures

Abstract

Using an analogue of the Guionnet-Jones-Shlaykhtenko construction for graphs we show that their construction applied to any subfactor planar algebra of finite depth yields an inclusion of interpolated free group factors with finite parameter, thereby giving another proof of their universality for finite depth planar algebras.