Free transport for finite depth subfactor planar algebras

TL;DR

This paper demonstrates that small perturbations of Temperley-Lieb diagrams induce trace-preserving embeddings of graded algebras associated with finite depth subfactor planar algebras, generating the same non-commutative probability tower.

Contribution

It introduces a method to generate the same tower of non-commutative probability spaces using perturbed Temperley-Lieb diagrams, extending the understanding of subfactor planar algebra embeddings.

Findings

Perturbed traces induce trace-preserving embeddings.

Generated towers recover the original subfactor planar algebra.

Embeddings are stable under small diagram perturbations.

Abstract

Given a finite depth subfactor planar algebra $\mathcal{P}$ endowed with the graded $*$-algebra structures $\{Gr_k^+ \mathcal{P}\}_{k\in\mathbb{N}}$ of Guionnet, Jones, and Shlyakhtenko, there is a sequence of canonical traces $Tr_{k,+}$ on $Gr_k^+\mathcal{P}$ induced by the Temperley-Lieb diagrams and a sequence of trace-preserving embeddings into the bounded operators on a Hilbert space. Via these embeddings the $*$-algebras $\{Gr_k^+\mathcal{P}\}_{k\in \mathbb{N}}$ generate a tower of non-commutative probability spaces $\{M_{k,+}\}_{k\in\mathbb{N}}$ whose inclusions recover $\mathcal{P}$ as its standard invariant. We show that traces $Tr_{k,+}^{(v)}$ induced by certain small perturbations of the Temperley-Lieb diagrams yield trace-preserving embeddings of $Gr_k^+\mathcal{P}$ that generate the same tower $\{M_{k,+}\}_{k\in\mathbb{N}}$.