Free transport for finite depth subfactor planar algebras

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 ${M_{k,+}}_{k\in\mathbb{N}}$.