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Free transport for finite depth subfactor planar algebras

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