Free transport for finite depth subfactor planar algebras

Given a finite depth subfactor planar algebra PP endowed with the graded ⁎-algebra structures {Grk+P}k∈N of Guionnet, Jones, and Shlyakhtenko, there is a sequence of canonical traces Trk,+Trk,+ on Grk+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 {Grk+P}k∈N generate a tower of non-commutative probability spaces {Mk,+}k∈N{Mk,+}k∈N whose inclusions recover PP as its standard invariant. We show that traces Trk,+(v) induced by certain small perturbations of the Temperley–Lieb diagrams yield trace-preserving embeddings of Grk+P that generate the same tower {Mk,+}k∈N{Mk,+}k∈N.