Free transport for finite depth subfactor planar algebras

Given a finite depth subfactot planar algebra P endowed with the graded *-algebra structures {Gr(k)(+) P}(k is an element of N) of Guionnet, Jones, and Shlyakhtenko, there is a sequence of canonical traces Tr-k,Tr-+ on Gr(k)(+) 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)(+) P}(k is an element of N) generate a tower of non-commutative probability spaces {M-k,M-+}(k is an element of N) whose inclusions recover P as its standard invariant. We show that traces Tr-k,+((nu)) induced by certain small perturbations of the Temperley-Lieb diagrams yield trace-preserving embeddings of Gr(k)(+) P, that generate the same tower {M-k,M-+}(k is an element of N). (C) 2015 Elsevier Inc. All rights reserved.