Free transport for finite depth subfactor planar algebras

Abstract

Given a finite depth subfactor planar algebra $\mathcal{P}$ endowed with the graded $\ast$-algebra structures $\{G{r}_k^{+}\mathcal{P}{\}}_{k\in \mathbb{N}}$ of Guionnet, Jones, and Shlyakhtenko, there is a sequence of canonical traces $T{r}_{k,+}$ on $G{r}_k^{+}\mathcal{P}$ 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 $\ast$-algebras $\{G{r}_k^{+}\mathcal{P}{\}}_{k\in \mathbb{N}}$ 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 $T{r}_{k,+}^{(v)}$ induced by certain small perturbations of the Temperley-Lieb diagrams yield trace-preserving embeddings of $G{r}_k^{+}\mathcal{P}$ that generate the same tower $\{M_{k,+}{\}}_{k\in \mathbb{N}}$.

23 pages, revised for publication

References in corpus (1)

• Random matrices, free probability, planar algebras and subfactors

Cited by in corpus (1)

• Free analysis and planar algebras