Free analysis and planar algebras

TL;DR

This paper explores the extension of Voiculescu's trace and free Gibbs states to Jones planar algebras, establishing conditions under which associated von Neumann algebras are factors with specific invariants.

Contribution

It introduces 2-cabled analogs of free states on planar algebras and links free Fisher information finiteness to the factor property of associated von Neumann algebras.

Findings

Finiteness of free Fisher information implies the von Neumann algebras are factors.

The standard invariant of the inclusion matches the original planar algebra.

Conditions for the von Neumann algebras to be non-$ ext{ extGamma}$ non-$L^2$ rigid factors.

Abstract

We study 2-cabled analogs of Voiculescu's trace and free Gibbs states on Jones planar algebras. These states are traces on a tower of graded algebras associated to a Jones planar algebra. Among our results is that, with a suitable definition, finiteness of free Fisher information for planar algebra traces implies that the associated tower of von Neumann algebras consists of factors, and that the standard invariant of the associated inclusion is exactly the original planar algebra. We also give conditions that imply that the associated von Neumann algebras are non-$\Gamma$ non-$L^2$ rigid factors.