## Proceedings of the Centre for Mathematics and its Applications

### Free Analysis and Planar Algebras

#### 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.

#### Article information

Dates
First available in Project Euclid: 21 February 2017

Permanent link to this document
https://projecteuclid.org/ euclid.pcma/1487646025

Mathematical Reviews number (MathSciNet)
MR3635669

Zentralblatt MATH identifier
06990152

#### Citation

Curran, S.; Dabrowski, Y.; Shlyakhtenko, D. Free Analysis and Planar Algebras. Proceedings of the 2014 Maui and 2015 Qinhuangdao Conferences in Honour of Vaughan F.R. Jones’ 60th Birthday, 115--142, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra AUS, 2017. https://projecteuclid.org/euclid.pcma/1487646025