## Proceedings of the Centre for Mathematics and its Applications

- Proc. Centre Math. Appl.
- Proceedings of the 2014 Maui and 2015 Qinhuangdao Conferences in Honour of Vaughan F.R. Jones’ 60th Birthday. Scott Morrison and David Pennys, eds. Proceedings of the Centre for Mathematics and its Applications, v. 46. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 2017), 115 - 142

### Free Analysis and Planar Algebras

S. Curran, Y. Dabrowski, and D. Shlyakhtenko

#### 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