Hilbert modules over a planar algebra and the Haagerup property

TL;DR

This paper constructs bimodules from subfactor planar algebras and demonstrates that Temperley-Lieb-Jones invariants possess the Haagerup property, offering a new proof of a known result.

Contribution

It introduces a method to build bimodules from Hilbert P-modules and applies it to prove the Haagerup property for Temperley-Lieb-Jones invariants.

Findings

Temperley-Lieb-Jones invariants have the Haagerup property

New diagrammatic proof of Popa and Vaes's result

Construction of bimodules from subfactor planar algebras

Abstract

Given a subfactor planar algebra P and a Hilbert P-module of lowest weight 0 we build a bimodule over the symmetric enveloping inclusion associated to P. As an application we prove diagrammatically that the Temperley-Lieb-Jones standard invariants have the Haagerup property. This provides a new proof of a result due to Popa and Vaes.