Hilbert modules over a planar algebra and the Haagerup property

Given a subfactor planar algebra PP and a Hilbert PP-module of lowest weight 0 we build a bimodule over the symmetric enveloping inclusion associated to PP. 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.