Representation theory for subfactors, $\lambda$-lattices and C*-tensor categories

TL;DR

This paper develops a representation theory for $mbda$-lattices and rigid C*-tensor categories, analyzing their approximation and rigidity properties, and providing new examples with property (T).

Contribution

It introduces a systematic framework for the representation theory of $mbda$-lattices and C*-tensor categories, including the universal C*-algebra, and identifies new subfactors with property (T).

Findings

All unitary representations of Temperley-Lieb-Jones $mbda$-lattices are classified.

Temperley-Lieb-Jones $mbda$-lattices have the Haagerup property.

First examples of subfactors with property (T) not derived from property (T) groups.

Abstract

We develop a representation theory for $\lambda$-lattices, arising as standard invariants of subfactors, and for rigid C*-tensor categories, including a definition of their universal C*-algebra. We use this to give a systematic account of approximation and rigidity properties for subfactors and tensor categories, like (weak) amenability, the Haagerup property and property (T). We determine all unitary representations of the Temperley-Lieb-Jones $\lambda$-lattices and prove that they have the Haagerup property and the complete metric approximation property. We also present the first subfactors with property (T) standard invariant and that are not constructed from property (T) groups.