  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.
