Realizing the Braided Temperley-Lieb-Jones C*-Tensor Categories as Hilbert C*-Modules

We associate to each Temperley-Lieb-Jones C*-tensor category TLJ( d) with parameter d in the discrete range {2 cos(pi/(k + 2)) : k = 1, 2,...}. {2} a certain C*-algebra B of compact operators. We use the unitary braiding on TLJ(delta) to equip the category ModB of (right) Hilbert B-modules with the structure of a braided C*-tensor category. We show that TLJ( d) is equivalent, as a braided C*-tensor category, to the full subcategory Mod fB of ModB whose objects are those modules which admit a finite orthonormal basis. Finally, we indicate how these considerations generalize to arbitrary finitely generated rigid braided C*-tensor categories.