Abstract
We associate to each Temperley–Lieb–Jones C*-tensor category {mathcal {T}}{mathcal {L}}{mathcal {J}}(delta ) with parameter delta in the discrete range {2cos (pi /(k+2)):,k=1,2,ldots }cup {2} a certain C*-algebra {mathcal {B}} of compact operators. We use the unitary braiding on {mathcal {T}}{mathcal {L}}{mathcal {J}}(delta ) to equip the category mathrm {Mod}_{{mathcal {B}}} of (right) Hilbert {mathcal {B}}-modules with the structure of a braided C*-tensor category. We show that {mathcal {T}}{mathcal {L}}{mathcal {J}}(delta ) is equivalent, as a braided C*-tensor category, to the full subcategory mathrm {Mod}_{{mathcal {B}}}^f of mathrm {Mod}_{{mathcal {B}}} 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.
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