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Realizing the Braided Temperley–Lieb–Jones C*-Tensor Categories as Hilbert C*-Modules
Communications in Mathematical Physics ( IF 2.102 ) Pub Date : 2020-05-23 , DOI: 10.1007/s00220-020-03729-w
Andreas Næs Aaserud, David E. Evans

We associate to each Temperley–Lieb–Jones C*-tensor category \({\mathcal {T}}{\mathcal {L}}{\mathcal {J}}(\delta )\) with parameter \(\delta \) in the discrete range \(\{2\cos (\pi /(k+2)):\,k=1,2,\ldots \}\cup \{2\}\) 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.
更新日期:2020-05-23

 

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