Realizing the braided Temperley-Lieb-Jones C*-tensor categories as Hilbert C*-modules

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 $\{2\cos (\pi /(k+2)):k=1,2,\dots \}\cup \{2\}$ 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.

In the latest version, we corrected a couple of typos and reformatted the bibliography. The paper will appear in essentially this form in Communications in Mathematical Physics (2020)

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• Associative-algebraic approach to logarithmic conformal field theories
• Enlarged symmetry algebras of spin chains, loop models, and S-matrices
• A fusion for the periodic Temperley-Lieb algebra and its continuum limit
• $K$-theory of AF-algebras from braided C*-tensor categories

Cited by in corpus (2)

• Subfactors and Mathematical Physics
• Remarks on anomalous symmetries of C*-algebras