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

## TL;DR

This paper constructs a C*-algebra associated with Temperley-Lieb-Jones categories and demonstrates their equivalence to a subcategory of Hilbert modules, providing a new realization of these categories as braided C*-tensor categories.

## Contribution

It introduces a C*-algebra framework for Temperley-Lieb-Jones categories and establishes their equivalence to a subcategory of Hilbert modules, extending to general finitely generated rigid braided C*-tensor categories.

## Key findings

- Equivalence of Temperley-Lieb-Jones categories with a subcategory of Hilbert modules.
- Construction of a C*-algebra of compact operators associated with these categories.
- Extension of the framework to arbitrary finitely generated rigid braided C*-tensor categories.

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