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

中文翻译：

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将编织 Temperley-Lieb-Jones C*-张量范畴实现为 Hilbert C*-Modules

我们将每个 Temperley–Lieb–Jones C* 张量类别 $${\mathcal {T}}{\mathcal {L}}{\mathcal {J}}(\delta )$$ TLJ ( δ ) 与参数 $ $\delta $$ δ 在离散范围 $$\{2\cos (\pi /(k+2)):\,k=1,2,\ldots \}\cup \{2\}$$ { 2 cos ( π / ( k + 2 ) ) : k = 1 , 2 , ... } ∪ { 2 } 某个 C*-代数 $${\mathcal {B}}$$ B 的紧算子。我们在 $${\mathcal {T}}{\mathcal {L}}{\mathcal {J}}(\delta )$$ TLJ ( δ ) 上使用幺正编织来装备类别 $$\mathrm {Mod} _{{\mathcal {B}}}$$（右）希尔伯特的 Mod B $${\mathcal {B}}$$ B 模块，具有编织 C* 张量类别的结构。我们证明 $${\mathcal {T}}{\mathcal {L}}{\mathcal {J}}(\delta )$$ TLJ ( δ ) 是等效的，作为编织的 C*-张量类别，到完整的子类别 $$\mathrm {Mod}_{{\mathcal {B}}}^f$$ Mod B f of $$\mathrm {Mod}_{{\mathcal {B}}}$$ Mod B其对象是那些承认有限正交基的模块。最后，我们指出这些考虑如何推广到任意有限生成的刚性编织 C* 张量类别。