We study inductive limit in the category of ternary ring of operator (TRO). The existence of inductive limit in this category is proved and its behaviour with quotienting is discussed. For a TRO V, if A(V) is the linking $$C^{*}$$-algebra generated by V, then we investigate whether it commutes with inductive limits of TROs, in the sense that if $$(V_n,f_n)$$ is an Inductive system then $$\varinjlim A(V_n)=A(\varinjlim V_n)$$. We show that some local properties such as simplicity, nuclearity and exactness behaves well with the inductive limit of TROs. We also discuss the commutativity of inductive limit of TROs with tensor products.

中文翻译：

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TRO 类中的电感限制

我们研究了算子三元环（TRO）范畴的归纳极限。证明了该范畴中归纳极限的存在，并讨论了其与商的行为。对于 TRO V，如果 A(V) 是由 V 生成的链接 $$C^{*}$$-代数，那么我们研究它是否与 TRO 的归纳限制相通，从某种意义上说，如果 $$(V_n, f_n)$$ 是一个归纳系统，则 $$\varinjlim A(V_n)=A(\varinjlim V_n)$$。我们展示了一些局部属性，如简单性、核性和精确性，在 TRO 的归纳限制下表现良好。我们还讨论了具有张量积的 TRO 的归纳极限的交换性。