Inductive limit in the category of TRO

Abstract

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.