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Abstract

We present a systematic development of inductive limits in the categories of ordered ${}^*$-vector spaces, Archimedean order unit spaces, matrix ordered spaces, operator systems and operator ${\rm C}^*$-systems. We show that the inductive limit intertwines the operation of passing to the maximal operator system structure of an Archimedean order unit space, and that the same holds true for the minimal operator system structure if the connecting maps are complete order embeddings. We prove that the inductive limit commutes with the operation of taking the maximal tensor product with another operator system, and establish analogous results for injective functorial tensor products provided the connecting maps are complete order embeddings. We identify the inductive limit of quotient operator systems as a quotient of the inductive limit, in case the kernels involved satisfy a lifting condition, implied by complete biproximinality. We describe the inductive limit of graph operator systems as operator systems of topological graphs, show that two such operator systems are completely order isomorphic if and only if their underlying graphs are isomorphic, identify the ${\rm C}^*$-envelope of such an operator system, and prove a version of Glimm’s Theorem on the isomorphism of UHF algebras in the category of operator systems.