A finite-dimensional realization in the sense of the Choi-Effros Theorem

Abstract Operator Systems over the Cone of Positive Semidefinite Matrices

We investigate which of several important abstract operator systems with the convex cone of positive semidefinite matrices at the first level is finitely generated, and which admits a finite-dimensional realization in the sense of the choi-effros theorem.We answer this question for all of the described systems completely.Our main contribution is that decomposable maps form a system which does not admit a finite-dimensional realization, though being finitely generated, whereas the system of doubly completely positive maps is not finitely generated, though having a finite-dimensional realization.This implies that there can not exist a finitary choi-type characterization of doubly completely positive maps.