On the Abstract Structure of Operator Systems and Applications to Quantum Information Theory

We introduce the notion of an abstract projection in an operator system and when a finite number of positive contractions in an operator system are all simultaneously abstract projections in that operator system. We extend this notion to Archimedean order unit spaces where we prove when a positive contraction is an abstract projection in some operator system, and furthermore when a finite number of positive contractions in an Archimedean order unit space are all simultaneously abstract projections in a single operator system. These methods are then used to provide new characterizations of both nonsignalling and quantum commuting correlations. In particular, we construct a universal Archimedean order unit space such that every quantum commuting correlation may be realized as the image of a unital linear positive map acting on the generators of that Archimedean order unit space. We also construct an Archimedean order unit space which is universal (in the same way) to nonsignalling correlations. We conclude with results concerning weak dual matrix ordered *-vector spaces and the operator systems they induce.