Mapping cones are operator systems

We investigate the relationship between mapping cones and matrix ordered *-vector spaces (that is, abstract operator systems). We show that to every mapping cone there is an associated operator system on the space of n-by-n complex matrices, and furthermore we show that the associated operator system is unique and has a certain homogeneity property. Conversely, we show that the cone of completely positive maps on any operator system with that homogeneity property is a mapping cone. We also consider several related problems, such as characterizing cones that are closed under composition on the right by completely positive maps, and cones that are also semigroups, in terms of operator systems.