Mapping Cones are Operator Systems

TL;DR

This paper establishes a deep connection between mapping cones and operator systems, showing a one-to-one correspondence and characterizing their properties through matrix-ordered *-vector spaces.

Contribution

It proves that each mapping cone corresponds uniquely to an operator system with homogeneity, and vice versa, advancing the understanding of their structural relationship.

Findings

Every mapping cone has an associated unique operator system.

The cone of completely positive maps on certain operator systems forms a mapping cone.

Characterizations of cones closed under composition and semigroup properties are provided.

Abstract

We investigate the relationship between mapping cones and matrix ordered *-vector spaces (i.e., 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.