Cones with a Mapping Cone Symmetry in the Finite-Dimensional Case

TL;DR

This paper introduces a new approach to the theory of cones with mapping cone symmetry in finite-dimensional operator algebras, simplifying proofs and generalizing key results such as duality and characterization of these cones.

Contribution

It presents a novel method based on an inner product and isometry properties, leading to faster proofs and broader applicability of results on mapping cones.

Findings

Dual of a mapping cone is also a mapping cone.

Characterization of cones with mapping cone symmetry via composition with dual elements.

Generalization of known results without symmetry or domain-target restrictions.

Abstract

In the finite-dimensional case, we present a new approach to the theory of cones with a mapping cone symmetry, first introduced by St{\o}rmer. Our method is based on a definition of an inner product in the space of linear maps between two algebras of operators and the fact that the Jamio{\l}kowski-Choi isomorphism is an isometry. We consider a slightly modified class of cones, although not substantially different from the original mapping cones by St{\o}rmer. Using the new approach, several known results are proved faster and often in more generality than before. For example, the dual of a mapping cone turns out to be a mapping cone as well, without any additional assumptions. The main result of the paper is a characterization of cones with a mapping cone symmetry, saying that a given map is an element of such cone if and only if the composition of the map with the conjugate of an arbitrary element in the dual cone is completely positive. A similar result was known in the case where the map goes from an algebra of operators into itself and the cone is a symmetric mapping cone. Our result is proved without the additional assumptions of symmetry and equality between the domain and the target space. We show how it gives a number of older results as a corollary, including an exemplary application.