Positive maps of second-order cones

Abstract

Let and let be the n-dimensional second-order cone, or Lorentz cone. A linear map M from to is called positive if M[L_m]⊂L_n . The set of positive maps forms a convex cone, the positive cone. Its dual cone, the separable cone, is the convex hull of tensor products x⊗y, where x∈L_n , y∈L_m . The structure of the positive and the separable cones is the subject of the present contribution. We study the action of the automorphism group of the positive cone and provide a complete classification of its orbits by constructing explicitly canonical forms. This enables us to identify and to classify the extreme rays of the positive cone, as well as their dual counterparts, the largest faces of the separable cone. We also characterize positive maps which are scalable to doubly stochastic ones.

Keywords:

• Second-order cone
• Positive maps
• Canonical forms
• Facial structure
• AMS Subject Classifications
• 15A48
• 15A69
• 47L07