A characterization of symmetric cones by an order-reversing property of the pseudoinverse maps

Abstract

When a homogeneous convex cone is given, a natural partial order is introduced in the cone. We shall show that a homogeneous convex cone is a symmetric cone if and only if Vinberg's *-map and its inverse reverse the order. Actually our theorem is formulated in terms of the family of pseudoinverse maps including the *-map, and states that the above order-reversing property is typical of the *-map of a symmetric cone which coincides with the inverse map of the Jordan algebra associated with the symmetric cone.