Infinite dimensional symmetric cones and gauge-reversing maps

Abstract

The famous Koecher-Vinberg theorem characterises the finite dimensional formally real Jordan algebras among the finite dimensional order unit spaces as the ones that have a symmetric cone. An alternative characterisation of symmetric cones was obtained by Walsh who showed that the symmetric cones correspond exactly to the finite dimensional order unit spaces for which there exists a gauge-reversing map from the interior of the cone to itself. In this paper we prove an infinite dimensional version of this characterisation of symmetric cones.