Infinite dimensional symmetric cones and gauge-reversing maps
Bas Lemmens, Mark Roelands, Marten Wortel
TL;DR
This work extends Walsh’s finite-dimensional gauge-reversing characterization of symmetric cones to infinite-dimensional order unit spaces, establishing a framework that links horofunctions, Funk and reverse-Funk Busemann points, and the Thompson metric to the self-duality and homogeneity of cones. The authors develop a robust structure, showing that a gauge-reversing map induces a d_T-symmetric, homogeneous, and self-dual cone, and that under reflexivity the cone is tied to a unital JH/JB-algebra (and conjecturally to JB-algebras) via an inner product making the cone the set of squares. The main contributions include a detailed horofunction-based methodology to recover inner-product structure, a correspondence between atoms and pure states through gauge-reversing maps, and a comprehensive account of how infinite-dimensional symmetric cones arise from order-unit spaces with these maps, clarifying the Jordan-algebraic underpinnings in the infinite-dimensional setting. This bridges operator-algebraic concepts and metric geometry to characterize symmetric cones beyond finite dimensions, with implications for the theory of JB(JH)-algebras and their geometric realizations.
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.