Symmetric cones, the Hilbert and Thompson metrics

Abstract

Symmetric cones can be endowed with at least two interesting non Riemannian metrics: the Hilbert and the Thompson metrics. It is trivial that the linear maps preserving the cone are isometries for those two metrics. Oddly enough those are not the only isometries in general. We give here a full description of the isometry groups for both the Hilbert and the Thompson metrics using essentially the theory of euclidean Jordan algebras. Those results were already proved for the symmetric cone of complexe positive hermitian matrices by L. Moln{\'a}r. In this paper however we do not make any assumption on the symmetric cone under scrutiny (it could be reducible and contain exceptional factors).