Hilbert and Thompson isometries on cones in JB-algebras

Hilbert's and Thompson's metric spaces on the interior of cones in JB-algebras are important examples of symmetric Finsler spaces. In this paper we characterize the Hilbert's metric isometries on the interiors of cones in JBW-algebras, and the Thompson's metric isometries on the interiors of cones in JB-algebras. These characterizations generalize work by Bosch\'e on the Hilbert and Thompson isometries on symmetric cones, and work by Hatori and Moln\'ar on the Thompson isometries on the cone of positive self-adjoint elements in a unital $C^*$-algebra. To obtain the results we develop a variety of new geometric and Jordan algebraic techniques.