Preservation of Order and Orthogonality on Preduals of Jordan Algebras

We study linear maps between preduals of JBW-algebras that preserve orthogonal decompositions of functionals. We generalize and strengthen results of Bunce and Wright (Pac J Math 158(2):265–272, 1993) obtained for von Neumann algebras by characterizing continuous orthogonally decomposable linear maps between preduals of JBW-algebras. In particular, we show that Banach space adjoint of such morphism \(\Phi \) $\mathrm{\Phi }$ is typically of the form \(\Phi ^*(y)= c\pi (y)\) ${\mathrm{\Phi }}^{\ast }(y)=c\pi (y)$ , where c is a central positive element and \(\pi \) $\pi$ is a Jordan homomorphism. Further we show that the order topology on preduals of JBW-algebras equals to the norm topology. This allows to relax continuity in characterization of orthogonally decomposable isomorphisms and show that order and orthogonality relation in preduals are complete invariants of JBW-algebras.