Isometric isomorphism of reflexive neutral strongly facially symmetric spaces

The problem on geometric characterization of state spaces of operator algebras is important in the theory of such algebras. In the mid-80's, Friedman and Russo introduced facially symmetric spaces for geometric characterization of the predual spaces of JBW*-triples that admit an algebraic structure. Many properties that are required in such characterizations are natural assumptions on state spaces of physical systems. These spaces are regarded as a geometric model for states in quantum mechanics. In the present article, we prove that, for all reflexive atomic neutral strongly facially symmetric spaces $X$ and $Y$, if a transform $P: M_X \rightarrow M_Y$ preserves both orthogonality between geometric triponents and the transition pseudo-probabilities then $P$ can be extended to an isometric isomorphism from $X^*$ to $Y^*$.