Isometric Isomorphism of Reflexive Neutral Strongly FaciallySymmetric 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 $${}^\ast $$ -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 $$P:{{M}_{X}}\to {{M}_{Y}} $$ preserves both orthogonality between geometric triponents and the transition pseudo-probabilities then $$P $$ can be extended to an isometric isomorphism from $${X}^{*} $$ to $${Y}^{*}$$ .