Isometric isomorphism of reflexive neutral strongly facially symmetric spaces

J. Kh. Seypullaev   ; K. B. Kalenbaev

Matematičeskie trudy, Tome 27 (2024) no. 3, pp. 99-110

Résumé

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 $Y^*$.

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