Isometric isomorphism of reflexive neutral strongly facially symmetric spaces

Seypullaev Jumabek Khamidullaevich
Scopus Author ID: 57202628285
1. Karakalpak State University named after Berdakh, Nukus, Uzbekistan
jumabek81@mail.ru
Kalenbaev Kamalatdin Bakhytbay Uly
1. V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, Tashkent, Uzbekistan
kkb97@mail.ru
The material was received by the Editorial Board: 17.04.2024

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 $Y^*$.

УДК 517.98

Keywords: $WFS$-space, $SFS$-space, symmetric face, geometric tripotent, Pierce projection.


References: Seypullaev J. Kh., Kalenbaev K. B. Isometric isomorphism of reflexive neutral strongly facially symmetric spaces. Mat. Trudy. 2024, 27, № 3. P. 99–110. DOI: 10.25205/1560-750X-2024-27-3-99-110