## Geometric characterization of real JBW-factors.(Russian. English summary)Zbl 1448.81065

Summary: One of the interesting problems in the theory of operator algebras is the geometric characterization of the state spaces of Jordan operator algebras. In the mid-1980s, Y. Friedman and B. Russo [Q. J. Math., Oxf. II. Ser. 37, 263-277 (1986; Zbl 0612.46073)] introduced the co-called facially symmetric spaces. The main purpose of introducing them is the geometric characterization of predual spaces of $$\mathrm{JB}^\ast$$-triples that admit an algebraic structure. Many of the properties required in these characterizations are natural assumptions for the state spaces of physical systems. Such spaces are considered as a geometric model for states of quantum mechanics. Y. Fridman and B. Russo showed that the predual space of a complex von Neumann algebra and more general $$\mathrm{JBW}^\ast$$-triple is a neutral strongly facially symmetric space. In this connection, Y. Friedman and B. Russo mainly studied neutral facially symmetric spaces, and in these spaces they obtained results that were previously known for the aforementioned predual spaces. In 2004, M. Neal and B. Russo gave geometric characterizations of the predual spaces of complex $$\mathrm{JBW}^\ast$$-triples in the class of facially symmetric spaces. At the same time, the description of real $$\mathrm{JBW}^\ast$$-triples remains an open question. The present paper is devoted to the study of predual spaces of real JBW-factors. It is proved that the predual space of a real JBW-factor is a strongly facially symmetric space if and only if it either is abelian or is a spin-factor.

### MSC:

 81P16 Quantum state spaces, operational and probabilistic concepts 46L30 States of selfadjoint operator algebras 17C10 Structure theory for Jordan algebras 17C90 Applications of Jordan algebras to physics, etc. 46A55 Convex sets in topological linear spaces; Choquet theory 46B20 Geometry and structure of normed linear spaces 46L10 General theory of von Neumann algebras 47L50 Dual spaces of operator algebras

### Keywords:

Banach space; facially symmetric space; JBW-algebra; JBW-factor; face

Zbl 0612.46073
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### References:

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