Geometric characterization of real JBW-factors

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 introduced the co-called facially symmetric spaces. The main purpose of introducing them is the geometric characterization of predual spaces of JB*-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 JBW*-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 JBW*-triples in the class of facially symmetric spaces. At the same time, the description of real JBW*-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.