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**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 |

### Citations:

Zbl 0612.46073
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\textit{M. M. Ibragimov} et al., Vladikavkaz. Mat. Zh. 20, No. 1, 61--68 (2018; Zbl 1448.81065)

### References:

[1] | Friedman Y., Russo B., “A geometric spectral theorem”, #Quart. J. Math. Oxford, #37:2 (1986), 263-277 · Zbl 0612.46073 |

[2] | Friedman Y., Russo B., “Affine structure of facially symmetric spaces”, #Math. Proc. Camb. Philos. Soc., #106:1 (1989), 107-124 · Zbl 0693.46010 |

[3] | Friedman Y., Russo B., “Some affine geometric aspects of operator algebras”, #Pacif. J. Math., #137:1 (1989), 123-144 · Zbl 0679.46048 |

[4] | Friedman Y., Russo B., “Geometry of the dual ball of the spin factor”, #Proc. Lon. Math. Soc. III Ser., #65:1 (1992), 142-174 · Zbl 0759.46061 |

[5] | Friedman Y., Russo B., “Classification of atomic facially symmetric spaces”, #Canad. J. Math., #45:1 (1993), 33-87 · Zbl 0803.46015 |

[6] | Neal M., Russo B., “State space of JB*-triples”, #Math. Ann., #328:4 (2004), 585-624 · Zbl 1051.46051 |

[7] | Ibragimov M. M., Kudayberegenov K. K., Seypullaev J. X., “Facially symmetric spaces and preduals of a hermitian part of von Neumann algebras”, #Russian Math., 2018, no. 5, 33-40 (in Russian) |

[8] | Ayupov Sh. A., #Classification and Representation of Ordered Jordan Algebras, Fan, Tashkent, 1986, 121 pp. (in Russian) · Zbl 0629.46045 |

[9] | Korobova K. B., Xudalov V. T., “On ordered structure of abstract spin-factor”, #Vladikavkaz Math. J., #6:1 (2004), 46-57 (in Russian) · Zbl 1097.46509 |

[10] | Yadgorov N. J., “Weakly and strongly facially symmetric spaces”, #Dokl. AN RUz, #5 (1996), 6-8 (in Russian) · Zbl 0955.46030 |

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