Geodesic
Geometry of the state space of modular Jordan algebras
Izvestiya. Mathematics , Tome 43 (1994) no. 3, pp. 581-592

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In the class of compact convex sets the normal state spaces of modular Jordan algebras are characterized. To wit,the state spaces are precisely those sets that are modular, spectral, and symmetric. 
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Sh. A. Ayupov; N. D. Yadgorov. Geometry of the state space of modular Jordan algebras. Izvestiya. Mathematics , Tome 43 (1994) no. 3, pp. 581-592. http://geodesic.mathdoc.fr/item/IM2_1994_43_3_a9/