# zbMATH — the first resource for mathematics

Boolean-valued analysis and $$JB$$-algebras. (English. Russian original) Zbl 0832.46045
Sib. Math. J. 35, No. 1, 114-122 (1994); translation from Sib. Mat. Zh. 35, No. 1, 124-134 (1994).
In the last decade Boolean-valued models of set theory find some applications to operator algebras and $$C^*$$-algebras in several papers of G. Takeuti, M. Ozawa and the author. The aim of this paper is to show that these Boolean-valued methods can also be used in the situation of JB$$^*$$-algebras, that is the Jordan analogues of $$B^*$$-algebras. Some results of restricted interest are proved.
Reviewer: B.Aupetit (Quebec)
##### MSC:
 46H70 Nonassociative topological algebras 46S20 Nonstandard functional analysis 17C65 Jordan structures on Banach spaces and algebras
##### Keywords:
Boolean-valued models of set theory; JB$$^*$$-algebras
Full Text:
##### References:
 [1] V. N. Berestovskii, ”Homogeneous manifolds with intrinsic metric. I,” Sibirsk. Mat. Zh.,29, No. 6, 17–29 (1988). · Zbl 0671.53036 [2] V. N. Berestovskii, ”Homogeneous spaces with intrinsic metric,” Dokl. Akad. Nauk SSSR,301, No. 2, 268–271 (1988). [3] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, Mathematical Theory of Optimal Processes [in Russian], Nauka, Moscow (1969). [4] H. Busemann, The Geometry of Geodesics [Russian translation], Fizmatgiz, Moscow (1962). · Zbl 0112.37202 [5] K. Leichtweiss, Convex Sets [Russian translation], Nauka, Moscow (1985). [6] V. N. Berestovskii, ”Homogeneous manifolds with intrinsic metric. II,” Sibirsk. Mat. Zh.,30, No. 2, 14–28 (1989). [7] V. N. Berestovskii, Homogeneous Spaces with Intrinsic Metric [in Russian], Diss. Dokt. Fiz.-Mat. Nauk, Inst. Mat. (Novosibirsk), Novosibirsk (1990). [8] A. D. Aleksandrov, ”über eine Verallgemeinerung der Riemannschen Geometrie,” Schrift. Inst. Math. der Deutschen Acad. Wiss., No. 1, 33–84 (1957). [9] A. M. Vershik and V. Ya. Gershkovich, ”Nonholonomic dynamical systems. Geometry of distributions and variational problems,” in: Sovrem. Probl. Mat. Fund. Naprav. (Itogi Nauki i Tekhniki),16 [in Russian], VINITI, Moscow, 1987, pp. 5–85. · Zbl 0797.58007 [10] L. S. Kirillova, ”Non-Riemmannian metrics and the maximum principle,” Dokl. Akad. Nauk UzSSR, No. 7, 9–11 (1986). [11] A. M. Vershik and O. A. Granichina, ”Reduction of nonholonomic variational problems to isoperimetric problems and connections in principal bundles,” Mat. Zametki,49, No. 5, 37–44 (1991). · Zbl 0734.49023 [12] R. Montgomery, Shortest Loops with a Fixed Holonomy [Preprint], MSRI (1988). [13] V. N. Berestovskii, ”Submetries of space forms of nonnegative curvature,” Sibirsk. Mat. Zh.,28, No. 4, 44–56 (1987). [14] Z. Ge, ”On a constrained variational problem and the spaces of horizontal paths,” Pacific J. Math.,149, No. 1, 61–94 (1991). · Zbl 0691.58021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.