Abstract
The present paper is devoted to the study of predual spaces of JBW-algebras. It is proved that the predual space of a JBW-algebra is a strongly facially symmetric space if and only if this algebra is the direct sum of an Abelian algebra and an algebra of type I2.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E.M. Alfsen and F.W. Shultz, “State spaces of Jordan algebras,” Acta. Math. 140 (3-4), 155–190 (1978).
E. M. Alfsen, H. Hanche-Olsen, and F. W. Shultz, “State spaces of C*-algebras,” Acta. Math. 144 (3-4), 267–305 (1980).
W. Kaup, “Riemannmapping theorem for bounded symmetric domains in complex Banach spaces,” Math. Z. 138 (4), 503–529 (1983).
W. Kaup, “Contractive projections on Jordan C*-algebras and generalizations,” Math. Scand. 54 (1), 95–100 (1984).
Y. Friedman and B. Russo, “Affine structure of facially symmetric spaces,” Math. Proc. Cambridge Philos. Soc. 106 (1), 107–124 (1989).
Y. Friedman and B. Russo, “Some affine geometric aspects of operator algebras,” Pacific. J. Math. 137 (1), 123–144 (1989).
Y. Friedman and B. Russo, “Geometry of the dual ball of the spin factor,” Proc. London Math. Soc. (3) 65(1), 142–174 (1992).
Y. Friedman and B. Russo, “Classification of atomic facially symmetric spaces,” Canad. J. Math. 45 (1), 33–87 (1993).
M. Neal and B. Russo, “State space of JB*-triples,” Math. Ann. 328 (4), 585–624 (2004).
M. M. Ibragimov, K. K. Kudaibergenov, S. Zh. Tleumuratov, and Zh. Kh. Seipullaev, “Geometric Description of the Preduals of Atomic Commutative von Neumann Algebras,” Mat. Zametki 93 (5), 728–735 (2013) [Math. Notes 93 (5), 715-721 (2013)].
N. Yadgorov, M. Ibragimov, and K. Kudaybergenov, “Geometric characterization of L1-spaces,” Studia Math. 219 (2), 21–27 (2013).
M. Ibragimov, K. Kudaibergenov, and Zh. Seipullaev, “Facially symmetric spaces and predual ones of Hermitian part of von Neumann algebras,” Izv. Vyssh. Uchebn. Zaved. Mat. (5), 33–40 (2018) [Russian Math. (Iz. VUZ) 62 (5), 27-33 (2018)].
Y. Friedman and B. Russo, “A geometric spectral theorem,” Quart. J. Math. Oxford Ser. (2) 37 (147), 263–277 (1986).
E. M. Alfsen and F. W. Shultz, Geometry of State Spaces of Operator Algebras (Birkha¨ user Boston, Boston, MA, 2003).
Sh. A. Ayupov, Classification and Representations of Ordered Jordan Algebras (Izd. Fan, Tashkent, 1986) [in Russian].
C. M. Edwards and G. T. Ruttimann, “The facial and inner ideal structure of a real JBW*-triple,” Math. Nachr. 222, 159–184 (2001).
N. Zh. Yadgorov, “Weakly and strongly facially symmetric spaces,” Dokl.Akad. Nauk Resp.Uzb., Mat. Tekh. Nauki Estestvozn. (5), 6–8 (1996).
Acknowledgments
The authors thank the referee for useful suggestions and comments.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Russian Text © The Author(s), 2020, published in Matematicheskie Zametki, 2020, Vol. 107, No. 4, pp. 539–549.
Rights and permissions
About this article
Cite this article
Kudaybergenov, K.K., Seipullaev, Z.K. Characterization of JBW-Algebras with Strongly Facially Symmetric Predual Space. Math Notes 107, 600–608 (2020). https://doi.org/10.1134/S000143462003027X
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S000143462003027X