manuscripta mathematica

, Volume 86, Issue 1, pp 311–335 | Cite as

On real forms of JB*-triples

  • José M. Isidro
  • W. Kaup
  • Angel Rodríguez Palacios


We introduce real JB*-triples as real forms of (complex) JB*-triples and give an algebraic characterization of surjective linear isometries between them. As main result we show: A bijective (not necessarily continuous) linear mapping between two real JB*-triples is an isometry if and only if it commutes with the cube mappinga→a 3={aaa}. This generalizes a result of Dang for complex JB*-triples. We also associate to every tripotent (i.e. fixed point of the cube mapping) and hence in particular to every extreme point of the unit ball in a real JB*-triple numerical invariants that are respected by surjective linear isometries.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alvermann, K.: Real normed Jordan algebras with involution, Arch. Math.47, 135–150 (1986)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Barton, T. J., Friedman, Y.: Bounded derivations of JB*-triples, Quart. J. Math. Oxford41, 255–268 (1990)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Barton, T. J., Timoney, R. M.: Weak*-continuity of Jordan triple products and applications, Math. Scand.59, 177–191 (1986)MathSciNetGoogle Scholar
  4. 4.
    Bonsall, F. F., Duncan, J.: Complete normed algebras. Ergebnisse der Math. und ihrer Grenzgebiete80, Springer-Verlag, Berlin-Heidelberg-New York (1973)MATHGoogle Scholar
  5. 5.
    Braun, R., Kaup, W., Upmeier, H.: A holomorphic characterization of Jordan C*-algebras, Math. Z.161, 277–290 (1978)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Bunce, L.J., Chu, C.H.: Real Contractive Projections on Commutative C*-algebras, preprintGoogle Scholar
  7. 7.
    Chu, C.H., Dang, T., Russo, B., Ventura, B.: Surjective isometries of real C*-algebras, J. London Math. Soc. 247, 97–118 (1993)MathSciNetGoogle Scholar
  8. 8.
    Dang, T.: Real isometries between JB*-triples, Proc. Amer. Math. Soc.114, 971–980 (1992)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Dang, T., Friedman, Y., Russo, B.: Affine geometric proofs of the Banach-Stone theorems of Kadison and Kaup, Rocky Mountain J. Math.20, 409–428 (1990)MathSciNetGoogle Scholar
  10. 10.
    Dang, T. C., Russo, B.: Real Banach-Jordan triples, Proc. Amer. Math. Soc.122, 135–145 (1994)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Dineen, S.: Complete holomorphic vector fields on the second dual of a Banach space, Math. Scand.59, 131–142 (1986)MathSciNetGoogle Scholar
  12. 12.
    Edwards, C. M., Rüttimann, G. T.: On the facial structure of the unit balls in a JBW*-triple and its predual, J. London Math. Soc.38, 317–332 (1988)MathSciNetGoogle Scholar
  13. 13.
    Friedman, Y., Russo, B.: Structure of the predual of a JBW*-triple, J. Reine Angew. Math.356, 67–89 (1985)MathSciNetGoogle Scholar
  14. 14.
    Friedman, Y., Russo, B.: The Gelfand-Naimark theorem for JB*-triples, Duke Math. J.53, 139–148, (1986)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Goodearl, K. R.: Notes on Real and Complex C*-algebras, Shiva Publ. 1982Google Scholar
  16. 16.
    Hanche-Olsen, H., Størmer, E.: Jordan operator algebras, Monographs Stud. Math.21 Pitman, Boston-London-Melbourne 1984MATHGoogle Scholar
  17. 17.
    Harris, L.A.: Bounded symmetric homogeneous domains in infinite dimensional spaces, In: Lecture Notes in Mathematics Vol. 364. Berlin-Heidelberg-New York: Springer 1973Google Scholar
  18. 18.
    Harris, L.A.: A Generalization of C*-algebras, Proc. Lond. Math. Soc.42, 331–361 (1981)Google Scholar
  19. 19.
    Harris, L.A., Kaup, W.: Linear algebraic groups in infinite dimensions, Illinois J. Math.21, 666–674 (1977)MathSciNetGoogle Scholar
  20. 20.
    Horn, G.: Characterization of the predual and ideal structure of a JBW*-triple, Math. Scand.61, 117–133 (1987)MathSciNetGoogle Scholar
  21. 21.
    Horn, G., Neher, E.: Classification of continuous JBW*-triples, Transactions Amer. Math. Soc.306, 553–578 (1988)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Isidro, J. M., Rodríguez, A.: Isometries of JB-algebras, to appear in Math. Scand.Google Scholar
  23. 23.
    Kaup, W.: Über die Klassifikation der symmetrischen Hermiteschen Manngfaltigkeiten unendlicher Dimension I, II, Math. Ann.257, 463–483 (1981);262, 503–529 (1983)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Kaup, W.: A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces, Math. Z.183, 503–529 (1983)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Kaup, W., Upmeier, H.: Jordan algebras and symmetric Siegel domains in Banach spaces, Math. Z.157, 179–200 (1977)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Koecher, M.: An elementary approach to bounded symmetric domains, Rice Univ. 1969Google Scholar
  27. 27.
    Loos, O.: Symmetric Spaces I, II, W. A. Benjamin 1969Google Scholar
  28. 28.
    Loos, O.: Bounded symmetric domains and Jordan pairs, Mathematical Lectures. Irvine: University of California at Irvine 1977Google Scholar
  29. 29.
    Loos, O.: Charakterisierung symmetrischer R-Räume durch ihre Einheitsgitter, Math. Z.189, 211–226 (1985)CrossRefMathSciNetGoogle Scholar
  30. 30.
    Sakai, S.: C*-algebras and W*-algebras, Berlin-Heidelberg-New York: Springer 1971Google Scholar
  31. 31.
    Sauter, J.: Über die Randstruktur beschränkter symmetrischer Gebiete, in preparationGoogle Scholar
  32. 32.
    Upmeier, H.: Symmetric Banach manifolds and Jordan C*-algebras, North Holland Math. Studies104, North Holland, Amsterdam 1985Google Scholar
  33. 33.
    Wright, J. D. M.: Jordan C*-algebras, Michigan Math. J.24, 291–302 (1977)CrossRefMathSciNetGoogle Scholar
  34. 34.
    Wright, J. D. M., Youngson, M. A.: On isometries of Jordan algebras, J. London Math. Soc.17, 339–344 (1978)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • José M. Isidro
    • 1
  • W. Kaup
    • 2
  • Angel Rodríguez Palacios
    • 3
  1. 1.Facultad de MatemáticasUniversidad de SantiagoSantiago de CompostelaSpain
  2. 2.Mathematisches InstitutUniversität TübingenTübingenGermany
  3. 3.Dep. Analisis Matematico Facultad de CienciasUniversidad de GranadaGranadaSpain

Personalised recommendations