Skip to main content
SpringerLink
Log in

On real forms of JB*-triples

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Alvermann, K.: Real normed Jordan algebras with involution, Arch. Math.47, 135–150 (1986)

    Article  MathSciNet  Google Scholar 

  2. Barton, T. J., Friedman, Y.: Bounded derivations of JB*-triples, Quart. J. Math. Oxford41, 255–268 (1990)

    Article  MathSciNet  Google Scholar 

  3. Barton, T. J., Timoney, R. M.: Weak*-continuity of Jordan triple products and applications, Math. Scand.59, 177–191 (1986)

    MathSciNet  Google Scholar 

  4. Bonsall, F. F., Duncan, J.: Complete normed algebras. Ergebnisse der Math. und ihrer Grenzgebiete80, Springer-Verlag, Berlin-Heidelberg-New York (1973)

    MATH  Google Scholar 

  5. Braun, R., Kaup, W., Upmeier, H.: A holomorphic characterization of Jordan C*-algebras, Math. Z.161, 277–290 (1978)

    Article  MathSciNet  Google Scholar 

  6. Bunce, L.J., Chu, C.H.: Real Contractive Projections on Commutative C*-algebras, preprint

  7. Chu, C.H., Dang, T., Russo, B., Ventura, B.: Surjective isometries of real C*-algebras, J. London Math. Soc. 247, 97–118 (1993)

    MathSciNet  Google Scholar 

  8. Dang, T.: Real isometries between JB*-triples, Proc. Amer. Math. Soc.114, 971–980 (1992)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  10. Dang, T. C., Russo, B.: Real Banach-Jordan triples, Proc. Amer. Math. Soc.122, 135–145 (1994)

    Article  MathSciNet  Google Scholar 

  11. Dineen, S.: Complete holomorphic vector fields on the second dual of a Banach space, Math. Scand.59, 131–142 (1986)

    MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  13. Friedman, Y., Russo, B.: Structure of the predual of a JBW*-triple, J. Reine Angew. Math.356, 67–89 (1985)

    MathSciNet  Google Scholar 

  14. Friedman, Y., Russo, B.: The Gelfand-Naimark theorem for JB*-triples, Duke Math. J.53, 139–148, (1986)

    Article  MathSciNet  Google Scholar 

  15. Goodearl, K. R.: Notes on Real and Complex C*-algebras, Shiva Publ. 1982

  16. Hanche-Olsen, H., Størmer, E.: Jordan operator algebras, Monographs Stud. Math.21 Pitman, Boston-London-Melbourne 1984

    MATH  Google Scholar 

  17. Harris, L.A.: Bounded symmetric homogeneous domains in infinite dimensional spaces, In: Lecture Notes in Mathematics Vol. 364. Berlin-Heidelberg-New York: Springer 1973

    Google Scholar 

  18. Harris, L.A.: A Generalization of C*-algebras, Proc. Lond. Math. Soc.42, 331–361 (1981)

    Google Scholar 

  19. Harris, L.A., Kaup, W.: Linear algebraic groups in infinite dimensions, Illinois J. Math.21, 666–674 (1977)

    MathSciNet  Google Scholar 

  20. Horn, G.: Characterization of the predual and ideal structure of a JBW*-triple, Math. Scand.61, 117–133 (1987)

    MathSciNet  Google Scholar 

  21. Horn, G., Neher, E.: Classification of continuous JBW*-triples, Transactions Amer. Math. Soc.306, 553–578 (1988)

    Article  MathSciNet  Google Scholar 

  22. Isidro, J. M., Rodríguez, A.: Isometries of JB-algebras, to appear in Math. Scand.

  23. Kaup, W.: Über die Klassifikation der symmetrischen Hermiteschen Manngfaltigkeiten unendlicher Dimension I, II, Math. Ann.257, 463–483 (1981);262, 503–529 (1983)

    Article  MathSciNet  Google Scholar 

  24. Kaup, W.: A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces, Math. Z.183, 503–529 (1983)

    Article  MathSciNet  Google Scholar 

  25. Kaup, W., Upmeier, H.: Jordan algebras and symmetric Siegel domains in Banach spaces, Math. Z.157, 179–200 (1977)

    Article  MathSciNet  Google Scholar 

  26. Koecher, M.: An elementary approach to bounded symmetric domains, Rice Univ. 1969

  27. Loos, O.: Symmetric Spaces I, II, W. A. Benjamin 1969

  28. Loos, O.: Bounded symmetric domains and Jordan pairs, Mathematical Lectures. Irvine: University of California at Irvine 1977

    Google Scholar 

  29. Loos, O.: Charakterisierung symmetrischer R-Räume durch ihre Einheitsgitter, Math. Z.189, 211–226 (1985)

    Article  MathSciNet  Google Scholar 

  30. Sakai, S.: C*-algebras and W*-algebras, Berlin-Heidelberg-New York: Springer 1971

    Google Scholar 

  31. Sauter, J.: Über die Randstruktur beschränkter symmetrischer Gebiete, in preparation

  32. Upmeier, H.: Symmetric Banach manifolds and Jordan C*-algebras, North Holland Math. Studies104, North Holland, Amsterdam 1985

    Google Scholar 

  33. Wright, J. D. M.: Jordan C*-algebras, Michigan Math. J.24, 291–302 (1977)

    Article  MathSciNet  Google Scholar 

  34. Wright, J. D. M., Youngson, M. A.: On isometries of Jordan algebras, J. London Math. Soc.17, 339–344 (1978)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Facultad de Matemáticas, Universidad de Santiago, 15706, Santiago de Compostela, Spain

    José M. Isidro

  2. Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076, Tübingen, Germany

    W. Kaup

  3. Dep. Analisis Matematico Facultad de Ciencias, Universidad de Granada, 18071, Granada, Spain

    Angel Rodríguez Palacios

Authors
  1. José M. Isidro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W. Kaup
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Angel Rodríguez Palacios
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Isidro, J.M., Kaup, W. & Palacios, A.R. On real forms of JB*-triples. Manuscripta Math 86, 311–335 (1995). https://doi.org/10.1007/BF02567997

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02567997

Keywords

Search

Navigation