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Relatively weakly open sets in closed balls of Banach spaces, and real JB*-triples of finite rank

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Abstract.

We prove that, given a real JB*-triple X, there exists a nonempty relatively weakly open subset of the closed unit ball of X with diameter less than 2 (if and) only if the Banach space of X is isomorphic to a Hilbert space. Moreover we give the structure of real JB*-triples whose Banach spaces are isomorphic to Hilbert spaces. Such real JB*-triples are also characterized in two different purely algebraic ways.

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References

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  3. Becerra, J., López, G., Rodríguez, A.: Relatively weakly open sets in closed balls of C*-algebras. J. London Math. Soc. 68, 753–761 (2003)

    Article  Google Scholar 

  4. Becerra, J., López, G., Rodríguez, A.: Isometric reflections on Banach spaces after a paper of A. Skorik and M. Zaidenberg. Rocky Mountain J. Math. 30, 63–83 (2000)

    Google Scholar 

  5. Benslimane, M., Merrachi, N.: Algèbres de Jordan Banach vérifiant |x||x-1|=1. J. Algebra 206, 129–134 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  7. Bunce, L. J., Chu, C. H.: Compact operations, multipliers and Radon-Nikodym property in JB*-triples. Pacific J. Math. 153, 249–265 (1992)

    MathSciNet  MATH  Google Scholar 

  8. Chu, C. H., Dang, T., Russo, B., Ventura, B.: Surjective isometries of real C*-algebras. J. London Math. Soc. 47, 87–110 (1993)

    Google Scholar 

  9. Cuartero, B., Galé, J. E.: Bounded degree of algebraic topological algebras. Commun. Algebra 22, 329–337 (1994)

    MathSciNet  MATH  Google Scholar 

  10. Deville, R., Godefroy, G., Zizler, V.: Smoothness and renormings in Banach spaces. Pitman Monographs and Surveys in Pure and Applied Math. 64, 1993

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

    MathSciNet  MATH  Google Scholar 

  12. Gooderal, K. R.: Notes on real and complex C*-algebras. Shiva Math. Series 5, Devon, 1982

  13. Hanche-olsen, H., Stormer, E.: Jordan operator algebras. Monograph Stud. Math. 21, Pitman, 1984

  14. Harmand, P., Werner, D., Werner, W.: M-ideals in Banach spaces and Banach algebras. Lecture Notes in Mathematics 1547, Springer-Verlag, Berlin, 1993

  15. Isidro, J. M., Kaup, W., Rodríguez, A.: On real forms of JB*-triples. Manuscripta Math. 86, 311–335 (1995)

    MathSciNet  MATH  Google Scholar 

  16. Isidro, J. M., Rodríguez, A.: On the definition of real W*-algebras. Proc. Am. Math. Soc. 124, 3407–3410 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jacobson, N.: Structure and representations of Jordan algebras. Am. Math. Soc. Coll. Publ. 39, Providence, Rhode Island 1968

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

    MathSciNet  MATH  Google Scholar 

  19. Kaup, W.: über die Klassifikation der symmetrischen hermiteschen Mannigfaltigkeiten unendlicher Dimension. I. Math. Ann 257, 463–486 (1981)

    MathSciNet  MATH  Google Scholar 

  20. Kaup, W.: On real Cartan factors. Manuscripta Math. 92, 191–222 (1997)

    MathSciNet  MATH  Google Scholar 

  21. Loos, O.: Jordan pairs. Lecture Notes in Mathematics 460, Springer-Verlag, Berlin, 1975

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

  23. Martínez, J., Peralta, A. M.: Separate weak*-continuity of the triple product in dual real JB*-triples. Math. Z. 234, 635–646 (2000)

    Article  MathSciNet  Google Scholar 

  24. Moreno, A., Rodríguez, A.: On the Zel’manovian classification of prime JB*-triples. J. Algebra 226, 577–613 (2000)

    Article  Google Scholar 

  25. Nygaard, O., Werner, D.: Slices in the unit ball of a uniform algebra. Archiv Math. 76, 441–444 (2001)

    MathSciNet  MATH  Google Scholar 

  26. Peralta, A. M., Rodríguez, A.: Grothendiekc’s inequalities for real and complex JBW*-triples. Proc. London Math. Soc. 83, 605–625 (2001)

    MathSciNet  MATH  Google Scholar 

  27. Peralta, A. M., Stachó, L. L.: Atomic decomposition of real JBW*-triples. Quart. J. Math. 52, 79–87 (2001)

    MATH  Google Scholar 

  28. Rickart, C. E.: General theory of Banach algebras. Kreiger, New York, 1974

  29. Rodríguez, A.: Nonassociative normed algebras spanned by hermitian elements. Proc. London Math. Soc. 47, 258–274 (1983)

    Google Scholar 

  30. Rodríguez, A.: One-sided division absolute valued algebras. Publ. Mat. 36, 925–954 (1992)

    Google Scholar 

  31. Sakai, S.: Weakly compact operators on operator algebras. Pacific J. Math. 14, 659–664 (1964)

    MATH  Google Scholar 

  32. Shvydkoy, R. V.: Geometric aspects of the Daugavet property. J. Funct. Anal. 176, 198–212 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  33. Youngson, M. A.: Non unital Banach Jordan algebras and C*-triple systems. Proc. Edinburgh Math. Soc. 24, 19–31 (1981)

    MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Universidad de Granada, Facultad de Ciencias, Departamento de Matemática Applicata, 18071, Granada, Spain

    Julio Becerra Guerrero

  2. Universidad de Granada, Facultad de Ciencias, Departamento de Análisis Matemático, 18071, Granada, Spain

    Ginés López Pérez, Antonio M. Peralta & A Rodríguez-Palacios

Authors
  1. Julio Becerra Guerrero
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  2. Ginés López Pérez
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  3. Antonio M. Peralta
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  4. A Rodríguez-Palacios
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Corresponding author

Correspondence to A Rodríguez-Palacios.

Additional information

Mathematics Subject Classification (2000): 46B04, 46B22, 46L05, 46L70

Partially supported by Junta de Andalucía grant FQM 0199.

Revised version: 30 September 2003

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Guerrero, J., Pérez, G., Peralta, A. et al. Relatively weakly open sets in closed balls of Banach spaces, and real JB*-triples of finite rank. Math. Ann. 330, 45–58 (2004). https://doi.org/10.1007/s00208-004-0537-y

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  • DOI: https://doi.org/10.1007/s00208-004-0537-y

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