manuscripta mathematica

, Volume 61, Issue 3, pp 297–314 | Cite as

Jordan axioms for C*-algebras

  • Angel Rodríguez Palacios
Article

Abstract

A complex Banach spaceA which is also an associative algebra provided with a conjugate linear vector space involution * satisfying (a2)*=(a*)2, ∥aa*a∥=∥a3 and ∥ab+ba∥≦2a∥∥b∥ for alla, b inA is shown to be a C*-algebra. The assumptions onA can be expressed in terms of the Jordan algebra obtained by symmetrization of the product ofA and are satisfied by any C*-algebra. Thus we obtain a purely Jordan characterization of C*-algebras.

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References

  1. 1.
    ALBERT,A.A.: Power-associative rings. Trans. Amer. Math. Soc.64, 552–593 (1948).Google Scholar
  2. 2.
    ALVERMANN,K.: The multiplicative triangle inequality in noncommutative JB- and JB*-algebras. Preprint, Technische Universität Braunschweig, 1985.Google Scholar
  3. 3.
    ALVERMANN,K. and JANSSEN,G.: Real and complex noncommutative Jordan Banach algebras. Math. Z.185, 105–113 (1984).Google Scholar
  4. 4.
    ARAKI,H. and ELLIOTT,G.A.: On the definition of C*-algebras. Publi. RIMS, Kyoto Univ.9, 93–112 (1973).Google Scholar
  5. 5.
    BONSALL,F.F. and DUNCAN,J.: Numerical ranges of operators on normed spaces and of elements of normed algebras. Lecture Notes Series 2, London Math. Soc., Cambridge 1971.Google Scholar
  6. 6.
    BRAUN,R.B.: A Gelfand-Neumark theorem for C*-alternative algebras. Math. Z.185, 225–242 (1984).Google Scholar
  7. 7.
    CUENCA,J.A. and RODRIGUEZ,A.: Isomorphisms of H*-algebras. Math. Proc. Camb. Phil. Soc.97, 93–99 (1985).Google Scholar
  8. 8.
    CUENCA,J.A. and RODRIGUEZ,A.: Structure theory for noncommutative Jordan H*-algebras. J. Algebra106, 1–14 (1987).Google Scholar
  9. 9.
    HERSTEIN,I.N.: Jordan homomorphisms. Trans. Amer. Math. Soc.81, 331–341 (1956).Google Scholar
  10. 10.
    KLEINFEL,E., KLEINFELD,M. and ROSIER,F.: A generalization of commutative and alternative rings. Canad. J. Math.22, 348–362 (1970).Google Scholar
  11. 11.
    MARTINEZ,J.: JV-algebras. Math. Proc. Camb. Phil. Soc.87, 47–50 (1980).Google Scholar
  12. 12.
    McCRIMMON,K.: The radical of a Jordan algebra. Proc. Nat. Acad. Sci. USA62, 671–678 (1969).Google Scholar
  13. 13.
    McCRIMMON,K.: On Herstein theorems relating Jordan and associative algebras. J. Algebra13, 382–392 (1969).Google Scholar
  14. 14.
    McCRIMMON,K.: Noncommutative Jordan rings. Trans. Amer. Math. Soc.158, 1–33 (1971).Google Scholar
  15. 15.
    PAYA,R., PEREZ,J. and RODRIGUEZ,A.: Noncommutative Jordan C*-algebras. Manuscripta Math.37 87–120 (1982).Google Scholar
  16. 16.
    PAYA,R., PEREZ,J. and RODRIGUEZ,A.: Type I factor representations of noncommutative JB*-algebras. Proc. London Math. Soc. (3)48, 428–444 (1984).Google Scholar
  17. 17.
    RODRIGUEZ,A.: Teorema de estructura de los Jordanisomorfismos de las C*-algebras. Rev. Mat. Hisp.-Amer. (4)37, 114–128 (1977).Google Scholar
  18. 18.
    RODRIGUEZ,A.: La continuidad del producto de Jordan implica la del ordinario en el caso completo semiprimo. In “Contribuciones en Probabilidad, Estadística Matemática, Enseñianza de la Matemática y Análisis”, 280–288. Secretariado de Publicaciones de la Universidad de Granada, 1979.Google Scholar
  19. 19.
    RODRIGUEZ,A.: A Vidav-Palmer theorem for Jordan C*-algebras and related topics. J. London Math. Soc. (2)22, 318–332 (1980).Google Scholar
  20. 20.
    RODRIGUEZ,A.: The uniqueness of the complete norm topology in complete normed nonassociative algebras. J. Funct. Anal.60, 1–15 (1985).Google Scholar
  21. 21.
    SCHAFER,R.D.: An introduction to nonassociative algebras. Academic Press, New York, 1966.Google Scholar
  22. 22.
    SCHAFER,R.D.: Generalized standard algebras. J. Algebra12, 376–417 (1969).Google Scholar
  23. 23.
    SINCLAIR,A.M.: Jordan homomorphisms and derivations on a semisimple Banach algebra. Proc. Amer. Math. Soc.24, 209–214 (1970).Google Scholar
  24. 24.
    SLATER,M.: Ideals in semiprime alternative rings. J. Algebra8, 60–76 (1968).Google Scholar
  25. 25.
    WRIGHT,J.D.M.: Jordan C*-algebras. Michigan Math. J.24, 291–302 (1977).Google Scholar
  26. 26.
    YOUNGSON,M.A.: A Vidav theorem for Jordan algebras. Math. Proc. Camb. Phil. Soc.84, 263–272 (1978).Google Scholar
  27. 27.
    YOUNGSON,M.A.: Hermitian operators on Banach Jordan algebras. Proc. Edinburgh Math. Soc.(2)22, 93–104 (1979).Google Scholar
  28. 28.
    YOUNGSON,M.A.: Non-unital Banach Jordan algebras and C*-triple systems. Proc. Edinburgh Math. Soc. (2)24, 19–30 (1981).Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Angel Rodríguez Palacios
    • 1
  1. 1.Departamento de Análisis MatemáticoUniversidad de GranadaGranadaSpain

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