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1995 | 113 | 1 | 81-100
Tytuł artykułu

On the behaviour of Jordan-algebra norms on associative algebras

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EN
Abstrakty
EN
We prove that for a suitable associative (real or complex) algebra which has many nice algebraic properties, such as being simple and having minimal idempotents, a norm can be given such that the mapping (a,b) ↦ ab + ba is jointly continuous while (a,b) ↦ ab is only separately continuous. We also prove that such a pathology cannot arise for associative simple algebras with a unit. Similar results are obtained for the so-called "norm extension problem", and the relationship between these results and the normed versions of Zel'manov's prime theorem for Jordan algebras are discussed.
Słowa kluczowe
Twórcy
  • Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
  • Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
  • Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain, apalacios@ugr.es
Bibliografia
  • [1] R. Arens, M. Goldberg and W. A. J. Luxemburg, Multiplicativity factors for seminorms II, J. Math. Anal. Appl. 170 (1992), 401-413.
  • [2] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, Berlin, 1973.
  • [3] M. Cabrera, A. Moreno and A. Rodríguez, Zel'manov's theorem for primitive Jordan-Banach algebras, preprint, Universidad de Granada, 1993.
  • [4] M. Cabrera and A. Rodríguez, Zel'manov's theorem for normed simple Jordan algebras with a unit, Bull. London Math. Soc. 25 (1993), 59-63.
  • [5] M. Cabrera and A. Rodríguez, Nondegenerately ultraprime Jordan-Banach algebras: a Zel'manovian treatment, Proc. London Math. Soc. 69 (1994), 576-604.
  • [6] A. Fernández, E. Garcia and A. Rodríguez, A Zel'manov prime theorem for JB*-algebras, J. London Math. Soc. 46 (1992), 319-335.
  • [7] A. Fernández and A. Rodríguez, Primitive noncommutative Jordan algebras with nonzero socle, Proc. Amer. Math. Soc. 96 (1986), 199-206.
  • [8] H. Hanche-Olsen and E. Størmer, Jordan Operator Algebras, Monographs Stud. Math. 21, Pitman, 1984.
  • [9] I. N. Herstein, Rings with Involution, Chicago Lectures in Math., The University of Chicago Press, Chicago, 1976.
  • [10] N. Jacobson, Structure of Rings, Amer. Math. Soc. Colloq. Publ. 37, Providence, R.I., 1968.
  • [11] N. Jacobson, Structure and Representations of Jordan Algebras, Amer. Math. Soc. Colloq. Publ. 39, Providence, R.I., 1968.
  • [12] N. Jacobson, Basic Algebra II, Freeman, San Francisco, 1980.
  • [13] W. S. Martindale, 3rd., Lie isomorphisms of prime rings, Trans. Amer. Math. Soc. 142 (1969), 437-455.
  • [14] K. McCrimmon, The Zel'manov approach to Jordan homomorphisms of associative algebras, J. Algebra 123 (1989), 457-477.
  • [15] J. M. Osborn and M. L. Racine, Jordan rings with nonzero socle, Trans. Amer. Math. Soc. 251 (1979), 375-387.
  • [16] J. Pérez, L. Rico, A. Rodríguez and A. R. Villena, Prime Jordan-Banach algebras with nonzero socle, Comm. Algebra 20 (1992), 17-53.
  • [17] C. E. Rickart, General Theory of Banach Algebras, Krieger, New York, 1974.
  • [18] A. Rodríguez, La continuidad del producto de Jordan implica la del ordinario en el caso completo semiprimo, in: Contribuciones en Probabilidad, Estadística Matemática, Enseñanza de la Matemática y Análisis, Secretariado de Publicaciones de la Universidad de Granada, Granada, 1979, 280-288.
  • [19] A. Rodríguez, A Vidav-Palmer theorem for Jordan C*-algebras and related topics, J. London Math. Soc. 22 (1980), 318-332.
  • [20] A. Rodríguez, Jordan axioms for C*-algebras, Manuscripta Math. 61 (1988), 297-314.
  • [21] A. Rodríguez, Continuity of densely valued homomorphisms into H*-algebras, Quart. J. Math. Oxford, to appear.
  • [22] A. Rodríguez, Jordan structures in Analysis, in: Jordan algebras: Proc. Conf. Oberwolfach, 1992, W. Kaup, K. McCrimmon and H. P. Petersson (eds.), de Gruyter, Berlin, 1994.
  • [23] A. Rodríguez, A. Slin'ko and E. Zel'manov, Extending the norm from Jordan-Banach algebras of hermitian elements to their associative envelopes, Comm. Algebra 22 (1994), 1435-1455.
  • [24] S. Shirali, On the Jordan structure of complex Banach *-algebras, Pacific J. Math. 27 (1968), 397-404.
  • [25] A. M. Slin'ko, On topological rings with involution, Uspekhi Mat. Nauk SSSR 41 (1986), 197-198 (in Russian).
  • [26] E. Zel'manov, Absolute zero-divisors and algebraic Jordan algebras, Siberian Math. J. 23 (1982), 100-116.
  • [27] E. Zel'manov, On prime Jordan algebras II, ibid. 24 (1983), 89-104.
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Bibliografia
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