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Properly algebraic and spectrum-finite ideals in Jordan systems

Published online by Cambridge University Press:  24 October 2008

Ottmar Loos
Ottmar Loos
Affiliation:
Institut für Mathematik, Universität Innsbruck, Technikerstr.25/7, A-6020 Innsbruck, Austria
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Extract

The two main results of this paper are:

(i) The set of properly algebraic elements of a Jordan system (algebra, triple system or pair) over an uncountable field is an ideal.

(ii) For a semiprimitive Banach Jordan system, the socle, the largest properly algebraic ideal, the largest properly spectrum-finite ideal and the largest von Neumann regular ideal all coincide.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 1 , July 1993 , pp. 149 - 161
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

REFERENCES

[1]Amitsue, A. S.. Algebras over infinite fields. Proc. Am. Math. Soc. 7 (1956), 35–48.CrossRefGoogle Scholar
[2]Benslimane, M., Jaa, O. and Kaidi, A.. The socle and the largest spectrum finite ideal. Quart. J. Math. Oxford 42 (1991), 1–7.CrossRefGoogle Scholar
[3]Fernández, A.. Noncommutative Jordan Riesz algebras. Quart. J. Math. Oxford 39 (1988), 67–80.Google Scholar
[4]Fernández, A., García, E. and Sánchez, E., von Neumann regular Jordan Banach triple systems. J. London Math. Soc. 42 (1990), 32–48.Google Scholar
[5]Fernández, A. and Rodríguez, A.. On the socle of a noncommutative Jordan algebra. Manuscr. math. 56 (1986), 269–278.CrossRefGoogle Scholar
[6]Hessenberger, G.. Banach–Jordan–Paare (Diplomarbeit, Universität Innsbruck, 1992).Google Scholar
[7]Jacobson, N. and Katz, J.. Generically algebraic quadratic Jordan algebras. Scripta math. 29 (1973), 215–227.Google Scholar
[8]Loos, O.. Jordan Pairs. Lecture Notes in Math., vol. 460 (Springer-Verlag, 1975).CrossRefGoogle Scholar
[9]Loos, O.. On the socle of a Jordan pair. Collect. Math. 40 (1989), 109–125.Google Scholar
[10]Loos, O.. Finiteness conditions in Jordan pairs. Math. Z. 206 (1991), 577–587.CrossRefGoogle Scholar
[11]Loos, O.. Recent results on finiteness conditions in Jordan pairs. In Proceedings of the 1992 Oberwolfach Conference on Jordan Algebras (to appear).Google Scholar
[12]Loos, O. and Neher, E.. Complementation of inner ideals in Jordan pairs. J. Algebra (to appear).Google Scholar
[13]Martínez, J.. Holomorphic functional calculus in Jordan-Banach algebras. Ann. Sci. Univ. ‘Blaise Pascal’, Clermont II, Sér. Math., Fasc. 27 (1991), 125–134.Google Scholar
[14]McCrimmon, K.. Homotopes of alternative algebras. Math. Ann. 191 (1971), 253–262.CrossRefGoogle Scholar
[15]McCrimmon, K.. Amitsur shrinkage of Jordan radicals. Comm. in Algebra 12 (1984), 777–826.CrossRefGoogle Scholar
[16]Petersson, H. P.. Zur Arithmetik der Jordan–Paare. Math. Z. 147 (1976), 139–161.CrossRefGoogle Scholar
[17]Rodríguez, A.. Jordan structures in analysis. In Proceedings of the 1992 Oberwolfach Conference on Jordan Algebras (to appear).Google Scholar
[18]Smyth, M. R. F.. Riesz theory in Banach algebras. Math. Z. 145 (1975), 145–155.CrossRefGoogle Scholar