Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-26T08:04:48.161Z Has data issue: false hasContentIssue false
  • English
  • Français

Algebraic ideals of semiprime Banach algebras

Published online by Cambridge University Press:  18 May 2009

S. Giotopoulos  and
M. Roumeliotis
S. Giotopoulos
Affiliation:
Department of Mathematics, Athens University, Panepistemiopolis, 15784 Athens, Greece
M. Roumeliotis
Affiliation:
Department of Mathematics, Athens University, Panepistemiopolis, 15784 Athens, Greece
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

If A is a semiprime Banach algebra, soc A, rad A the socle and radical of A, then Soc A ∩ rad A = (0). This elementary result enables us to prove some results concerning algebraic ideal and algebraic elements modulo the socle of A. We also deduce several conditions for A equivalent to the condition dim A <+∞.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 33 , Issue 3 , September 1991 , pp. 359 - 363
Copyright
Copyright © Glasgow Mathematical Journal Trust 1991

References

REFERENCES

1.Alexander, J. C., Compact Banach algebras, Proc. London Math. Soc. (3) 18 (1968), 1–18.CrossRefGoogle Scholar
2.Akemann, C. and Pedersen, G. K., Ideal perturbations in C*-algebras, Math. Scand. 41 (1977), 117–139.CrossRefGoogle Scholar
3.Allan, G. R., Embedding the algebra of formal power series in a Banach algebra, Proc. London Math. Soc. (3) 25 (1972), 329–340.CrossRefGoogle Scholar
4.Barnes, B., On the existence of minimal ideals in a Banach algebra, Trans. Amer. Math.Soc. 133 (1968), 511–517.CrossRefGoogle Scholar
5.Barnes, B., Algebraic elements of a Banach algebra modulo an ideal, Pacific J. Math. 117 (1985), 219–231.CrossRefGoogle Scholar
6.Barnes, B., Murphy, G., Smyth, M. and West, T., Riesz and Fredholm theory in Banach algebras, Research Notes in Mathematics 67 (Pitman, London, 1982)).Google Scholar
7.Bonsall, F. and Duncan, J., Complete normed algebras, (Springer-Verlag, 1973).CrossRefGoogle Scholar
8.Leoni Dalla, S.Giotopoulos and Nelli Katseli, The socle and finite dimensionality of a semiprime Banach algebra, Studia Math.. 92 (1989), 201–204.CrossRefGoogle Scholar
9.Dixon, P. G., Semiprime Banach algebras, J. London Math. Soc. (2) 6 (1973), 676–678.CrossRefGoogle Scholar
10.Dixon, P. G., Locally finite Banach algebras, J. London Math. Soc. (2) 8 (1974), 325–328.CrossRefGoogle Scholar
11.Kaplansky, I., Ring isomorphisms of Banach algebras, Canad. J. Math. 6 (1954), 374–381.CrossRefGoogle Scholar
12.Smyth, M. R. F., Riesz theory in Banach algebras, Math. Z. 145 (1975), 145–155.CrossRefGoogle Scholar
13.Tullo, A. W., Conditions on Banach algebras which imply finite dimensionality. Proc. Edinburgh Math. Soc. (2) 20 (1976), 1–5.CrossRefGoogle Scholar