Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-25T08:20:51.482Z Has data issue: false hasContentIssue false
  • English
  • Français

Commutative Banach Algebras with Idempotent Maximal Ideals

Published online by Cambridge University Press:  09 April 2009

R. J. Loy
R. J. Loy
Affiliation:
Monash University Clayton, VictoriaAustralia
Rights & Permissions [Opens in a new window]

Extract

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.

Let be a commutative Banach algebra over the complex field C, M an ideal of . Denote by M2 the set of all finite linear combinations of products of elements from M. M will be termed idempotent if M2 = M. The purpose of this paper is to investigate the structure of commutative Banach algebras in which all maximal ideals are idempotent.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 9 , Issue 3-4 , May 1969 , pp. 275 - 286
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Arens, R., ‘A Banach algebra generalization of conformal mappings of the disc’, Trans. Amer. Math. Soc. 81 (1956), 501–513.CrossRefGoogle Scholar
[2]Arens, R. and Kaplansky, I., ‘Topological representation of algebras’, Trans. Amer. Math. Soc. 63 (1948), 457–481.CrossRefGoogle Scholar
[3]Artin, E., Nesbitt, C. J. and Thrall, R. M., Rings with Minimum Condition (University of Michigan Press, Ann Arbor, 1961).Google Scholar
[4]Barnes, B. A., ‘Modular annihilator algebras’, Can. J. Math. 18 (1966), 566–578.CrossRefGoogle Scholar
[5]Cohen, P., ‘Factorization in group algebras’, Duke Math. J. 26 (1959), 199–205.CrossRefGoogle Scholar
[6]Feldman, C., ‘The Wedderburn principal theorem in Banach algebras’, Proc. Amer. Math. Soc. 2 (1951), 771–777.CrossRefGoogle Scholar
[7]Gelfand, I., Raikov, D. and Shilov, G., Commutative Normed Rings (Chelsea, 1964).Google Scholar
[8]Hoffman, K., Banach Spaces of Analytic Functions (Prentice-Hall Inc., 1962).Google Scholar
[9]Kaplansky, I., ‘Regular Banach algebras’, J. Indian Math. Soc. (N.S.), 12 (1948), 57–62.Google Scholar
[10]Kaplansky, I., ‘Functional Analysis’, in Some Aspects of Analysis and Probability (Surveys in Applied Mathematics, Vol. 4, John Wiley & Sons, New York, 1958), 3–36.Google Scholar
[11]Naimark, M. A., Normed Rings Noordoff, Groningen, Netherlands, 1964).Google Scholar
[12]Nakano, N., ‘Über idempotente Ideale in unendlichen algebraischen Zahlkörpern’, J. Sci. Hiroshima Univ. A, 17 (1953), 11–20.Google Scholar
[13]Satyanarayana, M., ‘Rings with primary ideals as maximal ideals’, Math. Scand. 20 (1967), 52–54.CrossRefGoogle Scholar
[14]Sherbert, D. R., ‘The structure of ideals and point derivations in Banach algebras of Lipschitz functions’, Trans. Amer. Math. Soc. 111 (1964), 240–272.CrossRefGoogle Scholar
[15]Singer, I. M. and Wermer, J., ‘Derivations on commutative normed algebras’, Math. Ann. 129 (1955), 260–264.CrossRefGoogle Scholar
[16]Yood, B., ‘Ideals in topological algebras’, Can. J. Math. 16 (1964), 28–45.CrossRefGoogle Scholar
[17]Zariski, O., and Samuel, P., Commutative Algebra, Vol.1 (Van Nostrand Inc., Princeton, 1963).Google Scholar