Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-19T06:31:58.625Z Has data issue: false hasContentIssue false
  • English
  • Français

The weak density of the non-invertible elements of a commutative algebra

Published online by Cambridge University Press:  17 April 2009

L. Narici ,
G. Bachman  and
E. Beckenstein
L. Narici
Affiliation:
St John's University, Jamaica. New York, USA
G. Bachman
Affiliation:
Polytechnic Institute of Brooklyn, Brooklyn, New York, USA
E. Beckenstein
Affiliation:
Polytechnic Institute of Brooklyn, Brooklyn, New York, USA.
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.

Let X be a commutative locally convex Hausdorff topological algebra with identity over a non-trivially valued field F. Let Mc denote the continuous nontrivial homomorphisms of X into F and M the set of all maximal ideals of X. If the spectrum of each element x in X is the set of scalars {f(x) | f ∈ Mc}, it is shown that the singular elements of X are weakly dense in X if and only if M is an infinite set.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 5 , Issue 3 , December 1971 , pp. 387 - 390
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Arens, Richard, “A generalization of normed rings”, Pacific J. Math. 2 (1952), 455–471.CrossRefGoogle Scholar
[2]Beckenstein, E., Bachman, G. and Narici, L., “Maximal ideals in topological algebras”, J. Analyse Math. 25 (1971), (to appear).Google Scholar
[3]Beckenstein, E., Bachman, G. and Narici, L., “Function algebras over valued fields”, (to appear).Google Scholar
[4]Graham, Colin C., “The weak density of the non-invertible elements of a commutative Banach algebra”, Bull. Austral. Math. Soc. 4 (1971), 179–182.CrossRefGoogle Scholar
[5]Michael, Ernest A., “Locally multiplicatively-convex topological algebras”, Mem. Amer. Math. Soc. 11 (Amer. Math. Soc., Providence, Rhode Island, 1952).Google Scholar
[6]Narici, Lawrence, “On nonarchimedean Banach algebras”, Arch. Math. 19 (1968), 428–435.CrossRefGoogle Scholar
[7]Narici, Lawrence, Beckenstein, Edward and Bachman, George, Functional analysis and valuation theory (Marcel Dekker, New York, 1971).Google Scholar
[8]Żelazko, W., “Metric generalizations of Banach algebras”, Rozprawy Mat. 47 (Polsk. Akad. Nauk, Inst. Mat., Warszawa, 1965).Google Scholar