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On a Characterization of Maximal Ideals

Published online by Cambridge University Press:  20 November 2018

Jamil A. Siddiqi
Jamil A. Siddiqi*
Affiliation:
Université de Sherbrooke, Sherbrooke, Québec
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Let A be a commutative complex Banach algebra with identity e. Gleason [1] (cf. also Kahane and Żelazko [2]) has given the following characterization of maximal ideals in A.

Theorem. A subspace X ⊂ A of codimension one is a maximal ideal in A if and only if it consists of non-invertible elements.

The proofs given by Gleason and by Kahane and Żelazko are both based on the use of Hadamard's factorization theorem for entire functions. In this note we show that this can be avoided by using elementary properties of analytic functions.

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 2 , June 1970 , pp. 219 - 220
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Gleason, A. M., A characterization of maximal ideals, J. Analys. Math. 19 (1967), 171-172.Google Scholar
2. Kahane, J.-P. and Żelazko, W., A characterization of maximal ideals in commutative Banach algebras, Studia Math. 29 (1968), 339-343.Google Scholar
3. Żelazko, W., A characterization of multiplicative linear functionals in complex Banach algebras, Studia Math. 30 (1968), 83-85.Google Scholar