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On the Gleason-Kahane-Żelazko theorem for associative algebras

Roitman, Moshe and Sasane, Amol (2022) On the Gleason-Kahane-Żelazko theorem for associative algebras. Results in Mathematics, 78 (1). ISSN 1422-6383

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Identification Number: 10.1007/s00025-022-01789-z

Abstract

The classical Gleason-Kahane-Żelazko Theorem states that a linear functional on a complex Banach algebra not vanishing on units, and such that Λ (1) = 1 , is multiplicative, that is, Λ (ab) = Λ (a) Λ (b) for all a, b∈ A. We study the GKŻ property for associative unital algebras, especially for function algebras. In a GKŻ algebra A over a field of at least 3 elements, and having an ideal of codimension 1, every element is a finite sum of units. A real or complex algebra with just countably many maximal left (right) ideals, is a GKŻ algebra. If A is a commutative algebra, then the localization A P is a GKŻ-algebra for every prime ideal P of A. Hence the GKŻ property is not a local-global property. The class of GKŻ algebras is closed under homomorphic images. If a function algebra A⊆ F X over a subfield F of C, contains all the bounded functions in F X, then each element of A is a sum of two units. If A contains also a discrete function, then A is a GKŻ algebra. We prove that the algebra of periodic distributions, and the unitisation of the algebra of distributions with support in (0 , ∞) satisfy the GKŻ property, while the algebra of compactly supported distributions does not.

Item Type: Article
Official URL: https://www.springer.com/journal/25
Additional Information: © 2022 The Authors
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 16 Nov 2022 10:54
Last Modified: 14 Dec 2022 08:30
URI: http://eprints.lse.ac.uk/id/eprint/117348

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