The Gleason--Kahane--Zelazko theorem: a gentle introduction
Abstract
After some personal recollections of Jean-Pierre Kahane, this paper gives a gentle introduction to the Gleason-Kahane-Zelazko theorem. The theorem states that a normalised complex-linear functional on a unital complex Banach algebra is multiplicative whenever it does not vanish on invertible elements. Neither continuity of the functional nor commutativity of the algebra is assumed. We give a detailed account of three proofs: the original argument of Gleason and, independently, Kahane and Zelazko; the polynomial proof of Roitman and Sternfeld; and a proof by the author. We explain the complex analysis behind each: a special case of Hadamard's factorisation theorem, a polynomial argument using the fundamental theorem of algebra, and the Pólya-Saxer theorem on zero-free entire functions and their derivatives. The elementary entire-function lemma needed for the first proof is proved in full; the Pólya-Saxer theorem is used without proof. We give full details of the algebraic argument common to the three proofs, which deduces multiplicativity from the preservation of squares, including the non-commutative case. We then discuss applications and counterexamples, and present an extension to Ransford's generalised spectra using the Roitman--Sternfeld argument. We conclude with a brief guide to the more recent literature.
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BibTeX
@misc{badea2026gleasonkahanezelazko,
title = {{The Gleason--Kahane--Zelazko theorem: a gentle introduction}},
author = {Catalin Badea},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39769},
eprint = {2610.39769},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39769},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Gleason--Kahane--Zelazko theorem: a gentle introduction
AU - Catalin Badea
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39769
UR - https://arxiv.org/abs/2610.39769
AB - After some personal recollections of Jean-Pierre Kahane, this paper gives a gentle introduction to the Gleason-Kahane-Zelazko theorem. The theorem states that a normalised complex-linear functional on a unital complex Banach algebra is multiplicative whenever it does not vanish on invertible elements. Neither continuity of the functional nor commutativity of the algebra is assumed. We give a detailed account of three proofs: the original argument of Gleason and, independently, Kahane and Zelazko; the polynomial proof of Roitman and Sternfeld; and a proof by the author. We explain the complex analysis behind each: a special case of Hadamard's factorisation theorem, a polynomial argument using the fundamental theorem of algebra, and the Pólya-Saxer theorem on zero-free entire functions and their derivatives. The elementary entire-function lemma needed for the first proof is proved in full; the Pólya-Saxer theorem is used without proof. We give full details of the algebraic argument common to the three proofs, which deduces multiplicativity from the preservation of squares, including the non-commutative case. We then discuss applications and counterexamples, and present an extension to Ransford's generalised spectra using the Roitman--Sternfeld argument. We conclude with a brief guide to the more recent literature.
ER -