{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39769",
  "author_total": 1,
  "authors": [
    {
      "name": "Catalin Badea"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bj7my3k2cmvky.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39769",
  "fulltext_url": null,
  "id": "bj7my3k2cmvky",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39769",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
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      "arxiv",
      "math.HO"
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      "math.NT"
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  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39769",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bj7my3k2cmvky.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39769",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39769"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The Gleason--Kahane--Zelazko theorem: a gentle introduction",
  "title_text": "The Gleason--Kahane--Zelazko theorem: a gentle introduction",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/bj7my3k2cmvky",
  "venue": "arXiv",
  "versions": []
}