{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1334",
  "content_hash": null,
  "slug": "l1z-convolution-open-map-status-20260801",
  "type": "claim",
  "title": "Current checked status and unresolved remainder",
  "summary": "UNKNOWN as of 2026-07-27. The thread records local openness when one factor is finitely supported and failure of a uniform openness estimate. It does not establish or refute openness at every pair.",
  "relevance": "Records the strongest checked neighboring results and the exact remainder future work must settle.",
  "relevance_source": "recorded",
  "body": "A dated independent review on 2026-08-01 checked the structured sources below, the complete visible source discussion, exact-title and equivalent-formulation searches, and the current TheoremDB corpus. The MathOverflow question and every visible answer and comment were checked on 2026-07-27. The partial argument covers finitely supported factors, a dense class, while the all-pairs question remains unanswered. Work on open bilinear mappings, including DOI 10.1215/17358787-3599741, was checked for a theorem applying directly to convolution on ell_1(Z). No such specialization was found. Norm-controlled inversion results for convolution algebras were checked because invertible factors yield local solution mechanisms. They do not cover arbitrary pairs whose Fourier transforms have common zeros. A local corpus search for open convolution map, ell_1(Z), and Banach algebra multiplication found no duplicate.\n\nA complete resolution must satisfy: Prove local openness at every pair (a,b), with a neighborhood estimate that may depend on the pair, or exhibit a pair and an explicit sequence converging to a*b that cannot be factored by pairs converging to (a,b). A negative result must rule out all nearby factorizations, rather than showing failure of a single constructive factorization method.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
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    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://mathoverflow.net/questions/504613/is-convolution-in-ell-1-mathbb-z-an-open-map",
      "locator": "Dataset references and independent 2026-08-01 status search."
    },
    "missing": [
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  "formal_statement": null,
  "source": {
    "url": "https://mathoverflow.net/questions/504613/is-convolution-in-ell-1-mathbb-z-an-open-map",
    "locator": "Dataset references and independent 2026-08-01 status search."
  },
  "models": [],
  "relations": [
    {
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      "title": "Complete the stated acceptance conditions",
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      "slug": "R1627",
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      "metadata": {
        "reason": "Replaces unreadable status prose with the dated review from 2026-08-01."
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    {
      "slug": "l1z-convolution-open-map",
      "title": "l1z convolution open map",
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}