{
  "abstract": "Kucher asked whether $\\ell_{\\infty}(E)$ is a Grothendieck space for every super-reflexive Banach space $E$. We answer this question affirmatively by proving the stronger result that $\\ell_{\\infty}(E)$ has Pełczyński's property $(V)$. Combined with the natural dual representation of $\\ell_{\\infty}(E)$, this yields that every non-weakly compact operator on $\\ell_{\\infty}(E)$ fixes a copy of $\\ell_{\\infty}$. The main ingredient is a general property-$(V)$ theorem for function modules over zero-dimensional compact Hausdorff spaces with stalks admitting a common modulus of uniform convexity. The proof is based on Gierz's integral representation of functionals and the weak compactness of the associated canonical measures. In the case of $\\ell_{\\infty}(E)$, the relevant stalks are ultrapowers of $E$, and super-reflexivity provides the uniform geometric control required by the theorem. We also derive several consequences for operators on $\\ell_{\\infty}(E)$, including weak compactness into separable spaces, reflexivity of separable quotients, and compactness of operators into Schur spaces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11180",
  "author_total": 4,
  "authors": [
    {
      "name": "Hui Li"
    },
    {
      "name": "Sijie Luo"
    },
    {
      "name": "Lian Wu"
    },
    {
      "name": "Dejian Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/csz5t126naw1m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11180",
  "fulltext_url": null,
  "id": "csz5t126naw1m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11180",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7221189498",
  "pdf_url": "https://arxiv.org/pdf/2610.11180",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/csz5t126naw1m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.11180",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11180"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Uniformly convex stalks and Kucher's $\\ell_{\\infty}(E)$ problem",
  "title_text": "Uniformly convex stalks and Kucher's ℓ_∞(E) problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/csz5t126naw1m",
  "venue": "arXiv",
  "versions": []
}