{
  "abstract": "We study the problem of geometric characterization of Banach-compact and $\\mathcal A$-compact operators on Hilbert $C^*$-modules in terms of uniform structures generated by seminorms. Previously, such a criterion for $\\mathcal A$-compactness of adjointable operators was obtained by Troitsky for countably generated range modules with a suitable system of seminorms. Using locally unital $C^*$-algebras as modules over themselves, we establish fundamental limitations of such characterizations. Namely, we show that for nonunital locally unital $C^*$-algebras not only does Troitsky's criterion fail to extend, but, more generally, no system of seminorms can characterize $\\mathcal A$-precompactness by total boundedness, nor Banach-compactness of bounded operators or $\\mathcal A$-compactness of adjointable operators by total boundedness of the images of the unit ball. Moreover, the same statement holds for Hilbert $C^*$-modules $\\mathcal N$ whenever the algebra $\\mathbf K(\\mathcal N)$ of $\\mathcal A$-compact operators is locally unital and nonunital. We also study Troitsky-type systems of seminorms and their relation to $\\mathcal A$-precompactness. We prove that every $\\mathcal A$-precompact subset is totally bounded with respect to the strongest of such structures. Consequently, for Hilbert $C^*$-modules admitting a standard frame, all the corresponding notions of total boundedness coincide.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39164",
  "author_total": 1,
  "authors": [
    {
      "name": "Denis Fufaev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7bwxtmxydse98.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39164",
  "fulltext_url": null,
  "id": "7bwxtmxydse98",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39164",
  "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"
    ],
    [
      "arxiv",
      "math.OA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39164",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7bwxtmxydse98.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39164",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39164"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Locally unital $C^*$-algebras and limitations of seminorm criteria for compact operators",
  "title_text": "Locally unital C^*-algebras and limitations of seminorm criteria for compact operators",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/7bwxtmxydse98",
  "venue": "arXiv",
  "versions": []
}