{
  "abstract": "For $0<θ<π$, let $K_θ=\\{e^{is}:|s|\\leθ\\}$ and $E_θ=\\operatorname{span}_{\\mathbb C}\\{1,z\\}\\subset C(K_θ)$. We prove that every contractive linear map from $E_θ$ into $\\mathcal B(H)$ admits a factorization through a unitary operator whose spectrum is contained in $K_θ$, and is therefore completely contractive. Consequently, $E_θ$ has a unique operator space structure. For $0<θ,φ<π$, we also show that $E_θ$ and $E_φ$ are isometrically isomorphic if and only if $θ=φ$, and that none of these spaces is isometrically isomorphic to $\\ell_1^2$ or $\\ell_\\infty^2$. Thus we obtain uncountably many mutually non-isometric two-dimensional complex Banach spaces with a unique operator space structure, giving a negative answer to a question of Pisier.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04608",
  "author_total": 1,
  "authors": [
    {
      "name": "Samya Kumar Ray"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b1r9e7bxv6sry.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04608",
  "fulltext_url": null,
  "id": "b1r9e7bxv6sry",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04608",
  "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"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7220650000",
  "pdf_url": "https://arxiv.org/pdf/2610.04608",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b1r9e7bxv6sry.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.04608",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04608"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Unique operator space structures on a family of two dimensional Banach spaces",
  "title_text": "Unique operator space structures on a family of two dimensional Banach spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/b1r9e7bxv6sry",
  "venue": "arXiv",
  "versions": []
}