{
  "abstract": "For $\\mathbb{K}=\\mathbb{R}$ or $\\mathbb{C}$ and every separable Banach space $V$, we use an oracle-relative two-sorted finite-extension construction, whose oracle records the local rational structure of $V$, to construct a separable Banach space $X_V$ such that $$ \\operatorname{Cal}(X_V)=\\mathcal{B}(X_V)/\\mathcal{K}(X_V)\\simeq \\begin{pmatrix} \\mathbb{K}&0\\\\ V^*&\\mathbb{K} \\end{pmatrix} $$ as Banach algebras. The identification of $V^*$ with the Jacobson radical is isometric. Consequently, every nonzero dual Banach space with a separable predual admits, after an equivalent renorming, a unital Banach-algebra structure isomorphic to the Calkin algebra of a separable Banach space. Moreover, there is a separable Banach space $X$ such that every separable Banach space is isometric to $\\mathcal{J}/\\mathcal{K}(X)$ for a closed two-sided ideal $\\mathcal{J}$ of $\\mathcal{B}(X)$, and the subspace--ideal correspondence preserves the canonical order.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32334",
  "author_total": 2,
  "authors": [
    {
      "name": "Rui Liu"
    },
    {
      "name": "Jie Shen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/apcmgtm0nhh97.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32334",
  "fulltext_url": null,
  "id": "apcmgtm0nhh97",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32334",
  "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": "W7214722743",
  "pdf_url": "https://arxiv.org/pdf/2609.32334",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/apcmgtm0nhh97.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2609.32334",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32334"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Duals of separable Banach Spaces as Calkin Algebras and Universal Ideal Quotients",
  "title_text": "Duals of separable Banach Spaces as Calkin Algebras and Universal Ideal Quotients",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/apcmgtm0nhh97",
  "venue": "arXiv",
  "versions": []
}