{
  "abstract": "For a unital Banach algebra $A$ with $\\|1_A\\|=1$ and a decreasing sequence $(I_r)$ of proper closed two-sided ideals of finite codimension, we introduce the bounded detection envelope $$\\widehat A:=\\operatorname{Env}^{b}_{(I_r)}(A)=\\varprojlim^{b}(A/I_r,π_r^s),\\qquad \\|(a_r)_r\\|=\\sup_r\\|a_r\\|.$$ Combining vector-valued Bourgain--Delbaen extensions, oracle Turing-machine recursion, and Argyros--Haydon operator analysis, we construct a separable Banach space $X$ such that, isometrically as unital Banach algebras, $$\\operatorname{Cal}(X)=\\mathcal{B}(X)/\\mathcal{K}(X)\\simeq\\widehat A. $$ We also characterize these envelopes as the unital Banach algebras with norm-one units admitting an isometric predual and countably many jointly faithful weak-star-to-norm continuous finite-dimensional unital representations. Applications include arbitrary finite-dimensional unital Banach algebras with norm-one units and their countable $\\ell_\\infty$-sums, as well as unitized coordinatewise algebras over normalized $1$-unconditional boundedly complete Schauder bases. These include incidence algebras, $\\ell_\\infty$, and the unitization of Tsirelson's original space. Further applications concern analytic operator algebras, Fourier--Stieltjes and Fourier multiplier algebras of countable discrete groups, and classical and quantum convolution algebras. In particular, we realize $H^\\infty(\\mathbb D)$, $\\operatorname{Mult}(H_2^2)$, $\\mathcal L_2$, $M_{\\mathrm{cb}}A(\\mathbb F_2)$, and $C^u(O_3^+)^*$ as Calkin algebras, with their canonical norms and products.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32330",
  "author_total": 2,
  "authors": [
    {
      "name": "Rui Liu"
    },
    {
      "name": "Jie Shen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4rbp7w2z8fw3a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32330",
  "fulltext_url": null,
  "id": "4rbp7w2z8fw3a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32330",
  "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": "W7214730314",
  "pdf_url": "https://arxiv.org/pdf/2609.32330",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4rbp7w2z8fw3a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2609.32330",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32330"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Inverse-Limit Detection Envelopes as Calkin Algebras",
  "title_text": "Inverse-Limit Detection Envelopes as Calkin Algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/4rbp7w2z8fw3a",
  "venue": "arXiv",
  "versions": []
}