{
  "abstract": "We construct in ZFC a unital, simple, infinite-dimensional $C^*$-algebra whose nonzero irreducible representations form a single unitary equivalence class, but which admits a faithful representation on a separable Hilbert space. The algebra contains a unital copy of the canonical anticommutation relation (CAR) algebra, and the normalized CAR trace has a unique extension among all states. This extension is tracial, and its GNS representation is separable and faithful, with weak closure the hyperfinite $\\mathrm{II}_1$ factor. In contrast, every nonzero irreducible representation acts on a Hilbert space of density $2^{\\aleph_0}$. The construction separates the added unitaries into shell terms, controlled by finite CAR relations, and rank-one defect terms. These relations determine every irreducible representation of the generated algebra and every extension of the CAR trace. The algebra has norm density $2^{\\aleph_0}$; consequently, the continuum hypothesis (CH) is equivalent over ZFC to the existence of a counterexample to Naimark's problem of norm density $\\aleph_1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26930",
  "author_total": 1,
  "authors": [
    {
      "name": "Ryotaro Tanaka"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a7kntw58gxqf1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26930",
  "fulltext_url": null,
  "id": "a7kntw58gxqf1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26930",
  "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",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7214193841",
  "pdf_url": "https://arxiv.org/pdf/2609.26930",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a7kntw58gxqf1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.26930",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26930"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A separably representable counterexample to Naimark's problem in ZFC",
  "title_text": "A separably representable counterexample to Naimark's problem in ZFC",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/a7kntw58gxqf1",
  "venue": "arXiv",
  "versions": []
}