{
  "abstract": "Dixmier asked in 1967 whether the spectra of simple separable non-type-I $C^*$-algebras all have the same Borel structure. We solve the formulation using ordinary isomorphisms of quotient Borel $σ$-algebras by proving that its universal assertion is independent of ZFC, relative to the consistency of ZFC. Under the continuum hypothesis, all separable non-type-I spectra are measurably isomorphic, without simplicity or unitality assumptions. In the model obtained by iterating Sacks forcing for $ω_2$ stages with countable supports over a GCH ground model, a fixed simple unital universal spectrum is not measurably isomorphic to the CAR spectrum. Neither consistency direction requires a large cardinal. The Borel-definable formulation, which requires isomorphisms to admit Borel lifts between irreducible representations, has a negative answer in ZFC. For the CAR algebra $A$ and a simple separable exact crossed product $B$ associated to $\\mathrm{SL}_3(\\mathbb{Z})$, there is no Borel-definable injection $\\widehat B\\to\\widehat A$. Nevertheless, all separable nuclear non-type-I spectra are Borel-definably isomorphic. We also determine the answer under several choiceless hypotheses. The axiom of determinacy together with dependent choice gives a stronger negative conclusion: the spectrum of $B$ does not even inject as a set into the spectrum of $A$. The symmetric Solovay model gives the same conclusion. Over ZF, a wellordering of the reals of order type $ω_1$ suffices for a positive answer. The finite-permutation Feferman--Levy model gives another positive answer for the full sigma-generated Borel structures. These comparisons concern the quotient structures themselves, without measure completion.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26319",
  "author_total": 1,
  "authors": [
    {
      "name": "Martino Lupini"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1bbekjydpet9z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26319",
  "fulltext_url": null,
  "id": "1bbekjydpet9z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26319",
  "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.LO"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7222260004",
  "pdf_url": "https://arxiv.org/pdf/2609.26319",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1bbekjydpet9z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2609.26319",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26319"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Solution to Dixmier's Problem about spectra of C*-algebras",
  "title_text": "Solution to Dixmier's Problem about spectra of C*-algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/1bbekjydpet9z",
  "venue": "arXiv",
  "versions": []
}