{
  "abstract": "We observe that the $2$-norm distance $d_{U,2}$ between the unitary orbits of normal elements in a $\\mathrm{II}_1$ factor $\\mathcal{M}$ is equal to the $2$-Wasserstein distance between the spectral measures induced by the trace $τ_\\mathcal{M}$. Using classification and optimal transport theory, we deduce an analogous $2$-norm equation for normal operators $x$ and $y$ in simple, separable, unital, nuclear, $\\mathcal{Z}$-stable $\\mathrm{C}^*$-algebras that are either monotracial, or real rank zero with finitely many extremal traces, provided that $σ(x)=σ(y)$ is convex. Consequently, $d_{U,2}$ equips the set of approximate unitary equivalence classes of contractive normal elements of $\\mathcal{M}$ with the structure of a compact length space. The same is true of the set of equivalence classes of embeddings into the Jiang-Su algebra $\\mathcal{Z}$ of classifiable tracial $2$-Wasserstein spaces over compact, convex planar domains.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.22585",
  "author_total": 1,
  "authors": [
    {
      "name": "Bhishan Jacelon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cw463bbpf9ee7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.22585",
  "fulltext_url": null,
  "id": "cw463bbpf9ee7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.22585",
  "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.MG"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7162124968",
  "pdf_url": "https://arxiv.org/pdf/2605.22585",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cw463bbpf9ee7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2605.22585",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.22585"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A $\\mathrm{C}^*$-algebraic Hoffman-Wielandt theorem",
  "title_text": "A C^*-algebraic Hoffman-Wielandt theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/cw463bbpf9ee7",
  "venue": "arXiv",
  "versions": []
}