{
  "abstract": "We introduce the notion of essential tracial rank at most one for simple $C^*$-algebras. We show that separable simple $C^*$-algebras with essential tracial rank at most one have several regularity properties, including stable rank one, strict comparison, and tracial approximate divisibility. A finite amenable simple $C^*$-algebras with this property is $\\mathcal Z$-stable. We also prove the converse. Thus this property gives a new characterization of separable simple finite amenable $\\mathcal Z$-stable $C^*$-algebras and shows that these algebras admit a one-dimensional tracial approximation. In the unital real-rank-zero case, this approximation can be strengthened to show that the algebra has tracial rank zero. This completes a long-standing project. As a byproduct, we prove that every tracial state on a unital separable simple finite amenable $C^*$-algebra is quasidiagonal and that the algebra itself is quasidiagonal. Consequently, every separable simple finite amenable $C^*$-algebra of finite nuclear dimension has finite decomposition rank. None of these results requires the UCT.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14711",
  "author_total": 1,
  "authors": [
    {
      "name": "Huaxin Lin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dtrh40a1f0ykt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14711",
  "fulltext_url": null,
  "id": "dtrh40a1f0ykt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14711",
  "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.OA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7221748030",
  "pdf_url": "https://arxiv.org/pdf/2610.14711",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dtrh40a1f0ykt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14711",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14711"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "C*-algebras of Essentially Tracially Rank One",
  "title_text": "C*-algebras of Essentially Tracially Rank One",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/dtrh40a1f0ykt",
  "venue": "arXiv",
  "versions": []
}