{
  "abstract": "We study the noncommutative generalization of measure-theoretic dominance and equivalence of states on $C^*$-algebras to explore quasi-invariance under group actions. For a dominated state, we derive an unbounded \"Radon-Nikodym\" derivative affiliated with the commutant algebra of the dominating state's GNS representation. Interestingly, this dominance is generally non-transitive because the product of the corresponding closed operators can be non-closable. When looking at group actions by $*$-automorphisms, the orbit of a fixed quasi-invariant state consists entirely of mutually equivalent states. However, the orbit closure may contain singular states, meaning the set of quasi-invariant states is closed under convex combinations but not topologically closed. The paper also provides a unitary implementation of the group action on the GNS Hilbert space-generalizing covariant representations and compares this approach with the Pedersen-Takesaki construction, where the Radon-Nikodym derivative sits in the centraliser instead of the commutant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00409",
  "author_total": 4,
  "authors": [
    {
      "name": "Ameur Dhahri"
    },
    {
      "name": "Francesco Fidaleo"
    },
    {
      "name": "Chul Ki Ko"
    },
    {
      "name": "Hyun Jae Yoo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/59knhysdgprhe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00409",
  "fulltext_url": null,
  "id": "59knhysdgprhe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00409",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7206151431",
  "pdf_url": "https://arxiv.org/pdf/2609.00409",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/59knhysdgprhe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.00409",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00409"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Dominance and equivalence for states on $C^*$-algebras: Quasi-Invariant states",
  "title_text": "Dominance and equivalence for states on C^*-algebras: Quasi-Invariant states",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/59knhysdgprhe",
  "venue": "arXiv",
  "versions": []
}