{
  "abstract": "Popa's relative dimension set $Λ(M,N)$ of an inclusion of $\\mathrm{II}_1$ factors $N\\subset M$ consists of the scalars $α$ for which some projection $p\\in M$ satisfies $E_N(p)=\\alpha1$. Popa computed it for index at most $4$, and for index $d>4$ outside the middle interval $(t_d,1-t_d)$, where $t_d(1-t_d)=d^{-1}$; whether the middle interval meets $Λ(M,N)$ for every irreducible inclusion of index $d>4$ has remained open. We show that for every extremal inclusion of finite index $d>4$, the whole interval $[t_d,1-t_d]$ is contained in $Λ(M,N)$. Consequently, $Λ(M,N)$ depends only on the index for all extremal inclusions of finite index, in particular for all irreducible ones. Extremality cannot be dropped: for $d>4$, Popa's locally trivial inclusions are exactly those whose relative dimension set misses the middle interval. Other non-extremal inclusions can contain the whole middle interval in their relative dimension set, or miss part of it.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02336",
  "author_total": 1,
  "authors": [
    {
      "name": "Julio Cáceres"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ccv9nb11rc3y7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02336",
  "fulltext_url": null,
  "id": "ccv9nb11rc3y7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02336",
  "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.OA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02336",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ccv9nb11rc3y7.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02336",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02336"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The relative dimension set of an extremal subfactor of finite index",
  "title_text": "The relative dimension set of an extremal subfactor of finite index",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/ccv9nb11rc3y7",
  "venue": "arXiv",
  "versions": []
}