{
  "abstract": "We investigate the relative position of von Neumann subalgebras through the Mashood--Taylor ($\\mathrm{d}_{\\mathrm{MT}}$) and Kadison--Kastler ($\\mathrm{d}_{\\mathrm{KK}}$) distances and the interior angle. For each $n\\in\\mathbb{N}$, we show that the hyperfinite $\\mathrm{II}_1$-factor $\\mathscr{R}$ contains an uncountable family of pairwise distinct regular $n\\times n$ spin model subfactors. In particular, this yields a continuous family $(\\mathscr{R}_{\\mathsf{H}_α})_{α\\in[0,π)}$ of $2\\times2$ spin model subfactors which, equipped with $\\mathrm{d}_{\\mathrm{MT}}$, is homeomorphic to the circle group $\\mathbb{T}$ and admits a natural topological group structure. We further establish \\[ \\mathrm{d}_{\\mathrm{MT}}(\\mathscr{R}_{\\mathsf{H}_α},\\mathscr{R}_{\\mathsf{H}_β}) =\\vert\\sin(α-β)\\vert. \\] We prove that two such spin model subfactors form a commuting square over their intersection if and only if they are maximally distant. More generally, commuting squares of $\\mathrm{II}_1$-factors, under natural index conditions, force maximal distance, yielding $\\mathrm{d}_{\\mathrm{KK}}=1=\\mathrm{d}_{\\mathrm{MT}}$. We also show that diffuse subalgebras which are orthogonal in the sense of Popa are maximally distant. In contrast, no two members of the above spin model family are Popa-orthogonal. Nevertheless, maximal distance between two such subfactors implies that their interior angle over the intersection is $π/2$, highlighting the distinction between angle orthogonality and Popa orthogonality. Finally, in $L(\\mathbb{F}_2)=L(\\langle a,b\\rangle)$, for $u\\in L(\\langle b\\rangle)$, we obtain \\[ \\mathrm{d}_{\\mathrm{MT}}(L(\\langle a\\rangle),uL(\\langle a\\rangle)u^*) =\\sqrt{1-\\vertτ(u)\\vert^4}, \\] and construct maximally distant masas not arising from subgroups of $\\mathbb{F}_2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.34856",
  "author_total": 2,
  "authors": [
    {
      "name": "Indrajit Ghosh"
    },
    {
      "name": "Sumit Kumar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cps09sn4edvxg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.34856",
  "fulltext_url": null,
  "id": "cps09sn4edvxg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.34856",
  "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",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7214784445",
  "pdf_url": "https://arxiv.org/pdf/2609.34856",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cps09sn4edvxg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2609.34856",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.34856"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Distances Between von Neumann Subalgebras: Spin Models, Commuting Squares, and Free Group Factors",
  "title_text": "Distances Between von Neumann Subalgebras: Spin Models, Commuting Squares, and Free Group Factors",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/cps09sn4edvxg",
  "venue": "arXiv",
  "versions": []
}