{
  "abstract": "We study the Kadison--Kastler distance and the interior angle between masas in finite-dimensional matrix algebras. In \\(\\mathbb{M}_2(\\mathbb{C})\\), we obtain an explicit formula for the Kadison--Kastler distance between two masas and show that every value in \\([0,1]\\) is attained by \\(\\mathrm{d}_{\\textrm{KK}}(Δ,uΔu^*)\\). We further prove that the distance attains its maximal value precisely when the relative unitary is a Hadamard unitary. In addition, for every $v$ in the groupoid normaliser of $Δ$, the Kadison--Kastler distance between \\(Δ\\) and \\(vΔv^*\\) assumes either its minimal or maximal possible value. We also establish the identity \\[ \\mathrm{d}_{\\textrm{KK}}(\\mathcal{A},\\mathcal{B})=\\sinα(\\mathcal{A},\\mathcal{B}) \\] for any two intermediate subalgebras \\(\\mathbb{C}\\subsetneq\\mathcal{A},\\mathcal{B}\\subsetneq\\mathbb{M}_2(\\mathbb{C})\\). Most notably, for any unitaries \\(u,v\\in\\mathbb{M}_2(\\mathbb{C})\\), we establish the equivalence of maximal Kadison--Kastler distance, maximal interior angle, maximal Connes--Størmer modified entropy, and minimal Popa probabilistic index: \\[ \\mathrm{d}_{\\textrm{KK}}(uΔu^*,vΔv^*)=1 \\iff h(uΔu^*,vΔv^*)=\\log 2 \\iff α(uΔu^*,vΔv^*)=\\fracπ{2} \\iff λ(uΔu^*,vΔv^*)=\\frac{1}{2}. \\] For general \\(\\mathbb{M}_n(\\mathbb{C})\\), we prove a unitary invariance property for the interior angle and derive an explicit formula for the angle between arbitrary masas of the form \\(uΔ^{(n)}u^*\\) and \\(vΔ^{(n)}v^*\\). As a consequence, the interior angle is \\(\\fracπ{2}\\) precisely when the relative unitary \\(u^*v\\) is a Hadamard unitary. Finally, we show that every value in \\(\\left[0,\\fracπ{2}\\right]\\) is attained as the interior angle between a pair of masas in \\(\\mathbb{M}_n(\\mathbb{C})\\).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38206",
  "author_total": 2,
  "authors": [
    {
      "name": "Indrajit Ghosh"
    },
    {
      "name": "Sumit Kumar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a59g4qfaxkd1n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38206",
  "fulltext_url": null,
  "id": "a59g4qfaxkd1n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38206",
  "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": "W7215071115",
  "pdf_url": "https://arxiv.org/pdf/2609.38206",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a59g4qfaxkd1n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2609.38206",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38206"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Kadison-Kastler Distance and Interior Angle for Masas: Connections with Entropy and Probabilistic Index",
  "title_text": "Kadison-Kastler Distance and Interior Angle for Masas: Connections with Entropy and Probabilistic Index",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/a59g4qfaxkd1n",
  "venue": "arXiv",
  "versions": []
}