{
  "abstract": "We introduce the geometric mean and the parallel sum of completely positive (CP) maps between von Neumann algebras, based on the Pusz--Woronowicz theory of positive sesquilinear forms. We provide a concrete characterization via a block matrix positivity condition and establish their fundamental properties, including the AM--GM--HM inequality with respect to the CP order. In finite-dimensional settings, our construction is compatible with the Choi--Jamiolkowski correspondence, under which the geometric mean of CP maps corresponds to the Kubo--Ando geometric mean of their Choi matrices. This yields a natural operator-theoretic framework for interpolating quantum channels. As an application, we obtain index-type inequalities for conditional expectations in subfactor theory. Finally, we establish a Lebesgue-type decomposition of CP maps via a parallel sum construction, thereby providing a unified framework that simultaneously generalizes Ando's decomposition of bounded positive operators and Kosaki's decomposition of normal positive functionals on von Neumann algebras.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.06019",
  "author_total": 1,
  "authors": [
    {
      "name": "Rui Okayasu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ftejx3wa2vspy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.06019",
  "fulltext_url": null,
  "id": "ftejx3wa2vspy",
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  "landing_url": "https://arxiv.org/abs/2605.06019",
  "language": null,
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  "metadata_licence": "CC0-1.0",
  "native_subjects": [
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      "arxiv",
      "math.FA"
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      "openalex",
      "2610|Advanced Operator Algebra Research"
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  "openalex": "W7160660874",
  "pdf_url": "https://arxiv.org/pdf/2605.06019",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ftejx3wa2vspy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:00Z",
      "native_id": "2605.06019",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.06019"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Geometric mean and Lebesgue-type decomposition of completely positive maps",
  "title_text": "Geometric mean and Lebesgue-type decomposition of completely positive maps",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/ftejx3wa2vspy",
  "venue": "arXiv",
  "versions": []
}