{
  "abstract": "The Radon--Nikodym theorem for completely positive maps identifies the order interval below a map with positive contractions in the commutant of its minimal Stinespring representation. In this paper, we study the behavior of projection-valued Radon--Nikodym derivatives (sharp submaps) under composition. We prove that sharpness loss is determined by the multiplicative defect of the induced pullback map. In the finite-dimensional case, we show that this defect is determined by the orthogonal complement of the composite Kraus relation space, denoted $\\mathcal{E}_{Λ,Ω}$. A sharp submap remains sharp if and only if $\\mathcal{E}_{Λ,Ω}$ reduces the lifted Radon--Nikodym projection. Furthermore, we establish a tensor-factor criterion for the global preservation of sharpness and explicitly quantify the sharpness-loss defect.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09915",
  "author_total": 1,
  "authors": [
    {
      "name": "Mohsen Kian"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eqm29z2bqwev8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09915",
  "fulltext_url": null,
  "id": "eqm29z2bqwev8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09915",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7212141379",
  "pdf_url": "https://arxiv.org/pdf/2609.09915",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eqm29z2bqwev8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09915",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09915"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharpness Loss for Radon--Nikodym Derivatives of Completely Positive Maps",
  "title_text": "Sharpness Loss for Radon--Nikodym Derivatives of Completely Positive Maps",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/eqm29z2bqwev8",
  "venue": "arXiv",
  "versions": []
}