{
  "abstract": "We develop a differential-geometric approach to contractive projections on complex Banach spaces based on Levi forms of plurisubharmonic functions. We show that if a projection $P$ contracts a $\\mathrm{C}^2$ plurisubharmonic function $Φ$, then, at any point of its range, the Levi form of $Φ$ defines a tangent Hilbertian geometry for which $P$ becomes an orthogonal projection. When the range is finite-dimensional and the relevant Levi forms are nondegenerate, comparing the Levi geometries associated with different points leads naturally to positive relative operators and associated one-parameter groups. We apply this framework to tracial noncommutative $\\mathrm{L}^4$-spaces. Exploiting the cubic form of the duality mapping, we characterize contractive projections by a Jordan triple orthogonality property and prove that a projection on $\\mathrm{L}^4(\\mathcal{M})$ is $2$-contractive if and only if it is completely contractive. When the ambient von Neumann algebra is finite-dimensional, we identify the relative operator associated with the left and right tangent forms on the range with a restriction of an inverse square root of a modular operator on a linking von Neumann algebra. This entails that the range of any $2$-contractive projection is completely isometric to a rectangular noncommutative $\\mathrm{L}^4$-space associated with a $\\mathrm{W}^*$-ternary ring of operators. As a consequence, every such projection is contractively decomposable. Finally, for every $1 < p < \\infty$ with $p \\neq 2$, we prove that every finite-rank $2$-contractive projection on a tracial noncommutative $\\mathrm{L}^p$-space is completely contractive. For $p > 2$, we obtain a weighted ternary description of its range in terms of a finite-dimensional $\\mathrm{W}^*$-$\\mathrm{TRO}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03375",
  "author_total": 1,
  "authors": [
    {
      "name": "Cédric Arhancet"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bfw79b9s8xgy4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03375",
  "fulltext_url": null,
  "id": "bfw79b9s8xgy4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03375",
  "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"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7220373566",
  "pdf_url": "https://arxiv.org/pdf/2610.03375",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bfw79b9s8xgy4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.03375",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03375"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A modular theory of contractive projections on Banach spaces with applications to noncommutative $\\mathrm{L}^p$-spaces",
  "title_text": "A modular theory of contractive projections on Banach spaces with applications to noncommutative L^p-spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/bfw79b9s8xgy4",
  "venue": "arXiv",
  "versions": []
}