{
  "abstract": "Let $\\mathcal{H}$ be a complex Hilbert space and $\\mathcal{B}(\\mathcal{H})$ be the algebra of all bounded linear operators on $\\mathcal{H}$. For $A \\in \\mathcal{B}(\\mathcal{H})$, we refer to the sequence $\\{|A^{n}|^{1/n}\\}_{n\\in\\mathbb{N}}$ as the normalized power sequence of $A$. In this article, we study the norm convergence property, i.e. convergence of normalized power sequence in the norm topology for operators belonging to type $\\mathrm{I}_k$ von Neumann algebras acting on a separable complex Hilbert space. By utilizing continuous upper-triangular forms via unitary conjugations, we construct specific projection-valued families to prove that every operator in a type $\\mathrm{I}_k$ von Neumann algebra is decomposable. As a consequence, this immediately establishes that every such operator possesses the norm convergence property, extending recent results known for matrices with complex-valued entries, compact operators on a separable Hilbert space, spectral operators, and Riesz operators. Finally, we provide a counterexample within the type $I_\\infty$ factor $\\mathcal{B}(l^2(\\mathbb{N}))$ to demonstrate that this convergence property generally fails when the dimension $k$ is infinite.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.18447",
  "author_total": 1,
  "authors": [
    {
      "name": "Ajay Kumar Karri"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e5a31qqjvwfyf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.18447",
  "fulltext_url": null,
  "id": "e5a31qqjvwfyf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.18447",
  "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.OA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7203902082",
  "pdf_url": "https://arxiv.org/pdf/2608.18447",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e5a31qqjvwfyf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2608.18447",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.18447"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Decomposability of Operators in Type $\\mathrm{I}_k$ von Neumann Algebras",
  "title_text": "Decomposability of Operators in Type I_k von Neumann Algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/e5a31qqjvwfyf",
  "venue": "arXiv",
  "versions": []
}