{
  "abstract": "Let $(\\mathscr{E}, \\langle \\cdot, \\cdot\\rangle)$ be a Hilbert $C^*$-module over a $C^*$-algebra $\\mathfrak{A}$. The space of adjointable operators on $\\mathscr{E}$ is denoted by $\\mathcal{L}\\left(\\mathscr{E}\\right)$. The sets of all states and pure states on $\\mathfrak{A}$ are denoted by $\\mathcal{S}\\left(\\mathfrak{A}\\right)$ and $\\mathcal{P}\\left( \\mathfrak{A}\\right)$, respectively. For $τ\\in\\mathcal{S}\\left( \\mathfrak{A}\\right)$, let us define $\\mathcal{N}^{\\mathscr{E}}_τ:=\\left\\lbrace x\\in\\mathscr{E}:τ\\left( \\left\\langle x,x\\right\\rangle\\right)=0 \\right\\rbrace$. The Hilbert completion of ${\\mathscr{E}}/{\\mathcal{N}^{\\mathscr{E}}_τ}$ is denoted by $\\mathscr{E}_τ$. For $T\\in\\mathcal{L}(\\mathscr{E})$, the operator $T_{\\mathscr{E}_τ}\\in\\mathbb{B}\\left( \\mathscr{E}_τ\\right)$, is defined by $T_{\\mathscr{E}_τ}\\left(x+\\mathcal{N}^{\\mathscr{E}}_τ\\right)=Tx+\\mathcal{N}^{\\mathscr{E}}_τ$. In this paper, we show that $\\mathscr{E}_τ={\\mathscr{E}}/{\\mathcal{N}^{\\mathscr{E}}_τ}$ when $\\mathfrak{A}$ either is a $C^*$-algebra of compact operators or is commutative. We introduce a quantity in the context of Hilbert $C^*$-modules, denoted by $π^{\\mathscr{E}}_k(\\cdot)$ for $k\\geq1$. We prove that $π^{\\mathscr{E}}_k(T)\\leq\\sup_{τ\\in\\mathcal{P}\\left( \\mathfrak{A}\\right)}\\left\\|T_{\\mathscr{E}_τ}\\right\\|_{\\left(k\\right)}\\leqπ^{\\mathscr{E}^{\\sharp}}_k(T_{\\mathscr{E}^{\\sharp}})$ for every $T\\in\\mathcal{L}\\left(\\mathscr{E}\\right)$, where the space $\\mathscr{E}^{\\sharp}$ is constructed as the extension of $\\mathscr{E}$ by the embedding of $\\mathfrak{A}$ into its enveloping von Neumann algebra $\\mathfrak{A}^{**}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": "Journal reference: Abedi, Sajjad; Moslehian, Mohammad Sal. Schatten norms on Hilbert $C^*$-modules via pure states. Math. Scand. 131 (2025), no. 3, 535--557",
  "arxiv": "2609.13944",
  "author_total": 2,
  "authors": [
    {
      "name": "Sajjad Abedi"
    },
    {
      "name": "Mohammad Sal Moslehian"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d7wrf57g14yzv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13944",
  "fulltext_url": null,
  "id": "d7wrf57g14yzv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13944",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2609.13944",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d7wrf57g14yzv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13944",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13944"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Schatten norms on Hilbert $C^*$-modules via pure states",
  "title_text": "Schatten norms on Hilbert C^*-modules via pure states",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/d7wrf57g14yzv",
  "venue": "arXiv",
  "versions": []
}