{
  "abstract": "The discretization of continuous frames in Hilbert spaces is well established, while the corresponding problem for Hilbert $C^*$-modules presents additional difficulties arising from the $C^*$-algebra-valued inner product and the structure of compact operators on Hilbert modules. In this paper, we develop an operator-selection framework for the discretization of continuous frames in Hilbert $C^*$-modules over finite-dimensional $C^*$-algebras. Our main result is a trace-controlled selection theorem for positive compact operators acting on countably generated Hilbert $C^*$-modules over finite-dimensional $C^*$-algebras. In particular, we establish a module version of the binary selection principle for positive trace-controlled compact operators. The proof combines finite-rank approximation, the matrix representation of finite-dimensional $C^*$-algebras, and operator estimates adapted to the $C^*$-module setting. We then apply these selection results to the discretization of continuous Hilbert $C^*$-module frames. Under suitable assumptions on the underlying metric measure space, with the coefficient algebra $\\mathcal A$ finite-dimensional, and assuming that every closed submodule of $\\mathcal H$ is orthogonally complemented. The resulting sampling family is a Hilbert $C^*$-module frame with explicit frame bounds and controlled separation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05073",
  "author_total": 1,
  "authors": [
    {
      "name": "Hicham Tarif"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0vqqmjd8d140p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05073",
  "fulltext_url": null,
  "id": "0vqqmjd8d140p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05073",
  "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",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7220746233",
  "pdf_url": "https://arxiv.org/pdf/2610.05073",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0vqqmjd8d140p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05073",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05073"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Trace-Controlled Operator Selection and Discretization of Continuous Frames in Hilbert \\(C^*\\)-Modules",
  "title_text": "Trace-Controlled Operator Selection and Discretization of Continuous Frames in Hilbert C^*-Modules",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/0vqqmjd8d140p",
  "venue": "arXiv",
  "versions": []
}