{
  "abstract": "This work initiates a systematic development of conjugation theory within the framework of Hilbert $C^*$-modules, extending classical complex symmetric operator theory beyond the Hilbert space setting. We first explore the foundational structure of $*$-conjugate-automorphisms $\\sharp$ on $C^*$-algebras and characterize their existence through isomorphisms with opposite algebras. We show that the set $\\mathbb{L}_c(\\mathscr{E})$ of all adjointable conjugate-linear operators on a Hilbert $C^*$-module $\\mathscr{E}$ possesses the structure of a Hilbert $\\mathbb{L}(\\mathscr{E})$-module. We also introduce the key notion of $\\sharp$-conjugation on a Hilbert $C^*$-module and fully characterize such modules in terms of real forms and real $C^*$-algebras. Moreover, we investigate the geometric properties of $\\sharp$-conjugate duals. A central achievement is the development of polar decomposition theory for semiregular linear (conjugate-linear) operators acting on a Hilbert $C^*$-module equipped with a $\\sharp$-conjugation $C$, thereby extending the celebrated Garcia--Putinar theorem [Trans. Amer. Math. Soc. 359 (2007), 3913--3931]. As a consequence, we show that any $C$-symmetric unitary operator factors as a product of two $\\sharp$-conjugations. A comparative analysis of alternative definitions shows that requiring the naive adjoint condition $\\langle Cx,y\\rangle = \\langle x,Cy\\rangle^*$ would force the underlying $C^*$-algebra to be essentially commutative, thus justifying the adopted approach. We supply various examples and counterexamples that extensively illustrate our results.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13931",
  "author_total": 3,
  "authors": [
    {
      "name": "Mahboubeh Faregh"
    },
    {
      "name": "Rasoul Eskandari"
    },
    {
      "name": "Mohammad Sal Moslehian"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8vpsmz6vbqm91.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13931",
  "fulltext_url": null,
  "id": "8vpsmz6vbqm91",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13931",
  "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/2609.13931",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8vpsmz6vbqm91.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13931",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13931"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Conjugations on Hilbert $C^*$-modules",
  "title_text": "Conjugations on Hilbert C^*-modules",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/8vpsmz6vbqm91",
  "venue": "arXiv",
  "versions": []
}