{
  "id": "2405.04683",
  "version": "v3",
  "published": "2024-05-07T21:53:39.000Z",
  "updated": "2024-05-14T18:26:30.000Z",
  "title": "Multicomplex Ideals, Modules and Hilbert Spaces",
  "authors": [
    "Derek Courchesne",
    "Sébastien Tremblay"
  ],
  "comment": "27 pages",
  "categories": [
    "math-ph",
    "math.MP",
    "math.RA"
  ],
  "abstract": "In this article we study some algebraic aspects of multicomplex numbers $\\mathbb M_n$. For $n\\geq 2$ a canonical representation is defined in terms of the multiplication of $n-1$ idempotent elements. This representation facilitates computations in this algebra and makes it possible to introduce a generalized conjugacy $\\Lambda_n$, i.e. a composition of the $n$ multicomplex conjugates $\\Lambda_n:=\\dagger_1\\cdots \\dagger_n$, as well as a multicomplex norm. The multicomplex algebra equipped with this norm and $\\Lambda_n$ as the involution forms a C$^*$-algebra. The ideals of the ring of multicomplex numbers are then studied in details. Free $\\mathbb M_n$-modules and their linear operators are considered. Finally, we develop Hilbert spaces on the multicomplex algebra.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2024-05-14T18:26:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hilbert spaces",
      "multicomplex ideals",
      "multicomplex algebra",
      "multicomplex numbers",
      "representation facilitates computations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}