{
  "id": "2007.05701",
  "version": "v1",
  "published": "2020-07-11T07:15:00.000Z",
  "updated": "2020-07-11T07:15:00.000Z",
  "title": "Isomorphisms of $BV(σ)$ spaces",
  "authors": [
    "Shaymaa Al-shakarchi",
    "Ian Doust"
  ],
  "comment": "12 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "In this paper we investigate the relationship between the properties of a compact set $\\sigma \\subseteq \\mathbb{C}$ and the structure of the space $BV(\\sigma)$ of functions of bounded variation (in the sense of Ashton and Doust) defined on $\\sigma$. For the subalgebras of absolutely continuous functions on $\\sigma$, it is known that for certain classes of compact sets one obtains a Gelfand--Kolmogorov type result: the function spaces $AC(\\sigma_1)$ and $AC(\\sigma_2)$ are isomorphic if and only if the domain sets $\\sigma_1$ and $\\sigma_2$ are homeomorphic. Our main theorem is that in this case the isomorphism must extend to an isomorphism of the $BV(\\sigma)$ spaces. An application is given to the spectral theory of $AC(\\sigma)$ operators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-11T07:15:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46J10",
      "26B30",
      "47B40"
    ],
    "keywords": [
      "isomorphism",
      "compact set",
      "gelfand-kolmogorov type result",
      "function spaces",
      "domain sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}