{
  "id": "1402.6564",
  "version": "v1",
  "published": "2014-02-26T15:04:07.000Z",
  "updated": "2014-02-26T15:04:07.000Z",
  "title": "Bourgain-Delbaen $\\mathcal{L}^{\\infty}$-sums of Banach spaces",
  "authors": [
    "Despoina Zisimopoulou"
  ],
  "comment": "29 pages, no figures",
  "categories": [
    "math.FA"
  ],
  "abstract": "Motivated by a problem stated by S.A.Argyros and Th. Raikoftsalis, we introduce a new class of Banach spaces. Namely, for a sequence of separable Banach spaces $(X_n,\\|\\cdot\\|_n)_{n\\in\\mathbb{N}}$, we define the Bourgain Delbaen $\\mathcal{L}^{\\infty}$-sum of the sequence $(X_n,\\|\\cdot\\|_n)_{n\\in\\mathbb{N}}$ which is a Banach space $\\mathcal{Z}$ constructed with the Bourgain-Delbaen method. In particular, for every $1\\leq p<\\infty$, taking $X_n=\\ell_p$ for every $n\\in\\mathbb{N}$ the aforementioned space $\\mathcal{Z}_p$ is strictly quasi prime and admits $\\ell_p$ as a complemented subspace. We study the operators acting on $\\mathcal{Z}_p$ and we prove that for every $n\\in\\mathbb{N}$, the space $\\mathcal{Z}^n_p=\\sum_{i=1}^n\\oplus \\mathcal{Z}_p$ admits exactly $n+1$, pairwise not isomorphic, complemented subspaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-26T15:04:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B03",
      "46B25",
      "46B28"
    ],
    "keywords": [
      "complemented subspace",
      "separable banach spaces",
      "bourgain delbaen",
      "bourgain-delbaen method",
      "strictly quasi prime"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.6564Z"
    }
  }
}