{
  "id": "2102.03081",
  "version": "v1",
  "published": "2021-02-05T10:02:43.000Z",
  "updated": "2021-02-05T10:02:43.000Z",
  "title": "Uniqueness of unconditional basis of infinite direct sums of quasi-Banach spaces",
  "authors": [
    "Fernando Albiac",
    "Jose L. Ansorena"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "This paper is devoted to providing a unifying approach to the study of the uniqueness of unconditional bases, up to equivalence and permutation, of infinite direct sums of quasi-Banach spaces. Our new approach to this type of problem permits us to show that a wide class of vector-valued sequence spaces have a unique unconditional basis up to a permutation. In particular, solving a problem from [F. Albiac and C. Ler\\'anoz, Uniqueness of unconditional bases in nonlocally convex $\\ell_1$-products, J. Math. Anal. Appl. 374 (2011), no. 2, 394--401] we show that if $X$ is quasi-Banach space with a strongly absolute unconditional basis then the infinite direct sum $\\ell_{1}(X)$ has a unique unconditional basis up to a permutation, even without knowing whether $X$ has a unique unconditional basis or not. Applications to the uniqueness of unconditional structure of infinite direct sums of non-locally convex Orlicz and Lorentz sequence spaces, among other classical spaces, are also obtained as a by-product of our work.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-05T10:02:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B15",
      "46B20",
      "46B42",
      "46B45",
      "46A16",
      "46A35",
      "46A40",
      "46A45"
    ],
    "keywords": [
      "infinite direct sum",
      "quasi-banach space",
      "unique unconditional basis",
      "uniqueness",
      "strongly absolute unconditional basis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}