{
  "id": "1909.06829",
  "version": "v1",
  "published": "2019-09-15T16:23:02.000Z",
  "updated": "2019-09-15T16:23:02.000Z",
  "title": "Projections and unconditional bases in direct sums of $\\ell_p$ spaces, $0<p\\le \\infty$",
  "authors": [
    "Fernando Albiac",
    "Jose L. Ansorena"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We show that every unconditional basis in a finite direct sum $\\bigoplus_{p\\in A} \\ell_p$, with $A\\subset (0,\\infty]$, splits into unconditional bases of each summand. This settles a 40 year old question raised in [A. Orty\\'nski, Unconditional bases in $\\ell_{p}\\oplus\\ell_{q},$ $0<p<q<1$, Math. Nachr. 103 (1981), 109-116]. As an application we obtain that for any $A\\subset (0,1]$ finite, the spaces $Z=\\bigoplus_{p\\in A} \\ell_p$, $Z\\oplus \\ell_{2}$, and $Z\\oplus c_{0}$ have a unique unconditional basis up to permutation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-15T16:23:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46A16",
      "46A35",
      "46A40",
      "46A45",
      "46B15"
    ],
    "keywords": [
      "projections",
      "unique unconditional basis",
      "finite direct sum",
      "old question",
      "application"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}