{
  "id": "2102.10088",
  "version": "v1",
  "published": "2021-02-19T18:41:06.000Z",
  "updated": "2021-02-19T18:41:06.000Z",
  "title": "The space $L_1(L_p)$ is primary for $1<p<\\infty$",
  "authors": [
    "Richard Lechner",
    "Pavlos Motakis",
    "Paul F. X. Müller",
    "Thomas Schlumprecht"
  ],
  "comment": "43 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "The classical Banach space $L_1(L_p)$ consists of measurable scalar functions $f$ on the unit square for which $$\\|f\\| = \\int_0^1\\Big(\\int_0^1 |f(x,y)|^p dy\\Big)^{1/p}dx < \\infty.$$ We show that $L_1(L_p)$ $(1 < p < \\infty)$ is primary, meaning that, whenever $L_1(L_p) = E\\oplus F$ then either $E$ or $F$ is isomorphic to $L_1(L_p)$. More generally we show that $L_1(X)$ is primary, for a large class of rearrangement invariant Banach function spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-19T18:41:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B09",
      "46B25",
      "46B28",
      "47A68"
    ],
    "keywords": [
      "rearrangement invariant banach function spaces",
      "classical banach space",
      "unit square",
      "large class",
      "measurable scalar functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}