{
  "id": "2003.11414",
  "version": "v1",
  "published": "2020-03-25T14:00:57.000Z",
  "updated": "2020-03-25T14:00:57.000Z",
  "title": "The number of closed ideals in $L(L_p)$",
  "authors": [
    "William B. Johnson",
    "Gideon Schechtman"
  ],
  "categories": [
    "math.FA",
    "math.OA"
  ],
  "abstract": "We show that there are $2^{2^{\\aleph_0}}$ different closed ideals in the Banach algebra $L(L_p(0,1))$, $1<p\\not= 2<\\infty$. This solves a problem in A. Pietsch's 1978 book \"Operator Ideals\". The proof is quite different from other methods of producing closed ideals in the space of bounded operators on a Banach space; in particular, the ideals are not contained in the strictly singular operators and yet do not contain projections onto subspaces that are non Hilbertian. We give a criterion for a space with an unconditional basis to have $2^{2^{\\aleph_0}}$ closed ideals in term of the existence of a single operator on the space with some special asymptotic properties. We then show that for $1<q<2$ the space ${\\frak X}_q$ of Rosenthal, which is isomorphic to a complemented subspace of $L_q(0,1)$, admits such an operator.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-25T14:00:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47L20",
      "46E30"
    ],
    "keywords": [
      "closed ideals",
      "special asymptotic properties",
      "banach space",
      "strictly singular operators",
      "banach algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}