{
  "id": "2012.08935",
  "version": "v1",
  "published": "2020-12-16T13:39:17.000Z",
  "updated": "2020-12-16T13:39:17.000Z",
  "title": "A new complemented subspace for the Lorentz sequence spaces, with an application to its lattice of closed ideals",
  "authors": [
    "Ben Wallis"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We show that every Lorentz sequence space $d(\\textbf{w},p)$ admits a 1-complemented subspace $Y$ distinct from $\\ell_p$ and containing no isomorph of $d(\\textbf{w},p)$. In the general case, this is only the second nontrivial complemented subspace in $d(\\textbf{w},p)$ yet known. We also give an explicit representation of $Y$ in the special case $\\textbf{w}=(n^{-\\theta})_{n=1}^\\infty$ ($0<\\theta<1$) as the $\\ell_p$-sum of finite-dimensional copies of $d(\\textbf{w},p)$. As an application, we find a sixth distinct element in the lattice of closed ideals of $\\mathcal{L}(d(\\textbf{w},p))$, of which only five were previously known in the general case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-16T13:39:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B20",
      "46B45",
      "47L20"
    ],
    "keywords": [
      "lorentz sequence space",
      "closed ideals",
      "application",
      "general case",
      "second nontrivial complemented subspace"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}