{
  "id": "1005.1260",
  "version": "v3",
  "published": "2010-05-07T17:29:49.000Z",
  "updated": "2010-09-06T16:55:18.000Z",
  "title": "M-ideals of homogeneous polynomials",
  "authors": [
    "Veronica Dimant"
  ],
  "journal": "Studia Mathematica 202 (2011), 81-104",
  "categories": [
    "math.FA"
  ],
  "abstract": "We study the problem of whether $\\mathcal{P}_w(^nE)$, the space of $n$-homogeneous polynomials which are weakly continuous on bounded sets, is an $M$-ideal in the space of continuous $n$-homogeneous polynomials $\\mathcal{P}(^nE)$. We obtain conditions that assure this fact and present some examples. We prove that if $\\mathcal{P}_w(^nE)$ is an $M$-ideal in $\\mathcal{P}(^nE)$, then $\\mathcal{P}_w(^nE)$ coincides with $\\mathcal{P}_{w0}(^nE)$ ($n$-homogeneous polynomials that are weakly continuous on bounded sets at 0). We introduce a polynomial version of property $(M)$ and derive that if $\\mathcal{P}_w(^nE)=\\mathcal{P}_{w0}(^nE)$ and $\\mathcal{K}(E)$ is an $M$-ideal in $\\mathcal{L}(E)$, then $\\mathcal{P}_w(^nE)$ is an $M$-ideal in $\\mathcal{P}(^nE)$. We also show that if $\\mathcal{P}_w(^nE)$ is an $M$-ideal in $\\mathcal{P}(^nE)$, then the set of $n$-homogeneous polynomials whose Aron-Berner extension do not attain the norm is nowhere dense in $\\mathcal{P}(^nE)$. Finally, we face an analogous $M$-ideal problem for block diagonal polynomials.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2010-09-06T16:55:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46G25",
      "46B04",
      "47L22",
      "46B20"
    ],
    "keywords": [
      "homogeneous polynomials",
      "block diagonal polynomials",
      "bounded sets",
      "aron-berner extension",
      "ideal problem"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1005.1260D"
    }
  }
}