{
  "id": "1501.01785",
  "version": "v1",
  "published": "2015-01-08T10:18:23.000Z",
  "updated": "2015-01-08T10:18:23.000Z",
  "title": "On complemented copies of $c_0(ω_1)$ in $C(K^n)$ spaces",
  "authors": [
    "Leandro Candido",
    "Piotr Koszmider"
  ],
  "categories": [
    "math.FA",
    "math.GN",
    "math.LO"
  ],
  "abstract": "Given a compact Hausdorff space $K$ we consider the Banach space of real continuous functions $C(K^n)$ or equivalently the $n$-fold injective tensor product $\\hat\\bigotimes_{\\varepsilon}C(K)$ or the Banach space of vector valued continuous functions $C(K, C(K, C(K ..., C(K)...)$. We address the question of the existence of complemented copies of $c_0(\\omega_1)$ in $\\hat\\bigotimes_{\\varepsilon}C(K)$ under the hypothesis that $C(K)$ contains an isomorphic copy of $c_0(\\omega_1)$. This is related to the results of E. Saab and P. Saab that $X\\hat\\otimes_\\varepsilon Y$ contains a complemented copy of $c_0$, if one of the infinite dimensional Banach spaces $X$ or $Y$ contains a copy of $c_0$ and of E. M. Galego and J. Hagler that it follows from Martin's Maximum that if $C(K)$ has density $\\omega_1$ and contains a copy of $c_0(\\omega_1)$, then $C(K\\times K)$ contains a complemented copy $c_0(\\omega_1)$. The main result is that under the assumption of $\\clubsuit$ for every $n\\in N$ there is a compact Hausdorff space $K_n$ of weight $\\omega_1$ such that $C(K)$ is Lindel\\\"of in the weak topology, $C(K_n)$ contains a copy of $c_0(\\omega_1)$, $C(K_n^n)$ does not contain a complemented copy of $c_0(\\omega_1)$ while $C(K_n^{n+1})$ does contain a complemented copy of $c_0(\\omega_1)$. This shows that additional set-theoretic assumptions in Galego and Hagler's nonseparable version of Cembrano and Freniche's theorem are necessary as well as clarifies in the negative direction the matter unsettled in a paper of Dow, Junnila and Pelant whether half-pcc Banach spaces must be weakly pcc.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-08T10:18:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "complemented copy",
      "compact hausdorff space",
      "infinite dimensional banach spaces",
      "half-pcc banach spaces",
      "fold injective tensor product"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150101785C"
    }
  }
}