{
  "id": "2202.11500",
  "version": "v1",
  "published": "2022-02-23T13:37:04.000Z",
  "updated": "2022-02-23T13:37:04.000Z",
  "title": "Quotient algebras of Banach operator ideals related to non-classical approximation properties",
  "authors": [
    "Henrik Wirzenius"
  ],
  "comment": "35 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "We investigate the quotient algebra $\\mathfrak{A}_X^{\\mathcal I}:=\\mathcal I(X)/\\overline{\\mathcal F(X)}^{||\\cdot||_{\\mathcal I}}$ for Banach operator ideals $\\mathcal I$ contained in the ideal of the compact operators, where $X$ is a Banach space that fails the $\\mathcal I$-approximation property. The main results concern the nilpotent quotient algebras $\\mathfrak A_X^{\\mathcal{QN}_p}$ and $\\mathfrak A_X^{\\mathcal{SK}_p}$ for the quasi $p$-nuclear operators $\\mathcal{QN}_p$ and the Sinha-Karn $p$-compact operators $\\mathcal{SK}_p$. The results include the following: (i) if $X$ has cotype 2, then $\\mathfrak A_X^{\\mathcal{QN}_p}=\\{0\\}$ for every $p\\ge 1$; (ii) if $X^*$ has cotype 2, then $\\mathfrak A_X^{\\mathcal{SK}_p}=\\{0\\}$ for every $p\\ge 1$; (iii) the exact upper bound of the index of nilpotency of $\\mathfrak A_X^{\\mathcal{QN}_p}$ and $\\mathfrak A_X^{\\mathcal{SK}_p}$ for $p\\neq 2$ is $\\max\\{2,\\left \\lceil p/2 \\right \\rceil\\}$, where $\\left \\lceil p/2 \\right \\rceil$ denotes the smallest $n\\in\\mathbb N$ such that $n\\ge p/2$; (iv) for every $p>2$ there is a closed subspace $X\\subset c_0$ such that both $\\mathfrak A_X^{\\mathcal{QN}_p}$ and $\\mathfrak A_X^{\\mathcal{SK}_p}$ contain a countably infinite decreasing chain of closed ideals. In addition, our methods yield a closed subspace $X\\subset c_0$ such that the compact-by-approximable algebra $\\mathfrak A_X=\\mathcal K(X)/\\mathcal A(X)$ contains two incomparable countably infinite chains of nilpotent closed ideals.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-23T13:37:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B28",
      "47L20",
      "47B10"
    ],
    "keywords": [
      "approximation property",
      "banach operator ideals",
      "non-classical approximation properties",
      "compact operators",
      "nilpotent quotient algebras"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}