{
  "id": "2301.03562",
  "version": "v1",
  "published": "2023-01-09T18:27:46.000Z",
  "updated": "2023-01-09T18:27:46.000Z",
  "title": "(Non-)amenability of $\\mathcal B(E)$ and Banach space geometry",
  "authors": [
    "Matthew Daws",
    "Matthias Neufang"
  ],
  "comment": "20 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $E$ be a Banach space, and $\\mathcal B(E)$ the algebra of all bounded linear operators on $E$. The question of amenability of $\\mathcal B(E)$ goes back to Johnson's seminal memoir \\cite{johnson} from 1972. We present the first general criteria applying to very wide classes of Banach spaces, given in terms of the Banach space geometry of $E$, which imply that $\\mathcal B(E)$ is non-amenable. We cover all spaces for which this is known so far (with the exception of one particular example), with much shorter proofs, such as $\\ell_p$ for $p \\in [1, \\infty]$ and $c_0$, but also many new spaces: the numerous classes of spaces covered range from all $\\mathcal{L}_p$-spaces for $p \\in (1, \\infty)$ to Lorentz sequence spaces and reflexive Orlicz sequence spaces, to the Schatten classes $S_p$ for $p \\in [1,\\infty]$, and to the James space $J$, the Schlumprecht space $S$, and the Tsirelson space $T$, among others. Our approach also highlights the geometric difference to the only space for which $\\mathcal B(E)$ \\emph{is} known to be amenable, the Argyros--Haydon space, which solved the famous scalar-plus-compact problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-09T18:27:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "banach space geometry",
      "amenability",
      "johnsons seminal memoir",
      "reflexive orlicz sequence spaces",
      "lorentz sequence spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}