{
  "id": "1609.08612",
  "version": "v1",
  "published": "2016-09-27T15:40:05.000Z",
  "updated": "2016-09-27T15:40:05.000Z",
  "title": "Representations of $p$-convolution algebras on $L^q$-spaces",
  "authors": [
    "Eusebio Gardella",
    "Hannes Thiel"
  ],
  "comment": "31 pages",
  "categories": [
    "math.FA",
    "math.OA"
  ],
  "abstract": "For a nontrivial locally compact group $G$, and $p\\in [1,\\infty)$, consider the Banach algebras of $p$-pseudofunctions, $p$-pseudomeasures, $p$-convolvers, and the full group $L^p$-operator algebra. We show that these Banach algebras are operator algebras if and only if $p=2$. More generally, we show that for $q\\in [1,\\infty)$, these Banach algebras can be represented on an $L^q$-space if and only if one of the following holds: (a) $p=2$ and $G$ is abelian; or (b) $|\\frac 1p - \\frac 12|=|\\frac 1q - \\frac 12|$. This result can be interpreted as follows: for $p,q\\in [1,\\infty)$, the $L^p$- and $L^q$-representation theories of a group are incomparable, except in the trivial cases when they are equivalent. As an application, we show that, for distinct $p,q\\in [1,\\infty)$, if the $L^p$ and $L^q$ crossed products of a topological dynamical system are isomorphic, then $\\frac 1p + \\frac 1q=1$. In order to prove this, we study the following relevant aspects of $L^p$-crossed products: existence of approximate identities, duality with respect to $p$, and existence of canonical isometric maps from group algebras into their multiplier algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-27T15:40:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "convolution algebras",
      "banach algebras",
      "representation",
      "operator algebra",
      "nontrivial locally compact group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}