{
  "id": "1002.1602",
  "version": "v3",
  "published": "2010-02-08T13:41:16.000Z",
  "updated": "2010-12-01T18:55:05.000Z",
  "title": "On Differentiable Vectors for Representations of Infinite Dimensional Lie Groups",
  "authors": [
    "Karl-Hermann Neeb"
  ],
  "comment": "44 pages, Lemma 5.2 and some typos corrected",
  "categories": [
    "math.RT",
    "math.FA"
  ],
  "abstract": "In this paper we develop two types of tools to deal with differentiability properties of vectors in continuous representations $\\pi \\: G \\to \\GL(V)$ of an infinite dimensional Lie group $G$ on a locally convex space $V$. The first class of results concerns the space $V^\\infty$ of smooth vectors. If $G$ is a Banach--Lie group, we define a topology on the space $V^\\infty$ of smooth vectors for which the action of $G$ on this space is smooth. If $V$ is a Banach space, then $V^\\infty$ is a Fr\\'echet space. This applies in particular to $C^*$-dynamical systems $(\\cA,G, \\alpha)$, where $G$ is a Banach--Lie group. For unitary representations we show that a vector $v$ is smooth if the corresponding positive definite function $\\la \\pi(g)v,v\\ra$ is smooth. The second class of results concerns criteria for $C^k$-vectors in terms of operators of the derived representation for a Banach--Lie group $G$ acting on a Banach space $V$. In particular, we provide for each $k \\in \\N$ examples of continuous unitary representations for which the space of $C^{k+1}$-vectors is trivial and the space of $C^k$-vectors is dense.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2010-12-01T18:55:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E65",
      "22E45"
    ],
    "keywords": [
      "infinite dimensional lie group",
      "differentiable vectors",
      "banach-lie group",
      "unitary representations",
      "smooth vectors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1002.1602N"
    }
  }
}