{
  "id": "1002.4792",
  "version": "v2",
  "published": "2010-02-25T14:19:07.000Z",
  "updated": "2010-10-12T18:32:48.000Z",
  "title": "On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups",
  "authors": [
    "Karl-Hermann Neeb"
  ],
  "comment": "Minor revisions",
  "categories": [
    "math.RT",
    "math.OA"
  ],
  "abstract": "Let $G$ be a 1-connected Banach-Lie group or, more generally, a BCH--Lie group. On the complex enveloping algebra $U_\\C(\\g)$ of its Lie algebra $\\g$ we define the concept of an analytic functional and show that every positive analytic functional $\\lambda$ is integrable in the sense that it is of the form $\\lambda(D) = \\la \\dd\\pi(D)v, v\\ra$ for an analytic vector $v$ of a unitary representation of $G$. On the way to this result we derive criteria for the integrability of *-representations of infinite dimensional Lie algebras of unbounded operators to unitary group representations. For the matrix coefficient $\\pi^{v,v}(g) = \\la \\pi(g)v,v\\ra$ of a vector $v$ in a unitary representation of an analytic Fr\\'echet-Lie group $G$ we show that $v$ is an analytic vector if and only if $\\pi^{v,v}$ is analytic in an identity neighborhood. Combining this insight with the results on positive analytic functionals, we derive that every local positive definite analytic function on a 1-connected Fr\\'echet--BCH--Lie group $G$ extends to a global analytic function.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-10-12T18:32:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E65",
      "22E45"
    ],
    "keywords": [
      "infinite dimensional lie groups",
      "unitary representation",
      "analytic vector",
      "positive analytic functional",
      "local positive definite analytic function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1002.4792N"
    }
  }
}