{
  "id": "math/0508195",
  "version": "v1",
  "published": "2005-08-11T10:05:34.000Z",
  "updated": "2005-08-11T10:05:34.000Z",
  "title": "A spectral gap property for random walks under unitary representations",
  "authors": [
    "Bachir Bekka",
    "Yves Guivarc'h"
  ],
  "comment": "19 pages",
  "categories": [
    "math.DS",
    "math.SP"
  ],
  "abstract": "Let $G$ be a locally compact group and $\\mu$ a probability measure on $G,$ which is not assumed to be absolutely continuous with respect to Haar measure. Given a unitary representation $(\\pi, \\cal H)$ of $G,$ we study spectral properties of the operator $\\pi(\\mu)$ acting on $\\cal H.$ Assume that $\\mu$ is adapted and that the trivial representation $1_G$ is not weakly contained in the tensor product $\\pi\\otimes \\bar\\pi.$ We show that $\\pi(\\mu)$ has a spectral gap, that is, for the spectral radius $r_{\\rm spec}(\\pi(\\mu))$ of $\\pi(\\mu),$ we have $r_{\\rm spec}(\\pi(\\mu))<1.$ This provides a common generalization of several previously known results. Another consequence is that, if $G$ has Kazhdan's Property (T), then $r_{\\rm spec}(\\pi(\\mu))<1$ for every unitary representation $\\pi$ of $G$ without finite dimensional subrepresentations. Moreover, we give new examples of so-called identity excluding groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-08-11T10:05:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A15",
      "37A30"
    ],
    "keywords": [
      "unitary representation",
      "spectral gap property",
      "random walks",
      "finite dimensional subrepresentations",
      "study spectral properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......8195B"
    }
  }
}