{
  "id": "0807.2940",
  "version": "v3",
  "published": "2008-07-18T10:18:39.000Z",
  "updated": "2011-11-18T21:17:24.000Z",
  "title": "On the commutant of $C(X)$ in $C^*$-crossed products by $\\mathbb{Z}$ and their representations",
  "authors": [
    "Christian Svensson",
    "Jun Tomiyama"
  ],
  "comment": "19 pages. Minor clarifications and improvements of exposition made. Some typos corrected",
  "journal": "Journal Of Functional Analysis, 2009, 256, 7, 2367-2386",
  "categories": [
    "math.OA",
    "math.DS"
  ],
  "abstract": "For the $C^*$-crossed product $C^*(\\Sigma)$ associated with an arbitrary topological dynamical system $\\Sigma = (X, \\sigma)$, we provide a detailed analysis of the commutant, in $C^* (\\Sigma)$, of $C(X)$ and the commutant of the image of $C(X)$ under an arbitrary Hilbert space representation $\\tilde{\\pi}$ of $C^* (\\Sigma)$. In particular, we give a concrete description of these commutants, and also determine their spectra. We show that, regardless of the system $\\Sigma$, the commutant of $C(X)$ has non-zero intersection with every non-zero, not necessarily closed or self-adjoint, ideal of $C^* (\\Sigma)$. We also show that the corresponding statement holds true for the commutant of $\\tilde{\\pi}(C(X))$ under the assumption that a certain family of pure states of $\\tilde{\\pi}(C^* (\\Sigma))$ is total. Furthermore we establish that, if $C(X) \\subsetneq C(X)'$, there exist both a $C^*$-subalgebra properly between $C(X)$ and $C(X)'$ which has the aforementioned intersection property, and such a $C^*$-subalgebra which does not have this property. We also discuss existence of a projection of norm one from $C^*(\\Sigma)$ onto the commutant of $C(X)$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-11-18T21:17:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47L65",
      "46H10",
      "37B05",
      "54H20"
    ],
    "keywords": [
      "crossed product",
      "arbitrary hilbert space representation",
      "corresponding statement holds true",
      "subalgebra",
      "pure states"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0807.2940S"
    }
  }
}