{
  "id": "math/0206041",
  "version": "v6",
  "published": "2002-06-05T14:56:32.000Z",
  "updated": "2002-12-28T17:59:28.000Z",
  "title": "Abstract harmonic analysis, homological algebra, and operator spaces",
  "authors": [
    "Volker Runde"
  ],
  "comment": "12 pages; a survey article; typos removed, references updated",
  "journal": "Contemp. Math. 328 (2003), 263-274",
  "categories": [
    "math.FA",
    "math.KT",
    "math.OA"
  ],
  "abstract": "In 1972, B. E. Johnson proved that a locally compact group $G$ is amenable if and only if certain Hochschild cohomology groups of its convolution algebra $L^1(G)$ vanish. Similarly, $G$ is compact if and only if $L^1(G)$ is biprojective: In each case, a classical property of $G$ corresponds to a cohomological propety of $L^1(G)$. Starting with the work of Z.-J. Ruan in 1995, it has become apparent that in the non-commutative setting, i.e. when dealing with the Fourier algebra $A(G)$ or the Fourier-Stieltjes algebra $B(G)$, the canonical operator space structure of the algebras under consideration has to be taken into account: In analogy with Johnson's result, Ruan characterized the amenable locally compact groups $G$ through the vanishing of certain cohomology groups of $A(G)$. In this paper, we give a survey of historical developments, known results, and current open problems.",
  "revisions": [
    {
      "version": "v6",
      "updated": "2002-12-28T17:59:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22D15",
      "22D25",
      "43A20",
      "43A30",
      "46H20",
      "46H25",
      "46L07",
      "46M18",
      "46M20",
      "47B47",
      "47L25",
      "47L50"
    ],
    "keywords": [
      "abstract harmonic analysis",
      "homological algebra",
      "locally compact group",
      "canonical operator space structure",
      "hochschild cohomology groups"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......6041R"
    }
  }
}