{
  "id": "0811.4432",
  "version": "v4",
  "published": "2008-11-28T05:54:10.000Z",
  "updated": "2010-03-29T19:33:34.000Z",
  "title": "Translation-finite sets, and weakly compact derivations from $\\lp{1}(\\Z_+)$ to its dual",
  "authors": [
    "Yemon Choi",
    "Matthew J. Heath"
  ],
  "comment": "v1: 14 pages LaTeX (preliminary). v2: 13 pages LaTeX, submitted. Some streamlining, renumbering and minor corrections. v3: appendix removed. v4: Modified appendix reinstated; 14 pages LaTeX. To appear in Bull. London Math. Soc.",
  "journal": "Bull. London Math. Soc. 42 (2010), no. 3, 429--440",
  "doi": "10.1112/blms/bdq003",
  "categories": [
    "math.FA",
    "math.CO"
  ],
  "abstract": "We characterize those derivations from the convolution algebra $\\ell^1({\\mathbb Z}_+)$ to its dual which are weakly compact. In particular, we provide examples which are weakly compact but not compact. The characterization is combinatorial, in terms of \"translation-finite\" subsets of ${\\mathbb Z}_+$, and we investigate how this notion relates to other notions of \"smallness\" for infinite subsets of ${\\mathbb Z}_+$. In particular, we show that a set of strictly positive Banach density cannot be translation-finite; the proof has a Ramsey-theoretic flavour.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2010-03-29T19:33:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "43A20",
      "43A46",
      "47B07"
    ],
    "keywords": [
      "weakly compact derivations",
      "translation-finite sets",
      "convolution algebra",
      "strictly positive banach density",
      "infinite subsets"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0811.4432C"
    }
  }
}