{
  "id": "1405.6403",
  "version": "v2",
  "published": "2014-05-25T16:26:52.000Z",
  "updated": "2015-02-25T21:07:08.000Z",
  "title": "Weak amenability for Fourier algebras of 1-connected nilpotent Lie groups",
  "authors": [
    "Yemon Choi",
    "Mahya Ghandehari"
  ],
  "comment": "v1: AMS LaTeX, 20 pages, submitted. v2: AMS LaTeX, 21 pages; significant revisions to Sections 4 and 5 following improvements suggested by the referee, and references added. To appear in JFA; this is the final accepted version (CC-BY licence)",
  "categories": [
    "math.FA"
  ],
  "abstract": "A special case of a conjecture raised by Forrest and Runde (Math. Zeit., 2005) asserts that the Fourier algebra of every non-abelian connected Lie group fails to be weakly amenable; this was aleady known to hold in the non-abelian compact cases, by earlier work of Johnson (JLMS, 1994) and Plymen (unpublished note). In recent work (JFA, 2014) the present authors verified this conjecture for the real ax+b group and hence, by structure theory, for any semsimple Lie group. In this paper we verify the conjecture for all 1-connected, non-abelian nilpotent Lie groups, by reducing the problem to the case of the Heisenberg group. As in our previous paper, an explicit non-zero derivation is constructed on a dense subalgebra, and then shown to be bounded using harmonic analysis. En route, we use the known fusion rules for Schr\\\"odinger representations to give a concrete realization of the \"dual convolution\" for this group as a kind of twisted, operator-valued convolution. We also give some partial results for solvable groups which give further evidence to support the general conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-25T16:26:52.000Z",
      "abstract": "A special case of a conjecture raised by Forrest and Runde (Math. Zeit., 2005) asserts that the Fourier algebra of every non-abelian connected Lie group fails to be weakly amenable; this was aleady known to hold in the non-abelian compact cases, by earlier work of Johnson (JLMS, 1994) and Plymen (unpublished note). In recent work (JFA, 2014) the present authors verified this conjecture for the real $ax+b$ group and hence, by structure theory, for any semsimple Lie group. In this paper we verify the conjecture for all $1$-connected, non-abelian nilpotent Lie groups, by reducing the problem to the case of the Heisenberg group. As in our previous paper, an explicit non-zero derivation is constructed on a dense subalgebra, and then shown to be bounded using harmonic analysis: we rely on the explicit Plancherel formula for the Heisenberg group and the known fusion rules for its Schr\\\"odinger representations. We also give some partial results for solvable groups which give further evidence to support the general conjecture.",
      "comment": "v1: AMS LaTeX, 20 pages, submitted",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-02-25T21:07:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B47"
    ],
    "keywords": [
      "fourier algebra",
      "weak amenability",
      "non-abelian connected lie group fails",
      "non-abelian nilpotent lie groups",
      "conjecture"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.6403C"
    }
  }
}