Rights statement: The final, definitive version of this article has been published in the Journal, Journal of Functional Analysis 268 (8), 2015, © ELSEVIER.
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Accepted author manuscript
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Final published version
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Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Weak amenability for Fourier algebras of 1-connected nilpotent Lie groups
AU - Choi, Yemon
AU - Ghandehari, Mahya
N1 - The final, definitive version of this article has been published in the Journal, Journal of Functional Analysis 268 (8), 2015, © ELSEVIER.
PY - 2015/4/15
Y1 - 2015/4/15
N2 - A special case of a conjecture raised by Forrest and Runde (2005) [10] asserts that the Fourier algebra of every non-abelian connected Lie group fails to be weakly amenable; this was already known to hold in the non-abelian compactcases, by earlier work of Johnson (1994) [13] and Plymen (unpublished note). In recent work (Choi and Ghandehari, 2014 [4]) the authors verified this conjecture for the real ax +b group and hence, by structure theory, for any semisimple Liegroup.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 Schrödinger 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.
AB - A special case of a conjecture raised by Forrest and Runde (2005) [10] asserts that the Fourier algebra of every non-abelian connected Lie group fails to be weakly amenable; this was already known to hold in the non-abelian compactcases, by earlier work of Johnson (1994) [13] and Plymen (unpublished note). In recent work (Choi and Ghandehari, 2014 [4]) the authors verified this conjecture for the real ax +b group and hence, by structure theory, for any semisimple Liegroup.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 Schrödinger 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.
KW - Dual convolution
KW - Fourier algebra
KW - Heisenberg group
KW - Weak amenability
U2 - 10.1016/j.jfa.2015.02.014
DO - 10.1016/j.jfa.2015.02.014
M3 - Journal article
VL - 268
SP - 2440
EP - 2463
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
SN - 0022-1236
IS - 8
ER -