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  • 1405.6403v3

    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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Weak amenability for Fourier algebras of 1-connected nilpotent Lie groups

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<mark>Journal publication date</mark>15/04/2015
<mark>Journal</mark>Journal of Functional Analysis
Issue number8
Volume268
Number of pages24
Pages (from-to)2440-2463
Publication StatusPublished
Early online date3/03/15
<mark>Original language</mark>English

Abstract

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 compact
cases, by earlier work of Johnson (1994) [13] and Plymen (unpublished note).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 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 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.

Bibliographic note

The final, definitive version of this article has been published in the Journal, Journal of Functional Analysis 268 (8), 2015, © ELSEVIER.