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

Choi, Yemon and Ghandehari, Mahya (2015) Weak amenability for Fourier algebras of 1-connected nilpotent Lie groups. Journal of Functional Analysis, 268 (8). pp. 2440-2463. ISSN 0022-1236

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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this was already known to hold in the non-abelian compact cases, by earlier work of Johnson (1994) [13] and Plymen (unpublished note).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.

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Journal Article
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Journal of Functional Analysis
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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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11 Mar 2015 08:54
Last Modified:
05 Jul 2020 04:20