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arXiv · 1405.6403

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

Yemon Choi·Mahya Ghandehari
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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this was aleady known to hold in the non-abelian compact cases, by earlier work of Johnson (JLMS, 1994) and Plymen (unpublished note).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 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 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.

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Yemon Choi, Mahya Ghandehari. 2014-05-25. Weak amenability for Fourier algebras of 1-connected nilpotent Lie groups. https://doi.org/10.1016/j.jfa.2015.02.014

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