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

作者：Choi Yemon^*; Ghandehari Mahya

来源：Journal of Functional Analysis, 2015, 268(8): 2440-2463.

DOI：10.1016/j.jfa.2015.02.014

• Dual convolution
• Fourier algebra
• Heisenberg group
• Weak amenability

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this was already known to hold in the non-abelian compact cases, by earlier work of Johnson (1994) [13] and Plymen (unpublished note).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

• 出版日期2015-4-15