Weak amenability for Fourier algebras of 1-connected nilpotent Lie groups
arXiv:1405.6403 · doi:10.1016/j.jfa.2015.02.014
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this was aleady known to hold in the non-abelian compact cases, by earlier work of Johnson (JLMS, 1994) and Plymen (unpublished note).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 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.
v1: AMS LaTeX, 20 pages, submitted. v2: AMS LaTeX, 21 pages; significant revisions to Sections 4 and 5 following improvements suggested by the referee, and references added. To appear in JFA; this is the final accepted author manuscript (CC-BY licence). v3: several minor typos fixed; metadata and acknowledgements updated