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

TL;DR

This paper proves that the Fourier algebra of all 1-connected, non-abelian nilpotent Lie groups is not weakly amenable, extending previous results and using harmonic analysis and explicit derivations.

Contribution

It verifies the weak amenability conjecture for all such nilpotent Lie groups by reducing to the Heisenberg group case and constructing explicit derivations.

Findings

Confirmed non-weak amenability for all 1-connected, non-abelian nilpotent Lie groups.

Constructed explicit non-zero derivations on dense subalgebras.

Used harmonic analysis and fusion rules to realize dual convolution explicitly.

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this was aleady known to hold in the non-abelian compact cases, by earlier work of Johnson (JLMS, 1994) and Plymen (unpublished note).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