Weak and cyclic amenability for Fourier algebras of connected Lie groups

TL;DR

This paper constructs explicit cyclic derivations on Fourier algebras of certain Lie groups, demonstrating their non-weak amenability, and extends these results to connected, semisimple, and nilpotent groups using harmonic analysis and Lie group structure theory.

Contribution

It provides the first explicit examples of non-zero cyclic derivations on Fourier algebras of specific Lie groups, proving their non-weak amenability.

Findings

Fourier algebra of the real ax+b group is not weakly amenable.

Fourier algebras of connected semisimple Lie groups are not weakly amenable.

Fourier algebra of the reduced Heisenberg group is not weakly amenable.

Abstract

Using techniques of non-abelian harmonic analysis, we construct an explicit, non-zero cyclic derivation on the Fourier algebra of the real $ax+b$ group. In particular this provides the first proof that this algebra is not weakly amenable. Using the structure theory of Lie groups, we deduce that the Fourier algebras of connected, semisimple Lie groups also support non-zero, cyclic derivations and are likewise not weakly amenable. Our results complement earlier work of Johnson (JLMS, 1994), Plymen (unpublished note) and Forrest--Samei--Spronk (IUMJ 2009). As an additional illustration of our techniques, we construct an explicit, non-zero cyclic derivation on the Fourier algebra of the reduced Heisenberg group, providing the first example of a connected nilpotent group whose Fourier algebra is not weakly amenable.