Weak amenability of Fourier algebras and local synthesis of the anti-diagonal

TL;DR

This paper proves that the Fourier algebra of a connected Lie group is weakly amenable only if the group is abelian, linking this property to the local synthesis of the anti-diagonal in the product group.

Contribution

It establishes a novel connection between weak amenability of Fourier algebras and the local synthesis of the anti-diagonal, providing a new criterion for abelianity of Lie groups.

Findings

Weak amenability of A(G) implies the anti-diagonal is a set of local synthesis.

Non-abelian connected Lie groups do not have weakly amenable Fourier algebras.

A(G) is weakly amenable iff the connected component of the identity is abelian.

Abstract

We show that for a connected Lie group $G$, its Fourier algebra $A(G)$ is weakly amenable only if $G$ is abelian. Our main new idea is to show that weak amenability of $A(G)$ implies that the anti-diagonal, $\check{\Delta}_G=\{(g,g^{-1}):g\in G\}$, is a set of local synthesis for $A(G\times G)$. We then show that this cannot happen if $G$ is non-abelian. We conclude for a locally compact group $G$, that $A(G)$ can be weakly amenable only if it contains no closed connected non-abelian Lie subgroups. In particular, for a Lie group $G$, $A(G)$ is weakly amenable if and only if its connected component of the identity $G_e$ is abelian.