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

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, $\checkΔ_G=\{(g,g^{-1}):g\in G\}$, 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.

The final version to appear in Adv. Math.;Proposition 3.4 and related comments from the previous version have been deleted; 25 pages