Weak amenability of Fourier algebras and local synthesis of the anti-diagonal
Résumé
We show that for a connected Lie group G, its Fourier algebra is weakly amenable only if G is abelian. Our main new idea is to show that weak amenability of implies that the anti-diagonal, , is a set of local synthesis for . We then show that this cannot happen if G is non-abelian. We conclude for a locally compact group G, that can be weakly amenable only if it contains no closed connected non-abelian Lie subgroups. In particular, for a Lie group G, is weakly amenable if and only if its connected component of the identity is abelian.