Since the work of Johnson characterizing amenability of a locally compact group G in terms of Banach algebra amenability of the convolution algebra L1(G), questions of characterizing various amenabilities of group related Banach algebras have been central theme of abstract harmonic analysis. For example, Ruan proved that operator space amenability of the Fourier algebra A(G) is equivalent to the amenability of G and Forrest/Runde showed that amenability of A(G) is equivalent to G being virtually abelian. In this talk we will focus on the weak amenability problem of Fourier algebras on Lie groups. We show that for a Lie group G, its Fourier algebra A(G) is weakly amenable if and only if its connected component of the identity Ge is abelian. Our main new idea is to show that for connected G, weak amenability of A(G) implies that the anti-diagonal of the product group G \times G, is a set of local synthesis for A(G\times G). We then to show that this cannot happen if G is non-abelian.
This is a joint work with Jean Ludwig (Metz), Ebrahim Samei (Saskatchewan) and Nico Spronk (Waterloo).