Weak Amenability of Fourier Algebras on Compact groups

We give for a compact group G, a full characterization of when its Fourier algebra A(G) is weakly amenable: when the connected component of the identity G(e) is abelian. This condition is also equivalent to the hyper-Tauberian property for A(G), and to having the anti-diagonal Delta = {(s, s(-1))) : s is an element of G} be a set of spectral synthesis for A(GxG). We extend Our results to some classes of non-compact, locally compact groups, including small invariant neighbourhood groups and maximally weakly almost periodic groups. We close by illustrating a curious relationship between amenability and weak amenability of A(G) for compact G, and (operator) amenability and (operator) weak amenability of A(Delta)(G), an algebra defined by the authors in [11].