Amenability, weak amenability and bounded approximate identities in multipliers of the Fourier algebra

We focus on problems concerning amenability and weak amenability in the algebras Acb(G) and AM(G) which are the closures of the Fourier algebras A(G) in the spaces of completely bounded multipliers of A(G) and the multipliers of A(G) respectively. We show that either algebra is weakly amenable if and only if the connected component Ge of G is abelian. We also show that if either algebra is amenable, then G has an open amenable subgroup. Moreover, if G is almost connected, then either algebra is amenable if and only if G is virtually abelian. Let A(G) be either Acb(G) or AM(G). Assume that A(G) has a bounded approximate identity. If G is a [SIN]-group and E is an element of R(G), the closed coset ring of G, then the ideal IA(G)(E) consisting of all functions in A(G) that vanish on E also has a bounded approximate identity. In particular, if A(G) is (weakly) amenable, then so is IA(G)(E).