Amenability and bounded approximate identities in ideals of A(G) (Q748812)

Let G be a locally compact group and A(G) the Fourier algebra. The author investigates under which condition a closed ideal J of A(G) possesses a bounded approximate identity (b.a.i.). This is related to amenability. E.g. G is amenable iff \(J_ e=\{f\in A(G):\) \(f(e)=0\}\) has a b.a.i. The same question is treated for \(W^*\)-closed and cofinal ideals. It is shown that G is amenable iff each homomorphism from A(G) with finite dimensional range is continuous. Finally, Banach modules, and, in particular, VN(G) as A(G)-module, are studied.