On ideals in the bidual of the Fourier algebra and related algebras

Let G be a compact nonmetrizable topological group whose local weight b(G) has uncountable cofinality. Let H be an amenable locally compact group, A(G×H) the Fourier algebra of G×H, and UC2(G×H) the space of uniformly continuous functionals in VN(G×H)=A∗(G×H). We use weak factorization of operators in the group von Neumann algebra VN(G×H) to prove that there exist at least 2b(G)2 left ideals of dimensions at least 2b(G)2 in A(G×H)∗∗ and in UC2∗(G×H). We show that every nontrivial right ideal in A(G×H)∗∗ and in UC2∗(G×H) has dimension at least 2b(G)2.