Arens regularity and weak sequential completeness for quotients of the Fourier algebra

This is a study of Arens regularity in the context of quotients of the Fourier algebra on a non-discrete locally compact abelian group (or compact group). (1) If a compact set E of G is of bounded synthesis and is the support of a pseudofunction, then A(E) is weakly sequentially complete. (This implies that every point of E is a Day point.) (2) If a compact set E supports a synthesizable pseudofunction, then A(E) has Day points. (The existence of a Day point implies that A(E) is not Arens regular.) We use be L-2-methods of proof which do not have obvious extensions to the case of A(p)(E). Related results, context (historical and mathematical), and open questions are given.