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

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... ular.) We use be L2-methods of proof which do not have obvious extensions to the case of Ap(E). Related results, context (historical and mathematical), and open questions are given.