Arens regularity and weak sequential completeness for quotients of the Fourier algebra (Q1593023)

Let \(G\) be a locally compact group, \(E\) a compact subset of \(G\), \(A(G)\) the Fourier algebra of \(G\), and \(A(E)\) the set of restrictions of elements of \(A(G)\) to \(E\), equipped with the quotient norm. The elements of \(A(G)^*\) are called pseudomeasures and those which lie in the norm closure of \(L^1(G)\) are called pseudofunctions. A pseudomeasure \(S\) is said to admit synthesis if it is the weak* limit of measures concentrated on the support of \(S\). A tenting sequence at a point \(a\in E\) is a sequence \(\{f_n\}\subset A(E)\) such that \(\|f_n\|\leq 1\) and \(f_n(a)=1\), \(\forall n\), and for any neighbourhood \(U\) of \(a\), \(\operatorname {supp} f_n\subset U\) for all but finitely many \(n\). A bounded sequence \(\{f_n\}\) in a Banach space is called a Sidon sequence if there exists \(\delta>0\) such that for any \(N>0\) and any complex numbers \(c_1,\dots,c_N\), \(\|\sum_{j=0}^N c_j f_j\|\geq\delta \sum_{j=0}^N|c_j|\). Finally, a point \(a\in E\) is called a Day point if there exists a tenting sequence at \(a\) which contains a Sidon subsequence. The main results of the paper are the following. (1) Theorem 4.1.3: Let \(G\) be non-discrete and either abelian or compact, and \(E\) a compact subset which admits a bounded spectral synthesis and is the support of a nonzero pseudofunction. Then \(A(E)\) is weakly sequentially complete. (This implies that every point of \(E\) is a Day point.) (2) Theorem 5.1.1: Let \(G\) be non-discrete and either abelian or compact, and assume that \(E\) supports a nonzero pseudofunction \(S\) which admits synthesis. Then every point of supp\( S\) is a Day point. (The existence of a Day point implies that \(A(E)\) is not Arens regular.) A number of related results, historical comments, and open questions are also included. In particular, in connection with the abelian case of (1) the author clarifies some claims of \textit{Y. Meyer} [Recent advances in spectral synthesis, Lect. Notes Math. 266, 239-253 (Berlin 1972; Zbl 0234.43003)].