Arens regularity and the second dual of certain quotients of the Fourier algebra (Q2716944)

Two multiplications can be defined in the second conjugate space \(A^{**}\) of a Banach algebra \(A\). If they coincide, \(A\) is called Arens regular (and \(A^{**}\) is then a Banach algebra). A useful criterion for this is that whenever \(\{g_n\}\), \(\{h_m\}\) are bounded sequences in \(A\) and \(S\in A^*\), the existence of the two iterated limits NEWLINE\[NEWLINE\lim_n\lim_m\langle S,g_n h_m \rangle,\quad \lim_m\lim_n \langle S,g_n h_m\rangleNEWLINE\]NEWLINE implies their equality. The present context is an LCA group \(G\) with dual \(\Gamma\). \(A(G):=L^1 (\Gamma)^\wedge\) with the inherited \(L^1\)-norm, and for closed \(E\subset G\), \(A(E)\) is the algebra of restrictions of \(A(G)\) to \(E\), endowed with the quotient norm. \(\widetilde A(E)\) is the set of functions on \(E\) that are uniform limits of \textit{bounded} sequences from \(A(E)\). The original algebra \(A(E)\) need not be closed in \(\widetilde A(E)\), but this is so whenever \(E\) is countable, as it is in much of this paper. The author eases the reader into his quite technical (existence) results by first proving a less ambitious fact about the circle \(\mathbb{T}\): it contains a compact, countable subset \(E\) such that (1) \(A(E)\) is Arens regular in every bounded multiplication; (2) \(\widetilde A(E)\) is not Arens regular, (3) \(A(E)^{**}\) is not Arens regular. [Earlier \textit{J. Pym}, J. Niger. Math. Soc. 2, 31-33 (1983; Zbl 0572.46044) had constructed an Arens regular, semisimple, commutative Banach algebra \(A\) for which \(A^{**}\) is not regular.] The proof of this is fairly transparent and is accomplished in about a page and a half. The generalization involves an infinite metrizable LCA group \(G\) in the role of \(\mathbb{T}\). En route to this the author introduces and studies \textit{blocked algebras}. (They look much like \(c_0\) and their duals are like \(\ell^1.)\) As to (3), he proves additionally that for a closed subset \(E\) of an arbitrary LCA group, \(A(E)^{**}\) is not Arens regular if (i) \(E\) contains arbitrarily long arithmetic progressions and (ii) either \(E\) contains cosets of arbitrarily large subgroups or there exist arbitrarily large \(E_1\) and \(E_2\) whose sum-set \(E_1+ E_2\) is contained in \(E\). The paper closes with a list of eleven open questions.