Arens regularity and the second dual of certain quotients of the Fourier algebra.(English)Zbl 0984.43003

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 $\lim_n\lim_m\langle S,g_n h_m \rangle,\quad \lim_m\lim_n \langle S,g_n h_m\rangle$ 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 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 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 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.

MSC:

 43A20 $$L^1$$-algebras on groups, semigroups, etc. 46J99 Commutative Banach algebras and commutative topological algebras

Zbl 0572.46044
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