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Independent families in Boolean algebras with some separation properties. (English) Zbl 1283.06018
A Boolean algebra \(\mathcal A\) is said to have the subsequential completeness property if for any countably infinite antichain \(\{ A_n : n \in \mathbb{N} \} \subseteq \mathcal{A}\) there exists an infinite \(M \subseteq \mathbb{N}\) such that in \(\mathcal{A}\) the supremum of \(\{ A_n : n \in M\}\) exists.
The authors prove that each infinite Boolean algebra \(\mathcal{A}\) which has the subsequential completeness property, has an independent family of cardinality \(\mathfrak{c}\). This essentially improves a result of Argyros from the 1980s. They use their result to show that for each Boolean algebra \(\mathcal{A}\) with the subsequential completeness property, its Stone space \(K_{\mathcal{A}}\) contains a copy of \(\beta(\mathbb{N})\) and the Banach space \(C(K_{\mathcal{A}})\) has \(l_{\infty}\) as a quotient.
The authors show that if a Boolean algebra \(\mathcal{A}\) satisfies the weak subsequential seperation property, then \(C(K_{\mathcal{A}})\) has the positive Grothendieck property. They discuss connections with Efimov’s problem.

06E05 Structure theory of Boolean algebras
03E05 Other combinatorial set theory
06E15 Stone spaces (Boolean spaces) and related structures
46B10 Duality and reflexivity in normed linear and Banach spaces
46E15 Banach spaces of continuous, differentiable or analytic functions
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