Independent families in Boolean algebras with some separation properties

We prove that any Boolean algebra with the subsequential completeness property contains an independent family of size ${mathfrak{c}}$ , the size of the continuum. This improves a result of Argyros from the 1980s, which asserted the existence of an uncountable independent family. In fact, we prove it for a bigger class of Boolean algebras satisfying much weaker properties. It follows that the Stone space ${K_mathcal{A}}$ of all such Boolean algebras ${mathcal{A}}$ contains a copy of the ?ech–Stone compactification of the integers ${betamathbb{N}}$ and the Banach space ${C(K_mathcal{A})}$ has l _∞ as a quotient. Connections with the Grothendieck property in Banach spaces are discussed.