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 , 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 of all such Boolean algebras contains a copy of the ech-Stone compactification of the integers and the Banach space has l (a) as a quotient. Connections with the Grothendieck property in Banach spaces are discussed.