Independent families in Boolean algebras with some separation properties

TL;DR

This paper demonstrates that Boolean algebras with the subsequential completeness property contain large independent families, leading to new insights into their Stone spaces and associated Banach spaces, extending previous results.

Contribution

It extends prior work by showing that a broader class of Boolean algebras contains continuum-sized independent families under weaker conditions.

Findings

Boolean algebras with subsequential completeness have continuum-sized independent families

Stone spaces of these algebras contain a copy of the Čech-Stone compactification of integers

Banach spaces of continuous functions on these spaces have $l_$ as a quotient

Abstract

We prove that any Boolean algebra with the subsequential completeness property contains an independent family of size continuum. This improves a result of Argyros from the 80ties 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 spaces of all such Boolean algebras contains a copy of the Cech-Stone compactification of the integers and the Banach space of contnuous functions on them has $l_\infty$ as a quotient. Connections with the Grothendieck property in Banach spaces are discussed.