Independent families in Boolean algebras with some separation properties

Piotr Koszmider,Saharon Shelah

doi:10.1007/s00012-013-0227-2

Abstract

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 Čech–Stone compactification of the integers $${\beta\mathbb{N}}$$ and the Banach space $${C(K_\mathcal{A})}$$ has l ∞ as a quotient. Connections with the Grothendieck property in Banach spaces are discussed.

Highlights

We prove that any Boolean algebra with the subsequential completeness property contains an independent family of size c, the size of the continuum
KA will stand for the Stone space of A and C(K) for the Banach space of real valued continuous functions on K. [A] will stand for the clopen subset {x ∈ KA : A ∈ x} for any A in a Boolean algebra A, and 1X for the characteristic function of X ⊆ KA
We would like to direct the attention of the reader to some links between the weak subsequential separation property and the Grothendieck property, which originated in the theory of Banach spaces

Summary

Independent families

A Boolean algebra A is said to have the weak subsequential separation property if for any countably infinite antichain in {An : n ∈ N} ⊆ A, there exists A ∈ A such that both of the sets. A Boolean algebra A is said to have the subsequential separation property if given any countably infinite antichain {An : n ∈ N} ⊆ A, there is an A ∈ A such that the set. Restricting continuous functions on KA to a copy of βN gives, by the Tietze extension theorem, a norm one linear operator onto C(βN), which is known to be isometric to l∞ It follows that many Banach spaces present in the literature have l∞ as a quotient, in particular the spaces of [5] or [9]. It is not difficult to generalize the proof of Theorem 1.4 to conclude that such K always contain βN as well

The Grothendieck property of Banach spaces

Efimov’s problem