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.
References
Brech C.: On the density of Banach spaces C(K) with the Grothendieck property. Proc. Amer. Math. Soc. 134, 3653–3663 (2006)
Diestel, J.: Sequences and Series in Banach Spaces. Graduate Texts in Mathematics, vol. 92. Springer, New York (1984)
Freniche F.J.: The Vitali–Hahn–Saks theorem for Boolean algebras with the subsequential interpolation property. Proc. Amer. Math. Soc. 92, 362–366 (1984)
Hart, K.P.: Efimov’s problem. In: Pearl, E. (ed.) Open Problems in Topology, vol. 2, pp. 171–177. Elsevier, Amsterdam (2007)
Haydon, R.: A nonreflexive Grothendieck space that does not contain \({\ell_{\infty}}\). Israel J. Math. 40, 65–73 (1981)
Haydon R.: Boolean rings that are Baire spaces. Serdica Math. J. 27, 91–106 (2001)
Haydon R., Odell E., Levy M.: On sequences without weak* convergent convex block subsequences. Proc. Amer. Math. Soc. 100, 94–98 (1987)
Koppelberg, S.: Handbook of Boolean Algebras, vol. 1. North–Holland, Amsterdam (1989)
Koszmider P.: Banach spaces of continuous functions with few operators. Math. Ann. 330, 151–183 (2004)
Schachermeyer, W.: On some classical measure-theoretic theorems for non-sigma-complete Boolean algebras. Dissertationes Math. (Rozprawy Mat.) 214 (1982)
Semadeni, Z.: Banach spaces of continuous functions. Vol. I. Monografie Matematyczne, Tom 55. PWN–Polish Scientific Publishers, Warsaw (1971)
Talagrand M.: Un nouveau C(K) qui possède la propriété de Grothendieck. Israel J. Math. 37, 181–191 (1980)
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by A. Dow.
The first author was partially supported by the National Science Center research grant 2011/01/B/ST1/00657. The second author was partially supported by the Israel Science Foundation, Grant no. 1053/11; this is paper no. 1015 on his publication list.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
About this article
Cite this article
Koszmider, P., Shelah, S. Independent families in Boolean algebras with some separation properties. Algebra Univers. 69, 305–312 (2013). https://doi.org/10.1007/s00012-013-0227-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00012-013-0227-2
2010 Mathematics Subject Classification
- Primary: 06E
- Secondary: 46E15
- 46B10
- 54D
Key words and phrases
- independent families
- Grothendieck property
- Efimov’s problem
- subsequential completeness property