An isometrically universal Banach space induced by a non-universal Boolean algebra
HTML articles powered by AMS MathViewer
- by Christina Brech and Piotr Koszmider PDF
- Proc. Amer. Math. Soc. 144 (2016), 2029-2036 Request permission
Abstract:
Given a Boolean algebra $A$, we construct another Boolean algebra $B$ with no uncountable well-ordered chains such that the Banach space of real-valued continuous functions $C(K_A)$ embeds isometrically into $C(K_B)$, where $K_A$ and $K_B$ are the Stone spaces of $A$ and $B$, respectively. As a consequence we obtain the following: If there exists an isometrically universal Banach space for the class of Banach spaces of a given uncountable density $\kappa$, then there is another such space which is induced by a Boolean algebra which is not universal for Boolean algebras of cardinality $\kappa$. Such a phenomenon cannot happen on the level of separable Banach spaces and countable Boolean algebras. This is related to the open question of whether the existence of an isometrically universal Banach space and of a universal Boolean algebra are equivalent on the nonseparable level (both are true on the separable level).References
- D. Amir and B. Arbel, On injections and surjections of continuous function spaces, Israel J. Math. 15 (1973), 301–310. MR 397380, DOI 10.1007/BF02787573
- Christina Brech and Piotr Koszmider, On universal Banach spaces of density continuum, Israel J. Math. 190 (2012), 93–110. MR 2956234, DOI 10.1007/s11856-011-0183-5
- Christina Brech and Piotr Koszmider, On universal spaces for the class of Banach spaces whose dual balls are uniform Eberlein compacts, Proc. Amer. Math. Soc. 141 (2013), no. 4, 1267–1280. MR 3008874, DOI 10.1090/S0002-9939-2012-11390-5
- Christina Brech and Piotr Koszmider, $\ell _\infty$-sums and the Banach space $\ell _\infty /c_0$, Fund. Math. 224 (2014), no. 2, 175–185. MR 3180586, DOI 10.4064/fm224-2-3
- Alan Dow and Klaas Pieter Hart, $\omega ^*$ has (almost) no continuous images, Israel J. Math. 109 (1999), 29–39. MR 1679586, DOI 10.1007/BF02775024
- Alan Dow and Klaas Pieter Hart, The measure algebra does not always embed, Fund. Math. 163 (2000), no. 2, 163–176. MR 1752102, DOI 10.4064/fm-163-2-163-176
- Alan Dow and Klaas Pieter Hart, A universal continuum of weight $\aleph$, Trans. Amer. Math. Soc. 353 (2001), no. 5, 1819–1838. MR 1707489, DOI 10.1090/S0002-9947-00-02601-5
- A. Dow, PFA and complemented subspaces of $\ell _\infty /c_0$, Preprint, July 2014.
- Mirna D amonja, Isomorphic universality and the number of pairwise nonisomorphic models in the class of Banach spaces, Abstr. Appl. Anal. , posted on (2014), Art. ID 184071, 11. MR 3214408, DOI 10.1155/2014/184071
- A. S. Esenin-Vol′pin, On the existence of a universal bicompactum of arbitrary weight, Doklady Akad. Nauk SSSR (N.S.) 68 (1949), 649–652 (Russian). MR 0031534
- Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos Santalucía, Jan Pelant, and Václav Zizler, Functional analysis and infinite-dimensional geometry, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 8, Springer-Verlag, New York, 2001. MR 1831176, DOI 10.1007/978-1-4757-3480-5
- W. Holsztyński, Continuous mappings induced by isometries of spaces of continuous function, Studia Math. 26 (1966), 133–136. MR 193491, DOI 10.4064/sm-26-2-133-136
- Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR 1940513
- Sabine Koppelberg, Handbook of Boolean algebras. Vol. 1, North-Holland Publishing Co., Amsterdam, 1989. Edited by J. Donald Monk and Robert Bonnet. MR 991565
- P. Koszmider, Universal objects and associations between classes of Banach spaces and classes of compact spaces. pp. 93–115. In Selected Topics in Combinatorial Analysis, eds., M. Kurilic, S. Todorcevic, Sbornik Radova, 17(25). Matematicki Institut SANU, Beograd, 2015.
- I. I. Parovičenko, On a universal bicompactum of weight $\aleph$, Dokl. Akad. Nauk SSSR 150 (1963), 36–39. MR 0150732
- Saharon Shelah and Alex Usvyatsov, Banach spaces and groups—order properties and universal models, Israel J. Math. 152 (2006), 245–270. MR 2214463, DOI 10.1007/BF02771986
- Stevo Todorčević, Gaps in analytic quotients, Fund. Math. 156 (1998), no. 1, 85–97. MR 1610488
Additional Information
- Christina Brech
- Affiliation: Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, 05314-970, São Paulo, Brazil
- MR Author ID: 792312
- Email: brech@ime.usp.br
- Piotr Koszmider
- Affiliation: Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warszawa, Poland
- MR Author ID: 271047
- Email: piotr.koszmider@impan.pl
- Received by editor(s): May 18, 2015
- Received by editor(s) in revised form: May 21, 2015
- Published electronically: September 9, 2015
- Additional Notes: The first author was partially supported by FAPESP grant (2012/24463-7) and by CNPq grant (307942/2012-0).
The research of the second author was partially supported by grant PVE Ciência sem Fronteiras - CNPq (406239/2013-4). - Communicated by: Mirna Džamonja
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 2029-2036
- MSC (2010): Primary 46B25, 03E35, 54D30
- DOI: https://doi.org/10.1090/proc/12862
- MathSciNet review: 3460164