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An isometrically universal Banach space induced by a non-universal Boolean algebra


Authors: Christina Brech and Piotr Koszmider
Journal: Proc. Amer. Math. Soc. 144 (2016), 2029-2036
MSC (2010): Primary 46B25, 03E35, 54D30
DOI: https://doi.org/10.1090/proc/12862
Published electronically: September 9, 2015
MathSciNet review: 3460164
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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).


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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
Email: brech@ime.usp.br

Piotr Koszmider
Affiliation: Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warszawa, Poland
Email: piotr.koszmider@impan.pl

DOI: https://doi.org/10.1090/proc/12862
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
Article copyright: © Copyright 2015 American Mathematical Society