Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On complementability of $c_0$ in spaces $C(K\times L)$
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by Jerzy Ka̧kol, Damian Sobota and Lyubomyr Zdomskyy;
Proc. Amer. Math. Soc. 152 (2024), 3777-3784
DOI: https://doi.org/10.1090/proc/16262
Published electronically: July 26, 2024

Abstract:

Using elementary probabilistic methods, in particular a variant of the Weak Law of Large Numbers related to the Bernoulli distribution, we prove that for every infinite compact spaces $K$ and $L$ the product $K\times L$ admits a sequence $\langle \mu _n\colon n\in \mathbb {N}\rangle$ of normalized signed measures with finite supports which converges to $0$ with respect to the weak* topology of the dual Banach space $C(K\times L)^*$. Our approach is completely constructive—the measures $\mu _n$ are defined by an explicit simple formula. We also show that this result generalizes the classical theorem of Cembranos [Proc. Amer. Math. Soc. 91 (1984), pp. 556–558] and Freniche [Math. Ann. 267 (1984), pp. 479–486] which states that for every infinite compact spaces $K$ and $L$ the Banach space $C(K\times L)$ contains a complemented copy of the space $c_0$.
References
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Bibliographic Information
  • Jerzy Ka̧kol
  • Affiliation: Faculty of Mathematics and Computer Science, A. Mickiewicz University, Poznań, Poland; \normalfont and Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic
  • MR Author ID: 96980
  • ORCID: 0000-0002-8311-2117
  • Email: kakol@amu.edu.pl
  • Damian Sobota
  • Affiliation: Kurt Gödel Research Center for Mathematical Logic, Department of Mathematics, Faculty of Mathematics, University of Vienna, Vienna, Austria
  • MR Author ID: 1044413
  • Email: ein.damian.sobota@gmail.com
  • Lyubomyr Zdomskyy
  • Affiliation: Kurt Gödel Research Center for Mathematical Logic, Department of Mathematics, Faculty of Mathematics, University of Vienna, Vienna, Austria
  • MR Author ID: 742789
  • Email: lzdomsky@gmail.com
  • Received by editor(s): June 8, 2022
  • Received by editor(s) in revised form: July 25, 2022
  • Published electronically: July 26, 2024
  • Additional Notes: The research of the first author was supported by the GAČR project 20-22230L and RVO: 67985840. The second and third authors were supported by the Austrian Science Fund FWF, Grants I 2374-N35, I 3709-N35, M 2500-N35
  • Communicated by: Stephen Dilworth
  • © Copyright 2024 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 152 (2024), 3777-3784
  • MSC (2020): Primary 46E15, 28A33, 46B09; Secondary 28C05, 28C15, 46E27
  • DOI: https://doi.org/10.1090/proc/16262
  • MathSciNet review: 4781973