On complementability of $c_0$ in spaces $C(K\times L)$
HTML articles powered by AMS MathViewer
- 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
- HTML | PDF | Request permission
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
- T. Banakh, J. Ka̧kol, and W. Śliwa, Josefson-Nissenzweig property for $C_p$-spaces, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113 (2019), no. 4, 3015–3030. MR 3999000, DOI 10.1007/s13398-019-00667-8
- Béla Bollobás, Random graphs, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 73, Cambridge University Press, Cambridge, 2001. MR 1864966, DOI 10.1017/CBO9780511814068
- Leandro Candido, Complementations in $C(K,X)$ and $\ell _\infty (X)$, Colloq. Math. 172 (2023), no. 1, 129–141. MR 4565998, DOI 10.4064/cm8868-10-2022
- Pilar Cembranos, $C(K,\,E)$ contains a complemented copy of $c_{0}$, Proc. Amer. Math. Soc. 91 (1984), no. 4, 556–558. MR 746089, DOI 10.1090/S0002-9939-1984-0746089-2
- Francisco J. Freniche, Barrelledness of the space of vector valued and simple functions, Math. Ann. 267 (1984), no. 4, 479–486. MR 742894, DOI 10.1007/BF01455966
- Jerzy Ka̧kol and Aníbal Moltó, Witnessing the lack of the Grothendieck property in $C(K)$-spaces via convergent sequences, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114 (2020), no. 4, Paper No. 179, 7. MR 4132567, DOI 10.1007/s13398-020-00914-3
- Surjit Singh Khurana, Grothendieck spaces, Illinois J. Math. 22 (1978), no. 1, 79–80. MR 458144
- Joram Lindenstrauss, On complemented subspaces of $m$, Israel J. Math. 5 (1967), 153–156. MR 222616, DOI 10.1007/BF02771101
- Walter Schachermayer, On some classical measure-theoretic theorems for non-sigma-complete Boolean algebras, Dissertationes Math. (Rozprawy Mat.) 214 (1982), 33. MR 673286
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