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Note on almost isometric ideals and local retracts in Banach and metric spaces

Part of: Model theory Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  27 August 2024

Leandro Candido Open the ORCID record for Leandro Candido [Opens in a new window] ,
Marek Cúth  and
Ondřej Smetana
Leandro Candido*
Affiliation:
Universidade Federal de S ao Paulo - UNIFESP, Instituto de Ciência e Tecnologia, Departamento de Matematica, São José dos Campos, SP, Brasil, ([email protected])
Marek Cúth
Affiliation:
Charles University, Faculty of Mathematics and Physics, Department of Mathematical Analysis, Sokolovská 83, 186 75 Prague 8, Czech Republic ([email protected], [email protected])
Ondřej Smetana
Affiliation:
Charles University, Faculty of Mathematics and Physics, Department of Mathematical Analysis, Sokolovská 83, 186 75 Prague 8, Czech Republic ([email protected], [email protected])
*
*Corresponding author.
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Abstract

We exhibit a new approach to the proofs of the existence of a large family of almost isometric ideals in nonseparable Banach spaces and existence of a large family of almost isometric local retracts in metric spaces. Our approach also implies the existence of a large family of nontrivial projections on every dual of a nonseparable Banach space. We prove three possible formulations of our results are equivalent. Some applications are mentioned which witness the usefulness of our novel approach.

Keywords

Almost isometric idealsalmost isometric local retractsseparable reductionrich familymethod of suitable models

MSC classification

Primary: 46B07: Local theory of Banach spaces 46B26: Nonseparable Banach spaces
Secondary: 03C30: Other model constructions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 25
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Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Abrahamsen, T. A.. Linear extensions, almost isometries, and diameter two. Extracta Math. 30 (2015), 135–151.Google Scholar
Abrahamsen, T. A., Lima, V. and Nygaard, O.. Almost isometric ideals in Banach spaces. Glasg. Math. J. 56 (2014), 395–407.CrossRefGoogle Scholar
Albiac, F., Ansorena, J. L., Cúth, M. and Doucha, M.. Embeddability of $\ell _p$ and bases in Lipschitz free $p$-spaces for $0< p\le 1$. J. Funct. Anal. 278 (2020), 108354.CrossRefGoogle Scholar
Bandyopadhyay, P., Dutta, S. and Sensarma, A.. Almost isometric ideals and non-separable Gurariy spaces. J. Math. Anal. Appl. 462 (2018), 279–284.CrossRefGoogle Scholar
Bandyopadhyay, P. and Sensarma, A.. A note on commutative and noncommutative Gurariĭ spaces. Arch. Math. (Basel) 118 (2022), 65–72.CrossRefGoogle Scholar
Correa, C., Cúth, M. and Somaglia, J.. Characterization of (semi-)Eberlein compacta using retractional skeletons. Studia Math. 263 (2022), 159–198.CrossRefGoogle Scholar
Correa, C., Cúth, M. and Somaglia, J.. Characterizations of weakly $\mathcal{K}$-analytic and Vašák spaces using projectional skeletons and separable PRI. J. Math. Anal. Appl., 515 (2022), 126389.CrossRefGoogle Scholar
Cúth, M.. Separable determination in Banach spaces. Fund. Math. 243 (2018), 9–27.CrossRefGoogle Scholar
Cúth, M., Doucha, M. and Wojtaszczyk, P. a.. On the structure of Lipschitz-free spaces. Proc. Amer. Math. Soc. 144 (2016), 3833–3846.CrossRefGoogle Scholar
Cúth, M. and Kalenda, O. F. K.. Rich families and elementary submodels. Cent. Eur. J. Math. 12 (2014), 1026–1039.Google Scholar
Garbulińska, J. and Kubiś, W.. Remarks on Gurarii spaces. Extra. Mathe. 26 (2011), 235–269.Google Scholar
García-Lirola, L., Procházka, A. and Rueda Zoca, A.. A characterisation of the Daugavet property in spaces of Lipschitz functions. J. Math. Anal. Appl. 464 (2018), 473–492.CrossRefGoogle Scholar
Godefroy, G. and Kalton, N. J., Lipschitz-free Banach spaces, Vol. 159 (2003), pp. 121–141. Dedicated to Professor Aleksander Pełczyński on the occasion of his 70th birthday.CrossRefGoogle Scholar
Godefroy, G., Kalton, N. J. and Saphar, P. D.. Unconditional ideals in Banach spaces. Studia Math. 104 (1993), 13–59.CrossRefGoogle Scholar
Gowers, W. T. and Maurey, B.. The unconditional basic sequence problem. J. Amer. Math. Soc. 6 (1993), 851–874.CrossRefGoogle Scholar
Hájek, P. and Novotný, M.. Some remarks on the structure of Lipschitz-free spaces. Bull. Belg. Math. Soc. Simon Stevin 24 (2017), 283–304.CrossRefGoogle Scholar
Heinrich, S. and Mankiewicz, P.. Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces. Studia Math. 73 (1982), 225–251.CrossRefGoogle Scholar
Kadets, V., Shvidkoy, R., Sirotkin, G. and Werner, D.. Banach spaces with the Daugavet property. Trans. Am. Math. Soc. 352 (2000), 855–873.CrossRefGoogle Scholar
Kalenda, O. F. K.. Projectional skeletons and Markushevich bases. Proc. Lond. Math. Soc. (3) 120 (2020), 514–586.CrossRefGoogle Scholar
Kalton, N. J.. Lipschitz and uniform embeddings into $\ell _\infty$. Fund. Math. 212 (2011), 53–69.CrossRefGoogle Scholar
Ka̧kol, J., Kubiś, W. a. and López-Pellicer, M., Descriptive topology in selected topics of functional analysis, Vol. 24 of Developments in Mathematics (Springer, New York, 2011).CrossRefGoogle Scholar
Koszmider, P., Shelah, S. and Świȩtek, M.. There is no bound on sizes of indecomposable Banach spaces. Adv. Math. 323 (2018), 745–783.CrossRefGoogle Scholar
Kubiś, W. a.. Banach spaces with projectional skeletons. J. Math. Anal. Appl. 350 (2009), 758–776.CrossRefGoogle Scholar
Kunen, K., Set theory, Vol. 102 of Studies in Logic and the Foundations of Mathematics (North-Holland Publishing Co., Amsterdam, 1983). An introduction to independence proofs, Reprint of the 1980 original.Google Scholar
López-Pérez, G. and Rueda Zoca, A.. $L$-orthogonality, octahedrality and Daugavet property in Banach spaces. Adv. Math. 383 (2021), 107719.CrossRefGoogle Scholar
Procházka, A. and Rueda Zoca, A.. A characterisation of octahedrality in Lipschitz-free spaces. Ann. Inst. Fourier (Grenoble) 68 (2018), 569–588.CrossRefGoogle Scholar
Quilis, A. and Zoca, A. R., (Almost isometric) local retracts in metric spaces arXiv:2311.13289 (2023).CrossRefGoogle Scholar
Rueda Zoca, A.. $L$-orthogonality in Daugavet centers and narrow operators. J. Math. Anal. Appl. 505 (2022), 125447.CrossRefGoogle Scholar
Sims, B. and Yost, D.. Linear Hahn-Banach extension operators. Proc. Edinburgh Math. Soc. (2) 32 (1989), 53–57.CrossRefGoogle Scholar
Smetana, O., Ideals of Banach spaces, Master Thesis, (2023).Google Scholar
Toruńczyk, H.. Characterizing Hilbert space topology. Fund. Math. 111 (1981), 247–262.CrossRefGoogle Scholar