Abstract
A Banach space is locally almost square if, for every y in its unit sphere, there exists a sequence \((x_n)\) in its unit sphere such that \(\lim \Vert y\pm x_n\Vert =1\). A Banach space is weakly almost square if, in addition, we require the sequence \((x_n)\) to be weakly null. It is known that these two properties are distinct, so we aim to investigate if local almost squareness implies a weaker version of the latter property by replacing the sequence with a net. In order to achieve this result, we restrict ourselves to Banach lattices and introduce a strengthening of local almost squareness by requiring that the sequence is in the positive cone of the lattice. As an application of such characterization, we prove that this positive variant of local almost squareness implies that every relatively weakly open set in the unit ball has diameter two, that is, the Banach space has the so called diameter two property. This in particular allows us also to generate new examples of Banach spaces enjoying the diameter two property.
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References
Abrahamsen, T.A., Langemets, J., Lima, V.: Almost square Banach spaces. J. Math. Anal. Appl. 434, 1549–1565 (2016)
Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis. A Hitchhiker’s Guide, 3rd edn. Springer, Berlin (2006)
Fremlin, D.H.: Tensor products of Banach lattices. Math. Ann. 211, 87–106 (1974)
Hardtke, J.-D.: Summands in locally almost square and locally octahedral spaces. Acta Comment. Univ. Tartu. Math. 22, 149–162 (2018)
Hardtke, J.-D.: Locally octahedral and locally almost square Köthe–Bochner spaces (2021). ar**v:2107.01180v1
Johnson, W.B., Lindenstrauss, J.: Basic concepts in the geometry of Banach spaces. In: Handbook of the Geometry of Banach Spaces, vol. 1, pp. 1–84. Elsevier, Amsterdam (2001)
Kaasik, J.K., Veeorg, T.: Weakly almost square Lipschitz-free spaces (2022). ar**v:2210.09158v1
Kubiak, D.: Some geometric properties of the Cesáro function spaces. J. Convex Anal. 21, 189–200 (2014)
Langemets, J., Lima, V., Rueda Zoca, A.: Almost square and octahedral norms in tensor products of Banach spaces. Rev. Real Acad. Ciencias Exactas Físicas Nat. Ser. A Mat. RACSAM 111, 841–853 (2017)
Acknowledgements
I would like to thank the anonymous referees for carefully reading this paper and for improving its exposition. Moreover, I’m most grateful to Johann Langemets and Aleksei Lissitsin for their help and support. The author certifies not to have affiliations with or involvement in any organization or entity with any financial interest, or non-financial interest in the subject matter or materials discussed in this manuscript.
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Ciaci, S. Locally almost square Banach lattices. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, 97 (2023). https://doi.org/10.1007/s13398-023-01434-6
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DOI: https://doi.org/10.1007/s13398-023-01434-6