Abstract
In this paper, we establish the stability of uaw-convergence under passing from sublattices. The various implications of this fact are presented through the paper. In particular, we show that if \((x_{\alpha })\) is an increasing net in a Banach lattice E and \(x_{\alpha }\overset{uaw}{\longrightarrow }0\) in E then \(x_{\alpha }\overset{un}{\longrightarrow }0\) in \(E^{''}\). Furthermore, we deduce some results concerning uaw-completeness. Additionally, we present a new characterizations of KB-spaces (resp. reflexive Banach lattices), using the concepts of uaw-convergence and un-convergence.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Reprint of the: Original, p. 2006. Springer, Berlin (1985)
Alpay, S., Altin, B., Tonyali, C.: On property (b) of vector lattices. Positivity 7, 135–139 (2003)
Brezis, H.: Functional Analysis. Universitext, Springer, Sobolev Spaces and Partial Differential Equations (2011)
Chen, Z.L., Wickstead, A.W.: Relative weak compactness of solid hulls in Banach lattices. Indag. Mathem. N.S. 9(2), 187–196 (1998)
Deng, Y., O’Brien, M., Troitsky, V.G.: Unbounded norm convergence in Banach lattices. Positivity 21(3), 963–974 (2017)
Fahri, K El., Khabaoui, H., H’mihane, J.: Some characterisation of L-weakly compat sets using the unbounded absolytely weakly convergene and applications. Positivity 26, 42 (2022). https://doi.org/10.1007/s11117-022-00912-2
Gao, N., Troitsky, V.G., Xanthos, F.: Uo-convergence and its applications to Cesàro means in Banach lattices. Isr. J. Math. 220, 649–689 (2017)
Kandic, M., Marabeh, M.A.A., Troitsky, V.G.: Unbounded norm topology in Banach lattices. J. Math. Anal. Appl. 451(1), 259–279 (2017)
Machrafi, N., El Fahri, K., Moussa, M.: A note on b-semicompact sets and operators. Rend. Circ. Mat. Palermo 65, 47–53 (2016). https://doi.org/10.1007/s12215-015-0217-7
Meyer-Nieberg, P.: Banach lattices. Universitext. Springer, Berlin (1991)
Nakano, H.: Ergodic theorems in semi-ordered linear spaces. Ann. Math. (2) 49, 538–556 (1948)
Zabeti, O.: Unbounded absolute weak convergence in Banach lattices. Positivity (2018). https://doi.org/10.1007/s11117-017-0524-7
Acknowledgements
The authors would like to thank the referee for his useful comments and suggestions to improve the quality of the paper.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Moktafi, H., Khabaoui, H. & El Fahri, K. Some results on unbounded absolute weak convergence. Acta Sci. Math. (Szeged) 90, 241–250 (2024). https://doi.org/10.1007/s44146-024-00111-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s44146-024-00111-3
Keywords
- Banach lattice
- Unbounded absolute weak convergence
- Unbounded norm convergence
- Order continuous norm
- KB-space