@misc{10481/91231,
year = {2023},
month = {10},
url = {https://hdl.handle.net/10481/91231},
abstract = {We prove some results on weakly almost square Banach spaces and their relatives. On the one hand, we discuss weak almost squareness in the setting of Banach function spaces. More precisely, let (Ω, Σ) be a measurable space, let E be a Banach lattice and let ν : Σ → E+ be a non-atomic countably additive measure having relatively norm compact range. Then the space L1(ν) is weakly almost square. This result applies to some abstract Cesàro function spaces. Similar arguments show that the Lebesgue–Bochner space L1(µ, Y) is weakly almost square for any Banach space Y and for any non-atomic finite measure µ. On the other hand, we make some progress on the open question of whether there exists a locally almost square Banach space, which fails the diameter two property. In this line, we prove that if X is any Banach space containing a complemented isomorphic copy of c0, then for every 0 < ε < 1, there exists an equivalent norm |⋅| on X satisfying the following: (i) every slice of the unit ball B(X,|⋅|) has diameter 2; (ii) B(X,|⋅|) contains non-empty relatively weakly open subsets of arbitrarily small diameter and (iii) (X, |⋅|) is (r, s)-SQ for all 0 < r, s < (1−ε/1+ε).},
organization = {Grants PID2021-122126NB-C32 and PID2021-122126NB-C31 funded by MCIN/AEI/10.13039/501100011033 and "ERDF: A way of making Europe"},
organization = {Grant 21955/PI/22 funded by Fundación Séneca - ACyT Región de Murcia},
organization = {Grants FQM-0185 and PY20 00255
funded by Junta de Andalucía},
organization = {Generalitat Valenciana project CIGE/2022/97},
publisher = {Cambridge University Press},
keywords = {Almost squareness},
keywords = {Slice},
keywords = {Weakly open set},
title = {On weakly almost square banach spaces},
doi = {10.1017/S0013091523000536},
author = {Rodríguez Ruiz, José and Rueda Zoca, Abraham},
}