Uniformly convex stalks and Kucher's ℓ∞(E) problem
Abstract
Kucher asked whether ℓ∞(E) is a Grothendieck space for every super-reflexive Banach space E. We answer this question affirmatively by proving the stronger result that ℓ∞(E) has Pełczyński's property (V). Combined with the natural dual representation of ℓ∞(E), this yields that every non-weakly compact operator on ℓ∞(E) fixes a copy of ℓ∞. The main ingredient is a general property-(V) theorem for function modules over zero-dimensional compact Hausdorff spaces with stalks admitting a common modulus of uniform convexity. The proof is based on Gierz's integral representation of functionals and the weak compactness of the associated canonical measures. In the case of ℓ∞(E), the relevant stalks are ultrapowers of E, and super-reflexivity provides the uniform geometric control required by the theorem. We also derive several consequences for operators on ℓ∞(E), including weak compactness into separable spaces, reflexivity of separable quotients, and compactness of operators into Schur spaces.
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BibTeX
@misc{li2026uniformly,
title = {{Uniformly convex stalks and Kucher's $\ell_\infty(E)$ problem}},
author = {Hui Li and Sijie Luo and Lian Wu and Dejian Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11180},
eprint = {2610.11180},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11180},
}
RIS
TY - UNPB
TI - Uniformly convex stalks and Kucher's ℓ_∞(E) problem
AU - Hui Li
AU - Sijie Luo
AU - Lian Wu
AU - Dejian Zhou
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11180
UR - https://arxiv.org/abs/2610.11180
AB - Kucher asked whether ℓ_∞(E) is a Grothendieck space for every super-reflexive Banach space E. We answer this question affirmatively by proving the stronger result that ℓ_∞(E) has Pełczyński's property (V). Combined with the natural dual representation of ℓ_∞(E), this yields that every non-weakly compact operator on ℓ_∞(E) fixes a copy of ℓ_∞. The main ingredient is a general property-(V) theorem for function modules over zero-dimensional compact Hausdorff spaces with stalks admitting a common modulus of uniform convexity. The proof is based on Gierz's integral representation of functionals and the weak compactness of the associated canonical measures. In the case of ℓ_∞(E), the relevant stalks are ultrapowers of E, and super-reflexivity provides the uniform geometric control required by the theorem. We also derive several consequences for operators on ℓ_∞(E), including weak compactness into separable spaces, reflexivity of separable quotients, and compactness of operators into Schur spaces.
ER -