PreprintCC BY 4.0arXiv
Grothendieck and ℓ∞-Grothendieck subspaces of ℓ∞
Abstract
Let c=2ℵ0. We prove that ℓ∞ contains 2c pairwise non-isomorphic non-reflexive Grothendieck subspaces. They may all be chosen to contain the canonical copy of c0 and to have infinite-dimensional reflexive quotient. This is optimal and answers by the authors in 2021. We also show that the relative ℓ∞-Grothendieck property depends on the specified copy of c0 and is not an isomorphic invariant of the subspace. More generally, for every closed E⊆ℓ∞ there is an ℓ∞-Grothendieck subspace isomorphic to ℓ∞⊕∞ E; taking E=c0 answers a problem by González et al.
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BibTeX
@misc{gonzalez2026grothendieck,
title = {{Grothendieck and $\ell_\infty$-Grothendieck subspaces of $\ell_\infty$}},
author = {Manuel González and Tomasz Kania},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20170},
eprint = {2609.20170},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20170},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Grothendieck and ℓ_∞-Grothendieck subspaces of ℓ_∞
AU - Manuel González
AU - Tomasz Kania
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.20170
UR - https://arxiv.org/abs/2609.20170
AB - Let c=2^ℵ_0. We prove that ℓ_∞ contains 2^c pairwise non-isomorphic non-reflexive Grothendieck subspaces. They may all be chosen to contain the canonical copy of c_0 and to have infinite-dimensional reflexive quotient. This is optimal and answers by the authors in 2021. We also show that the relative ℓ_∞-Grothendieck property depends on the specified copy of c_0 and is not an isomorphic invariant of the subspace. More generally, for every closed E⊆ℓ_∞ there is an ℓ_∞-Grothendieck subspace isomorphic to ℓ_∞⊕_∞ E; taking E=c_0 answers a problem by González et al.
ER -