Grothendieck and ℓ∞‐Grothendieck Subspaces of ℓ∞

Q: What does this research mean for the field?

The space ℓ∞ contains pairwise non-isomorphic non-reflexive Grothendieck subspaces, and the relative ℓ∞-Grothendieck property depends on the specified copy of ℓ∞ rather than being an isomorphic invariant. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

Manuel González; Tomasz Kania

doi:10.1002/mana.70255

September 9, 2026Mathematische NachrichtenOpen Access

Grothendieck and ℓ∞‐Grothendieck Subspaces of ℓ∞

View Full Paper

Ask AI

Bookmark

Share

Key Points

To determine the existence, structure, and isomorphism invariance of Grothendieck and relative Grothendieck subspaces within the infinite-dimensional sequence space l-infinity.
Conducted structural functional analysis on the sequence space l-infinity over index sets of cardinality at least the continuum.
Constructed explicit non-reflexive subspaces containing the canonical copy of c0 with infinite-dimensional reflexive quotients.
Analyzed the embedding dependence of relative l-infinity-Grothendieck properties for direct-sum subspaces.
Proved that l-infinity contains an optimal family of 2^Gamma pairwise non-isomorphic, non-reflexive Grothendieck subspaces containing the canonical copy of c0, answering Problem 24 of González and Kania.
Demonstrated that the relative l-infinity-Grothendieck property is not an isomorphic invariant, showing that for every closed subspace Y in l-infinity there exists an l-infinity-Grothendieck subspace isomorphic to Y direct sum l-infinity.
Resolved Problem 1 of González et al. by confirming the existence of an l-infinity-Grothendieck subspace isomorphic to c0 direct sum l-infinity.

Ask about this paper

Answers are grounded in this paper and cite their sources.

Implication

Theoretical analysis reveals optimal families of non-isomorphic Grothendieck subspaces in sequence spaces, highlighting that the relative Grothendieck property is not an isomorphic invariant.

Abstract

Let . We prove that contains pairwise non‐isomorphic non‐reflexive Grothendieck subspaces. They may all be chosen to contain the canonical copy of and to have an infinite‐dimensional reflexive quotient. This is optimal and answers Problem 24 of González and Kania [ Japanese Journal of Mathematics 16 (2021), 247–313]. We also show that the relative ‐Grothendieck property depends on the specified copy of and is not an isomorphic invariant of the subspace. More generally, for every closed there is an ‐Grothendieck subspace isomorphic to ; taking answers Problem 1 of González et al. [ Journal of Mathematical Analysis and Applications 497 (2021), Article 124857].

Discussion

No takes yet. Share an insight, caveat, or question.

Cite This Study

González et al. (2026) studied this question.

synapsesocial.com/papers/6aa27bbd58559d80afc751d9 https://doi.org/10.1002/mana.70255

View Full Paper

Ask AI

Bookmark

Share

Grothendieck and ℓ∞‐Grothendieck Subspaces of ℓ∞

Key Points

Ask about this paper

Implication

Abstract

Discussion

Cite This Study

Grothendieck and ℓ∞‐Grothendieck Subspaces of ℓ∞

Key Points

Ask about this paper

Implication

Abstract

Discussion

Cite This Study

Keep reading

Keep reading