A forgotten theorem of Pełczyński: (λ+)-injective spaces need not be λ-injective -- the case λ∈ (1,2]

2022/01/19 by Tomasz Kania, Kania, Tomasz, Grzegorz Lewicki +1
Mathematics · #46B25 Secondary 46E15 #54G05 #Advanced Banach Space Theory #Advanced Topics in Algebra #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Primary 46B04

paper · pdf · doi:10.48550/arxiv.2201.07837

openalex publication_date 2022/01/19 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28

Abstract

Isbell and Semadeni [Trans. Amer. Math. Soc. 107 (1963)] proved that every infinite-dimensional 1-injective Banach space contains a hyperplane that is (2+ε)-injective for every ε > 0, yet is is not 2-injective and remarked in a footnote that Pełczyński had proved for every λ> 1 the existence of a (λ+ ε)-injective space (ε > 0) that is not λ-injective. Unfortunately, no trace of the proof of Pełczyński's result has been preserved. In the present paper, we establish the said theorem for λ∈ (1,2] by constructing an appropriate renorming of ℓ_∞. This contrasts (at least for real scalars) with the case λ= 1 for which Lindenstrauss [Mem. Amer. Math. Soc. 48 (1964)] proved the contrary statement.

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