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

0565258 - MÚ 2024 RIV PL eng J - Journal Article
Kania, Tomasz - Lewicki, G.
A forgotten theorem of Pełczyński: (λ+)-injective spaces need not be λ-injective—the case λ∈(1,2].
Studia mathematica. Roč. 268, č. 3 (2023), s. 311-317. ISSN 0039-3223. E-ISSN 1730-6337
Institutional support: RVO:67985840
Keywords : injective Banach space * minimal projection
OECD category: Pure mathematics
Impact factor: 0.8, year: 2022
Method of publishing: Limited access
https://doi.org/10.4064/sm220119-25-6

Isbell and Semadeni [Trans. Amer. Math. Soc. 107 (1963)] proved that every infinite-dimensional 1-injective Banach space contains a hyperplane that is (2+ epsilon)-injective for every epsilon > 0, yet is not 2-injective, and remarked in a footnote that Pelczynski had proved for every lambda > 1 the existence of a (lambda + epsilon)-injective space (epsilon > 0) that is not lambda-injective. Unfortunately, no trace of the proof of Pelczynski's result has been preserved. In the present paper, we establish that result for lambda is an element of (1, 2] by constructing an appropriate renorming of l(infinity). This contrasts (at least for real scalars) with the case lambda = 1 for which Lindenstrauss [Mem. Amer. Math. Soc. 48 (1964)] proved the contrary statement.
Permanent Link: https://hdl.handle.net/11104/0336827