A forgotten theorem of Pełczyński: $(\lambda +)$-injective spaces need not be $\lambda $-injective—the case $\lambda \in (1,2]$

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 Pe lczy´nski 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 Pe lczy´nski’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.