A forgotten theorem of Pe{\l}czy\'n\'ski: $(\lambda+)$-injective spaces need not be $\lambda$-injective -- the case $\lambda\in (1,2]$

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 lczyński had proved for every λ > 1 the existence of a (λ + ε)-injective (ε > 0) that is not λ-injective. Unfortunately, no trace of the proof of Pe lczyński’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.