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 $(2+\epsilon )$-injective for every $\epsilon >0$, yet is is \emph{not} $2$-injective and remarked in a footnote that Pełczyński 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 Pełczyński's result has been preserved. In the present paper, we establish the said theorem for $\lambda \in (1,2]$ by constructing an appropriate renorming of ${\ell }_{\infty }$. 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.

7 pp; to appear in Studia Mathematica