A forgotten theorem of Pełczyński: -injective spaces need not be \lambda -injective—the case \lambda \in (1,2]
Volume 268 / 2023
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+\varepsilon )-injective for every \varepsilon \gt 0, yet is not 2-injective, and remarked in a footnote that Pełczyński had proved for every \lambda \gt 1 the existence of a (\lambda + \varepsilon )-injective space (\varepsilon \gt 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 that result 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.
