$(\hat{I}{»}^{+})$-injective Banach spaces

Abstract

In a companion paper (Studia Math., 2023), we proved for every $\hat{I}»\in (1,2]$ the existence of a $(\hat{I}{»}^{+})$-injective renorming of ${\ell }_{\infty }$ that is not $\hat{I}»$-injective, thereby establishing a~forgotten theorem of Pełczyński in that range. The complementary range $\hat{I}»\in (2,\infty )$ was left open. In the present paper, we resolve this remaining case: for every $\hat{I}»>2$ we construct a Banach space that is $(\hat{I}{»}^{+})$-injective but not $\hat{I}»$-injective, completing Pełczyński's theorem for all $\hat{I}»>1$. The construction uses a single device: the `zero-sum' subspace $\hat{I}{\pounds }_N(Y)\subset Z_{\infty }^N$, which multiplies the relative projection constant by $\hat{I}{¼}_N=2-2/N$ while preserving non-attainment. Iterating this operation reduces the problem to the range $(1,2]$ already covered by the companion paper. Since the ambient spaces arising in the iteration are finite ${\ell }_{\infty }$-sums of ${\ell }_{\infty }$, the resulting examples may be realised as subspaces of~${\ell }_{\infty }$. We also prove that if two Banach spaces are each isometrically isomorphic to their own square and each is isometric to a $1$-complemented subspace of the other, then their Banach--Mazur distance is at most $9+6\sqrt{3}$. Consequently, we obtain the estimate $\mathrm{dist}(L_{\infty }[0,1],{\ell }_{\infty })\le 9+6\sqrt{3}$, thereby improving a recent result of Korpalski and Plebanek.

7 pp