$(\lambda^+)$-injective Banach spaces

Abstract

In a companion paper (Studia Math., 2023), we proved for every $\lambda\in(1,2]$ the existence of a $(\lambda^+)$-injective renorming of $\ell_\infty$ that is not $\lambda$-injective, thereby establishing a~forgotten theorem of Pe{\l}czy\'nski in that range. The complementary range $\lambda\in(2,\infty)$ was left open. In the present paper, we resolve this remaining case: for every $\lambda>2$ we construct a Banach space that is $(\lambda^+)$-injective but not $\lambda$-injective, completing Pe{\l}czy\'nski's theorem for all $\lambda>1$. The construction uses a single device: the `zero-sum' subspace $\Sigma_N(Y)\subset Z_\infty^N$, which multiplies the relative projection constant by $\mu_N=2-2/N$ 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 $\operatorname{dist}(L_\infty[0,1],\ell_\infty)\le 9+6\sqrt{3}$, thereby improving a recent result of Korpalski and Plebanek.