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$(λ^+)$-injective Banach spaces

Tomasz Kania, Grzegorz Lewicki

Abstract

In a companion paper (Studia Math., 2023), we proved for every $λ\in(1,2]$ the existence of a $(λ^+)$-injective renorming of $\ell_\infty$ that is not $λ$-injective, thereby establishing a~forgotten theorem of Pełczyński in that range. The complementary range $λ\in(2,\infty)$ was left open. In the present paper, we resolve this remaining case: for every $λ>2$ we construct a Banach space that is $(λ^+)$-injective but not $λ$-injective, completing Pełczyński's theorem for all $λ>1$. The construction uses a single device: the `zero-sum' subspace $Σ_N(Y)\subset Z_\infty^N$, which multiplies the relative projection constant by $μ_N=2-2/N$ 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.

$(λ^+)$-injective Banach spaces

Abstract

In a companion paper (Studia Math., 2023), we proved for every the existence of a -injective renorming of that is not -injective, thereby establishing a~forgotten theorem of Pełczyński in that range. The complementary range was left open. In the present paper, we resolve this remaining case: for every we construct a Banach space that is -injective but not -injective, completing Pełczyński's theorem for all . The construction uses a single device: the `zero-sum' subspace , which multiplies the relative projection constant by while preserving non-attainment. Iterating this operation reduces the problem to the range already covered by the companion paper. Since the ambient spaces arising in the iteration are finite -sums of , the resulting examples may be realised as subspaces of~. We also prove that if two Banach spaces are each isometrically isomorphic to their own square and each is isometric to a -complemented subspace of the other, then their Banach--Mazur distance is at most . Consequently, we obtain the estimate , thereby improving a recent result of Korpalski and Plebanek.
Paper Structure (5 sections, 9 theorems, 26 equations, 1 figure)

This paper contains 5 sections, 9 theorems, 26 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries: injectivity and projection constants
  3. The zero-sum construction
  4. Proof of Theorem A
  5. An optimised quantitative decomposition: proof of Theorem B

Key Result

Corollary 1

$d_{\mathrm{BM}}(L_\infty[0,1],\ell_\infty)\leqslant 9+6\sqrt{3}\approx 19.39$.

Figures (1)

  • Figure 1: The extension property defining $\lambda$-injectivity: every bounded operator $T\colon E\to X$ defined on a subspace $E\subset Z$ extends across the inclusion $\iota$ with norm loss at most the factor $\lambda$.

Theorems & Definitions (20)

  • Corollary
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 10 more