How many miles from $L_\infty$ to $\ell_\infty$?

TL;DR

This work investigates the Banach-Mazur distance between the classical spaces $L_ olinebreak_ olinebreakinfty[0,1]$ and $\\ell_ olinebreak_ olinebreakinfty$, establishing new lower and upper bounds for $d_{ m BM}$. It develops a general measure-theoretic framework: for a norm-increasing isomorphism $T:C(K)\to C(L)$, the associated measures $\\nu_y=T^*\\delta_y$ yield 1-norming families whose total variation provides distortion control, formalized in a key lemma. Using this, the authors prove a general lower bound $d_{ m BM}(C(K),C(L))\ge 3+2\sqrt{2}$ when $K$ is zero-dimensional without isolated points and $L$ is a compactification of $\\omega$, and they obtain a concrete lower bound $d_{ m BM}(L_ olinebreak_ olinebreakinfty[0,1],\\ell_ olinebreakinfty) > 7.41$ via a refined construction (including a new averaging argument). They further derive an explicit upper bound $d_{ m BM}(L_ olinebreak_ olinebreakinfty[0,1],\\ell_ olinebreakinfty) \le (3+\sqrt{2})^2$ by composing Pełczyński-type decompositions and exploiting $1$-injectivity, thereby sandwiching the distance but leaving a sizable gap to close. The results illuminate how structural properties of $C(K)$ spaces and decomposition techniques constrain isomorphisms between $L_ olinebreak_ olinebreakinfty[0,1]$ and $\\ell_ olinebreakinfty$, with implications for explicit distance estimates among common function spaces.

Abstract

The classical Banach spaces $L_\infty[0,1]$ and $\ell_\infty$ are isomorphic. We present here some lower and upper bounds for their Banach-Mazur distance.

Theorems & Definitions (20)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 3.1
• proof
• Lemma 4.1
• proof
• Lemma 4.2
• proof