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Surjective linear isometries of Alexiewicz-normed $L^\infty$ spaces

Nuno J. Alves

TL;DR

The paper classifies surjective linear isometries between Alexiewicz-normed spaces $(L^{\infty}(K),\|\cdot\|_A)$ on compact sets of positive measure by a Banach–Stone type representation: each isometry is given by a sign and an increasing lipeomorphism of measure intervals, yielding a weighted composition formula with $\phi=\sigma_K\circ\psi\circ\pi_M$ and $J_M(Tf)=\varepsilon\,(J_K f)\circ \phi$ while $(Tf)(y)=\varepsilon\,(f\circ\phi)(y)\,(\psi'\circ\pi_M)(y)$ a.e. The paper embeds $L^{\infty}(K)$ into $\mathrm{Lip}_0([0,|K|])$ via the measure projection $\pi_K$ and selector $\sigma_K$, identifying its completion with $C_0([0,|K|])$ and enabling a Banach–Stone analysis on $C_0$ spaces. It then introduces fiber- and gap-compatibility conditions that precisely determine when the induced variable change lifts to a homeomorphism or a lipeomorphism of the underlying sets, respectively, and provides a constructive parametrization of isometries in terms of $(\varepsilon,\psi)$. Finally, the results are packaged in a categorical framework, showing an equivalence between a category of measure-interval data and the category of Alexiewicz-normed $L^{\infty}$ spaces, with subcategories encoding the geometric compatibility needed for stronger regularity of the change of variables.

Abstract

We characterize the surjective linear isometries between spaces of essentially bounded functions on compact subsets of the real line of positive Lebesgue measure, equipped with the Alexiewicz norm. We prove a Banach--Stone type representation: every such isometry is induced by a sign and an increasing lipeomorphism between the associated measure intervals, and admits an explicit weighted composition formula. In particular, any two such Alexiewicz-normed spaces are linearly isometric. We further introduce geometric compatibility conditions on the compact sets (fiber and gap compatibility) which characterize when the induced map between the underlying sets is a homeomorphism or a lipeomorphism.

Surjective linear isometries of Alexiewicz-normed $L^\infty$ spaces

TL;DR

The paper classifies surjective linear isometries between Alexiewicz-normed spaces on compact sets of positive measure by a Banach–Stone type representation: each isometry is given by a sign and an increasing lipeomorphism of measure intervals, yielding a weighted composition formula with and while a.e. The paper embeds into via the measure projection and selector , identifying its completion with and enabling a Banach–Stone analysis on spaces. It then introduces fiber- and gap-compatibility conditions that precisely determine when the induced variable change lifts to a homeomorphism or a lipeomorphism of the underlying sets, respectively, and provides a constructive parametrization of isometries in terms of . Finally, the results are packaged in a categorical framework, showing an equivalence between a category of measure-interval data and the category of Alexiewicz-normed spaces, with subcategories encoding the geometric compatibility needed for stronger regularity of the change of variables.

Abstract

We characterize the surjective linear isometries between spaces of essentially bounded functions on compact subsets of the real line of positive Lebesgue measure, equipped with the Alexiewicz norm. We prove a Banach--Stone type representation: every such isometry is induced by a sign and an increasing lipeomorphism between the associated measure intervals, and admits an explicit weighted composition formula. In particular, any two such Alexiewicz-normed spaces are linearly isometric. We further introduce geometric compatibility conditions on the compact sets (fiber and gap compatibility) which characterize when the induced map between the underlying sets is a homeomorphism or a lipeomorphism.
Paper Structure (8 sections, 19 theorems, 169 equations)

This paper contains 8 sections, 19 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. Preliminary material
  3. Measure projections and selector maps
  4. Alexiewicz isometric isomorphism
  5. Proof of Theorem \ref{['thm:ABS']}
  6. Fiber "03C8-compatibility and homeomorphic "03D5
  7. Gap "03C8-compatibility and lipeomorphic "03D5
  8. A categorical formulation

Key Result

Theorem 1.1

Let $M,K \subseteq \mathbb{R}$ be compact sets of positive measure. A linear map is a surjective isometry if and only if there exist $\varepsilon\in\{\pm1\}$ and an increasing lipeomorphism such that the map $\phi: M \to K$ given by satisfies, for every $f\in L^{\infty}(K)$, Equivalently, for a.e. $y\in M$,

Theorems & Definitions (43)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 3.1
  • ...and 33 more