On phase-isometries between the unit spheres of the Banach space of continuous real-valued functions

TL;DR

This paper characterizes surjective phase-isometries between the unit spheres of spaces of continuous functions vanishing at infinity, showing they are essentially weighted composition operators involving homeomorphisms and sign functions.

Contribution

It proves that all such phase-isometries are equivalent to weighted composition operators with specific sign and homeomorphism components, extending understanding of isometric structures in function spaces.

Findings

Surjective phase-isometries are weighted composition operators.

Existence of a sign function and a homeomorphism linking the spaces.

Characterization applies to spaces of continuous functions vanishing at infinity.

Abstract

For a locally compact Hausdorff space $L$, we denote by $C_0(L,\mathbb{R})$ the Banach space of all continuous real-valued functions on $L$ vanishing at infinity, endowed with the supremum norm. In this paper, we prove that every surjective phase-isometry $T\colon S(C_0(X,\mathbb{R}))\to S(C_0(Y,\mathbb{R}))$ between the unit spheres of $C_0(X,\mathbb{R})$ and $C_0(Y,\mathbb{R})$ is a variant of a weighted composition operator in the following sense: there exist a function $\varepsilon\colon S(C_0(X,\mathbb{R}))\to\{-1,1\}$,a continuous function $\alpha\colon Y\to \{-1,1\}$ and a homeomorphism $\sigma\colon Y\to X$ such that $T(f)(q)=\varepsilon(f)\alpha(q)f(\sigma(q))$ for every $f\in S(C_0(X,\mathbb{R}))$ and $q\in Y$.