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

Yuta Enami, Izuho Matsuzaki

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 $α\colon Y\to \{-1,1\}$ and a homeomorphism $σ\colon Y\to X$ such that $T(f)(q)=\varepsilon(f)α(q)f(σ(q))$ for every $f\in S(C_0(X,\mathbb{R}))$ and $q\in Y$.

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

Abstract

For a locally compact Hausdorff space , we denote by the Banach space of all continuous real-valued functions on vanishing at infinity, endowed with the supremum norm. In this paper, we prove that every surjective phase-isometry between the unit spheres of and is a variant of a weighted composition operator in the following sense: there exist a function ,a continuous function and a homeomorphism such that for every and .
Paper Structure (5 sections, 17 theorems, 68 equations)

This paper contains 5 sections, 17 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rearrangement of $T$
  4. The proof of the main theorem
  5. Remarks

Key Result

Theorem 1.1

Assume that $X$ and $Y$ are locally compact Hausdorff spaces and that $T\colon S(C_0(X, \mathbb R))\to S(C_0(Y, \mathbb R))$ is a surjective phase-isometry. Then there exist a function $\varepsilon\colon S(C_0(X, \mathbb R))\to\left\{ -1,1 \right\}$, a continuous function $\alpha\colon Y\to\left\{ for all $f\in S(C_0(X, \mathbb R))$ and $q\in Y$.

Theorems & Definitions (36)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1: Tan and Gao tangao
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4: Tan, Zhang and Huang TZH
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • ...and 26 more