On ε-phase-isometries between the positive cones of continuous function spaces On ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}-phase-isometries between the positive

• Published in: Indian journal of pure and applied mathematics Vol. 56; no. 2; pp. 728 - 736
• Main Authors: Wang, Wenting, An, Aimin
• Format: Journal Article
• Language: English
• Published: New Delhi Indian National Science Academy 01.06.2025
• Subjects: Applications of Mathematics; Mathematics; Mathematics and Statistics; Numerical Analysis; Original Research; Primary 46B04; phase-isometries; Continuous function space; 46B20; Banach-Stone theorem; Linear isometries
• ISSN: 0019-5588; 0975-7465
• DOI: 10.1007/s13226-023-00514-y

Let K and T be compact Hausdorff spaces, C + ( K ) = { f ∈ C ( K ) : f ( k ) ≥ 0 for all k ∈ K } be the positive cone of C ( K ). In this paper, we prove that if K is a compact Hausdorff perfectly normal space, then for every ε -phase-isometry F : C + ( K ) → C + ( T ) , there are nonempty closed subset S ⊂ T and an additive isometry V : C + ( K ) → C + ( S ) defined as V ( f ) = lim n → ∞ F ( 2 n f ) | S 2 n for each f ∈ C + ( K ) satisfying that ‖ F ( f ) | S - V ( f ) ‖ ≤ 3 2 ε , for all f ∈ C + ( K ) . Moreover, if F is almost surjective, then there exists a unique homeomorphism γ : T → K such that | F ( f ) ( t ) - f ( γ ( t ) ) | ≤ 3 2 ε , t ∈ T , f ∈ C + ( K ) .