On $$\varepsilon $$ ε -phase-isometries between the positive cones of continuous function spaces

Wenting Wang () and Aimin An ()
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Wenting Wang: Lanzhou University of Technology
Aimin An: Lanzhou University of Technology

Indian Journal of Pure and Applied Mathematics, 2025, vol. 56, issue 2, 728-736

Abstract: Abstract Let K and T be compact Hausdorff spaces, $$C_+(K)=\{f\in C(K): f(k)\ge 0\; \mathrm{for\; all\;}\; k\in K\}$$ 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 $$\varepsilon $$ ε -phase-isometry $$F:C_+(K)\rightarrow C_+(T)$$ F : C + ( K ) → C + ( T ) , there are nonempty closed subset $$S\subset T$$ S ⊂ T and an additive isometry $$V:C_+(K)\rightarrow C_+(S)$$ V : C + ( K ) → C + ( S ) defined as $$V(f)=\lim _{n\rightarrow \infty }\frac{F(2^nf)|_S}{2^n}$$ V ( f ) = lim n → ∞ F ( 2 n f ) | S 2 n for each $$f\in C_+(K)$$ f ∈ C + ( K ) satisfying that $$\begin{aligned} \Vert F(f)|_S-V(f)\Vert \le \frac{3}{2}\varepsilon ,\;\mathrm{for\;all}\; f\in C_+(K). \end{aligned}$$ ‖ 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 $$\gamma :T\rightarrow K$$ γ : T → K such that $$\begin{aligned} | F(f)(t)-f(\gamma (t))|\le \frac{3}{2}\varepsilon ,\;\;t\in T,\;f\in C_+(K). \end{aligned}$$ | F ( f ) ( t ) - f ( γ ( t ) ) | ≤ 3 2 ε , t ∈ T , f ∈ C + ( K ) .

Keywords: $$\varepsilon $$ ε -phase-isometries; Linear isometries; Banach-Stone theorem; Continuous function space; Primary 46B04; 46B20 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s13226-023-00514-y

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