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

• Published in: Indian journal of pure and applied mathematics Vol. 56; no. 2; pp. 728 - 736
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer Nature B.V 01.06.2025
• Subjects: Function space; Mathematics
• 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)≥0forallk∈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)=limn→∞F(2nf)|S2n for each f∈C+(K) satisfying that ‖F(f)|S-V(f)‖≤32ε,forallf∈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))|≤32ε,t∈T,f∈C+(K).