On ε-phase-isometries between the positive cones of continuous function spaces - Wang, Wenting; An, Aimin (Indian journal of pure and applied mathematics, 2025)
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).