On a variant of Tingley's problem for some function spaces

Abstract

Let (Ω,A,μ) and (Γ,B,ν) be two arbitrary measure spaces, and p∈[1,∞]. SetS(Lp(μ))+:={f∈Lp(μ):‖f‖p=1;f≥0μ-a.e.}, that is, the positive part of the unit sphere of Lp(μ). We show that every surjective isometry Φ:S(Lp(μ))+→S(Lp(ν))+ can be extended (necessarily uniquely) to an isometric order isomorphism from Lp(μ) onto Lp(ν). A Lamperti form, i.e., a weighted composition like form, of Φ is provided, when (Γ,B,ν) is localizable (in particular, when it is σ-finite). On the other hand, we show that for compact Hausdorff spaces X and Y, if Φ is a surjective isometry from the positive part of the unit sphere of C(X) to that of C(Y), then there is a homeomorphism τ:Y→X satisfying Φ(f)(y)=f(τ(y)) for f∈S(C(X))+ and y∈Y.