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

Let (Ω , A , μ) and (Γ , B , ν) be two arbitrary measure spaces, and p ∈ [ 1 , ∞ ]. Set S (L p (μ)) + : = { f ∈ L p (μ) : ‖ f ‖ p = 1 ; f ≥ 0 μ -a.e. } , that is, the positive part of the unit sphere of L p (μ). We show that every surjective isometry Φ : S (L p (μ)) + → S (L p (ν)) + can be extended (necessarily uniquely) to an isometric order isomorphism from L p (μ) onto L p (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. [ABSTRACT FROM AUTHOR]