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

Let $(Ω, \mathfrak{A}, μ)$ and $(Γ, \mathfrak{B}, ν)$ be two arbitrary measure spaces, and $p\in [1,\infty]$. Set $L^p(μ)_+^\mathrm{sp}:= \{f\in L^p(μ): \|f\|_p =1; f\geq 0\ μ\text{-a.e.} \}$ i.e., the positive part of the unit sphere of $L^p(μ)$. We show that every metric preserving bijection $Φ: L^p(μ)_+^\mathrm{sp} \to L^p(ν)_+^\mathrm{sp}$ 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 $(Γ, \mathfrak{B}, ν)$ 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 metric preserving bijection from the positive part of the unit sphere of $C(X)$ to that of $C(Y)$, then there is a homeomorphism $τ:Y\to X$ satisfying $Φ(f)(y) = f(τ(y))$ ($f\in C(X)_+^\mathrm{sp}; y\in Y$).

No reviews yet.

No comments yet.