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

Abstract

Let $(\Omega ,\mathfrak{A},\mu )$ and $(\Gamma ,\mathfrak{B},\nu )$ be two arbitrary measure spaces, and $p\in [1,\infty ]$. Set $$L^p(\mu {)}_{+}^{\mathrm{sp}}:=\{f\in L^p(\mu ):\Vert f{\Vert }_p=1;f\ge 0\mu \text{-a.e.}\}$$ i.e., the positive part of the unit sphere of $L^p(\mu )$. We show that every metric preserving bijection $\Phi :L^p(\mu {)}_{+}^{\mathrm{sp}}\to L^p(\nu {)}_{+}^{\mathrm{sp}}$ can be extended (necessarily uniquely) to an isometric order isomorphism from $L^p(\mu )$ onto $L^p(\nu )$. A Lamperti form, i.e., a weighted composition like form, of $\Phi$ is provided, when $(\Gamma ,\mathfrak{B},\nu )$ is localizable (in particular, when it is $\sigma$-finite). On the other hand, we show that for compact Hausdorff spaces $X$ and $Y$, if $\Phi$ 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 $\tau :Y\to X$ satisfying $\Phi (f)(y)=f(\tau (y))$ ($f\in C(X{)}_{+}^{\mathrm{sp}};y\in Y$).