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

TL;DR

This paper characterizes metric-preserving bijections on positive unit spheres in $L^p$ spaces and continuous functions, showing they extend to isometries and are represented by homeomorphisms, respectively.

Contribution

It establishes the extension of metric-preserving bijections to isometric order isomorphisms in $L^p$ spaces and characterizes such maps as weighted compositions, also providing a homeomorphic representation for continuous functions.

Findings

Metric-preserving bijections extend uniquely to isometric order isomorphisms in $L^p$ spaces.

In localizable measure spaces, these maps have a Lamperti form.

For continuous functions on compact spaces, such bijections correspond to homeomorphisms.

Abstract

Let $(\Omega, \mathfrak{A}, \mu)$ and $(\Gamma, \mathfrak{B}, \nu)$ be two arbitrary measure spaces, and $p\in [1,\infty]$. Set $$L^p(\mu)_+^\mathrm{sp}:= \{f\in L^p(\mu): \|f\|_p =1; f\geq 0\ \mu\text{-a.e.} \}$$ 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 $\Phi(f)(y) = f(\tau(y))$ ($f\in C(X)_+^\mathrm{sp}; y\in Y$).