A variant of Tingley's problem for positive unit spheres of continuous functions that vanish at infinity

TL;DR

The paper characterizes surjective isometries between positive unit spheres of continuous functions vanishing at infinity, showing they are induced by homeomorphisms and extend to linear isometries.

Contribution

It proves that surjective isometries on positive unit spheres of $C_0$ spaces are induced by homeomorphisms and extend to linear isometries, generalizing Tingley's problem.

Findings

Surjective isometries correspond to homeomorphisms between spaces.

Such isometries extend to surjective real-linear isometries.

Characterization of surjective phase-isometries on positive spheres.

Abstract

Let $S(C_0(X))^+$ and $S(C_0(Y))^+$ denote the positive parts of the unit spheres of $C_0(X)$ and $C_0(Y)$, where $X$ and $Y$ are locally compact Hausdorff spaces. We prove that every surjective isometry from $S(C_0(X))^+$ onto $S(C_0(Y))^+$ is a composition operator induced by a homeomorphism between $X$ and $Y$ . As a consequence, such a map extends to a surjective reallinear isometry from $C_0(X)$ onto $C_0(Y)$. We also characterize surjective phase-isometries on the positive unit sphere.