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

TL;DR

The paper resolves a non-unital version of Tingley’s problem for positive unit spheres by showing that every surjective isometry $\Phi:S(C_0(X))^+\to S(C_0(Y))^+$ is induced by a homeomorphism $\sigma:Y\to X$, i.e., $\Phi(f)=f\circ\sigma$ for all $f\in S(C_0(X))^+$. This leads to a unique extension of $\Phi$ to a surjective real-linear isometry between $C_0(X)$ and $C_0(Y)$. The authors develop a framework based on the sets $D_X(f)$ and peak functions to extract the underlying space map, ensuring the needed continuity. They further apply these results to characterize surjective phase-isometries on $S(C_0(X))^+$, showing they are exactly composition operators induced by a homeomorphism, with the extension to a real-linear isometry. The work broadens the scope of Tingley-type rigidity phenomena to non-unital function spaces and provides tools for analyzing isometries via peak-set and zero-set structures.

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.

Theorems & Definitions (23)

• Theorem 3.1
• Corollary 3.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof