# On the Mazur--Ulam property for the space of Hilbert-space-valued continuous functions

## TL;DR

This paper proves that the space of continuous functions from a compact space into a Hilbert space has the Mazur--Ulam property, meaning every isometry on the sphere extends uniquely to a linear isometry, using JB*-triple theory.

## Contribution

It establishes the Mazur--Ulam property for $C(K,H)$ spaces with new techniques involving JB*-triple structures and facial analysis.

## Key findings

- Every surjective isometry on the sphere extends to a linear isometry.
- Characterization of the facial structure of the unit ball in JB*-triples.
- Application of JB*-triple theory to function space isometries.

## Abstract

Let $K$ be a compact Hausdorff space and let $H$ be a real or complex Hilbert space with dim$(H_\mathbb{R})\geq 2$. We prove that the space $C(K,H)$ of all $H$-valued continuous functions on $K$, equipped with the supremum norm, satisfies the Mazur--Ulam property, that is, if $Y$ is any real Banach space, every surjective isometry $\Delta$ from the unit sphere of $C(K,H)$ onto the unit sphere of $Y$ admits a unique extension to a surjective real linear isometry from $C(K,H)$ onto $Y$. Our strategy relies on the structure of $C(K)$-module of $C(K,H)$ and several results in JB$^*$-triple theory. For this purpose we determine the facial structure of the closed unit ball of a real JB$^*$-triple and its dual space.