Exploring new solutions to Tingley's problem for function algebras

TL;DR

This paper provides new positive solutions to Tingley's problem for certain function algebra subspaces, showing that surjective isometries between their unit spheres extend to linear isometries, with applications to abelian JB*-triples.

Contribution

It introduces two novel positive results extending surjective isometries to linear isometries in specific function algebra contexts, including spaces of continuous functions on principal T-bundles.

Findings

Surjective isometries between unit spheres extend to real linear isometries.

Results apply to uniformly closed function algebras on locally compact spaces.

Extensions are established for spaces of continuous functions with T-equivariance.

Abstract

In this note we present two new positive answers to Tingley's problem in certain subspaces of function algebras. In the first result we prove that every surjective isometry between the unit spheres, $S(A)$ and $S(B)$, of two uniformly closed function algebras $A$ and $B$ on locally compact Hausdorff spaces can be extended to a surjective real linear isometry from $A$ onto $B$. In a second goal we study surjective isometries between the unit spheres of two abelian JB$^*$-triples represented as spaces of continuous functions of the form $$C^{\mathbb{T}}_0 (X) := \{ a \in C_0(X) : a (\lambda t) = \lambda a(t) \hbox{ for every } (\lambda, t) \in \mathbb{T}\times X\},$$ where $X$ is a (locally compact Hausdorff) principal $\mathbb{T}$-bundle. We establish that every surjective isometry $\Delta: S(C_0^{\mathbb{T}}(X))\to S(C_0^{\mathbb{T}}(Y))$ admits an extension to a surjective real linear isometry between these two abelian JB$^*$-triples.