Every commutative JB$^*$-triple satisfies the complex Mazur--Ulam property

TL;DR

This paper proves that all commutative JB$^*$-triples satisfy the complex Mazur--Ulam property, showing that surjective isometries on their unit spheres extend to linear isometries, using their representation as function spaces.

Contribution

It establishes the complex Mazur--Ulam property for all commutative JB$^*$-triples through a representation as function spaces and extension of isometries.

Findings

Surjective isometries on the unit sphere extend to linear isometries.

Representation of commutative JB$^*$-triples as $C_0^ ext{T}(L)$ spaces.

Validation of the complex Mazur--Ulam property for these spaces.

Abstract

We prove that every commutative JB$^*$-triple satisfies the complex Mazur--Ulam property. Thanks to the representation theory, we can identify commutative JB$^*$-triples as spaces of complex-valued continuous functions on a principal $\mathbb{T}$-bundle $L$ in the form $$C_0^\mathbb{T}(L):=\{a\in C_0(L):a(\lambda t)=\lambda a(t)\text{ for every } (\lambda,t)\in\mathbb{T}\times L\}.$$ We prove that every surjective isometry from the unit sphere of $C_0^\mathbb{T}(L)$ onto the unit sphere of any complex Banach space admits an extension to a surjective real linear isometry between the spaces.