Every commutative JB${}^{\ast }$-triple satisfies the complex Mazur--Ulam property

Abstract

We prove that every commutative JB${}^{\ast }$-triple satisfies the complex Mazur--Ulam property. Thanks to the representation theory, we can identify commutative JB${}^{\ast }$-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.