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arXiv · 2201.06307

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

David Cabezas·María Cueto-Avellaneda·Daisuke Hirota·Takeshi Miura·Antonio M. Peralta
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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.

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David Cabezas, María Cueto-Avellaneda, Daisuke Hirota, Takeshi Miura, Antonio M. Peralta. 2022-01-17. Every commutative JB$^*$-triple satisfies the complex Mazur--Ulam property. https://arxiv.org/abs/2201.06307

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