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

2022/01/17 by Cabezas, David, Cueto-Avellaneda, María, Hirota, Daisuke +2
#17C65 #46B04 #46B20 #46J10 #46J15 #47B49 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2201.06307

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 \mathbbT-bundle L in the form C₀^\mathbbT(L):=\a∈ C₀(L):a(λt)=λa(t) for every (λ,t)∈\mathbbT× L\. We prove that every surjective isometry from the unit sphere of C₀^\mathbbT(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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