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 T-bundle L in the formfor every (λ, t) ∈ T × L}. We prove that every surjective isometry from the unit sphere of C T 0 (L) onto the unit sphere of any complex Banach space admits an extension to a surjective real linear isometry between the spaces.
In 2022, Hatori [14] gave a sufficient condition for complex Banach spaces to have the complex Mazur-Ulam property. In this paper, we introduce a class of complex Banach spaces B that do not satisfy the condition but enjoy the property that every surjective isometry on the unit sphere of such B admits an extension to a surjective real linear isometry on the whole space B. Typical examples of Banach spaces studied in this note are the spaces Lip([0, 1]) of all Lipschitz complex-valued functions on [0, 1] and C 1 ([0, 1]) of all continuously differentiable complex-valued functions on [0, 1] equipped with the norm |f (0)| + f ′ ∞.
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