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

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

doi:10.1007/s43034-022-00204-6

Cited by 3 publications

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References 21 publications

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“…Many mathematicians have been involved in the study of the Mazur-Ulam property for several Banach spaces (cf. [3,4,6,14,16,26,27]). A refinement of the Mazur-Ulam property was introduced by Hatori in [14].…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Tingley's problem for complex Banach spaces which do not satisfy the Hausdorff distance condition

David¹,

Cueto-Avellaneda²,

Enami³

et al. 2022

Preprint

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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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Section: Introduction and The Main Resultsmentioning

confidence: 99%

Tingley's problem for complex Banach spaces which do not satisfy the Hausdorff distance condition

David¹,

Cueto-Avellaneda²,

Enami³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

Tingley’s problem for complex Banach spaces which do not satisfy the Hausdorff distance condition

David

Cueto-Avellaneda

Enami

et al. 2023

Banach J. Math. Anal.

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Linear orthogonality preservers between function spaces associated with commutative JB ^⋆ -triples

Cabezas

Peralta

2022

Linear and Multilinear Algebra

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It is known, by Gelfand theory, that every commutative JB * -triple admits a representation as a space of continuous functions of the formwhere L is a principal T-bundle and T denotes the unit circle in C. We provide a description of all orthogonality preserving (non-necessarily continuous) linear maps between commutative JB * -triples. We show that each linear orthogonality preserver T : C T 0 (L 1 ) → C T 0 (L 2 ) decomposes in three main parts on its image, on the first part as a positive-weighted composition operator, on the second part the points in L 2 where the image of T vanishes, and a third part formed by those points s in L 2 such that the evaluation mapping δs • T is non-continuous. Among the consequences of this representation, we obtain that every linear bijection preserving orthogonality between commutative JB * -triples is automatically continuous and biorthogonality preserving.

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

Abstract: We prove that every commutative JB$$^*$$ ∗ -triple, represented as a space of continuous functions $$C_0^{\mathbb {T}}(L),$$ C 0 T ( L ) , … Show more

Cited by 3 publications

References 21 publications

Tingley's problem for complex Banach spaces which do not satisfy the Hausdorff distance condition

Tingley's problem for complex Banach spaces which do not satisfy the Hausdorff distance condition

Tingley’s problem for complex Banach spaces which do not satisfy the Hausdorff distance condition

Linear orthogonality preservers between function spaces associated with commutative JB ^⋆ -triples

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

Abstract: We prove that every commutative JB$$^*$$ ∗ -triple, represented as a space of continuous functions $$C_0^{\mathbb {T}}(L),$$ C 0 T ( L ) , … Show more

Cited by 3 publications

References 21 publications

Tingley's problem for complex Banach spaces which do not satisfy the Hausdorff distance condition

Tingley's problem for complex Banach spaces which do not satisfy the Hausdorff distance condition

Tingley’s problem for complex Banach spaces which do not satisfy the Hausdorff distance condition

Linear orthogonality preservers between function spaces associated with commutative JB ⋆ -triples

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Product

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Linear orthogonality preservers between function spaces associated with commutative JB ^⋆ -triples