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

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

doi:10.48550/arxiv.2201.06307

Cited by 3 publications

(6 citation statements)

References 14 publications

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“…On the other hand we have not enough examples of Banach spaces which satisfy the conditon. Cabezas, Cueto-Avellaneda, Hirota, Miura and Peralta [14] preved that every commutative JB * triple satisfies the complex Mazur-Ulam property. We do not know if a commutative JB * triple satisfies the condition ( * ) or not.…”

Section: Final Remarksmentioning

confidence: 98%

“…If it will be proved, then (14) will hold for every g ∈ Ball B such that 0 < |p(g)| < 1 since lim n→∞ a n = 1. Combining with the results for α = 0 or |α| = 1, it will follows that (13) holds for every f ∈ Ball(B) and 0 < r < 1.…”

Section: Banach Spaces Which Satisfy the Condition ( * )mentioning

confidence: 99%

“…In the proof of Theorem 4.5 in [25], it is crucial that a uniform algebra contains the constants. Cabezas, Cueto-Avellaneda, Hirota, Miura and Peralta [14,Corollary 3.2] proved that C 0 (Y, C) satisfies the complex Mazur-Ulam property. Cueto-Avellaneda, Hirota, Miura and Peralta [17,Theorem 2.1] have proved that each surjective isometry between the unit spheres of two uniformly closed algebras on locally compact Hausdorff spaces which separates the points without common zeros admits an extension to a surjective real linear isometry between these algebras.…”

Section: Banach Spaces Which Satisfy the Condition ( * )mentioning

confidence: 99%

“…We do not know if a commutative JB * triple satisfies the condition ( * ) or not. If it would satisfy the condition ( * ), we would get an alternative proof of a theorem of Cabezas, Cueto-Avellaneda, Hirota, Miura and Peralta [14,Theorem 3.1]. It is interesting to exhibit examples of Banach spaces which satisfy the condition ( * ).…”

Section: Final Remarksmentioning

confidence: 99%

“…Cueto-Avellaneda, Hirota, Miura and Peralta [17] recently showed that each surjective isometry between the unit spheres of two uniformly closed algebras on locally compact Hausdorff spaces which separates the points without common zeros admits an extension to a surjective real linear isometry between these algebras. Very recently, Cabezas, Cueto-Avellaneda, Hirota, Miura and Peralta [14] proved the complex Mazur-Ulam property for a commutative JB*-triple. Both results concerns the spaces of continuous functions without constants.…”

Section: Introductionmentioning

confidence: 99%

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The Mazur-Ulam property for a Banach space which satisfies a separation condition

Hatori¹

2022

Preprint

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We study C-rich spaces, lush spaces, and C-extremely regular spaces concerning with the Mazur-Ulam property. We show that a uniform algebra and the real part of a uniform algebra with the supremum norm are C-rich spaces, hence lush spaces. We prove that a uniformly closed subalgebra of the algebra of complex-valued continuous functions on a locally compact Hausdorff space which vanish at infinity is C-extremely regular provided that it separates the points of the underlying space and has no common zeros. In section 3 we exhibit descriptions on the Choquet bounday, the Šilov bounday, the strong boundary points. We also recall the definition that a function space strongly separates the points in the underlying space. We need to avoid the confusion which appears because of the variety of names of these concepts; they sometimes differs from authors to authors. After some preparation, we study the Mazur-Ulam property in sections 4 through 6. We exhibit a sufficient condition on a Banach space which has the Mazur-Ulam property and the complex Mazur-Ulam property (Propositions 4.11 and 4.12). In section 5 we consider a Banach space with a separation condition ( * ) (Definition 5.1). We prove that a real Banach space satisfying ( * ) has the Mazur-Ulam propety (Theorem 6.1), and a complex Banach space satisfying ( * ) has the complex Mazur-Ulam property (Theorem 6.3). Applying the results in the previous sections we prove that an extremely Cregular complex linear subspace has the complex Mazur-Ulam property (Corollary 6.4) in section 6. As a consequence we prove that any closed subalgebra of the algebra of all complex-valued continuous functions defined on a locally compact Hausdorff space has the complex Mazur-Ulam property (Corollary 6.5).

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Section: Final Remarksmentioning

confidence: 98%

Section: Banach Spaces Which Satisfy the Condition ( * )mentioning

confidence: 99%

Section: Banach Spaces Which Satisfy the Condition ( * )mentioning

confidence: 99%

Section: Final Remarksmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Mazur-Ulam property for a Banach space which satisfies a separation condition

Hatori¹

2022

Preprint

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

Cabezas¹,

Peralta²

2022

Preprint

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

Cited by 3 publications

References 14 publications

The Mazur-Ulam property for a Banach space which satisfies a separation condition

The Mazur-Ulam property for a Banach space which satisfies a separation condition

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

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

Cited by 3 publications

References 14 publications

The Mazur-Ulam property for a Banach space which satisfies a separation condition

The Mazur-Ulam property for a Banach space which satisfies a separation condition

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

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

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