An optimal nonlinear extension of Banach–Stone theorem

Galego, Elói Medina; Silva, André Luis Porto da

doi:10.1016/j.jfa.2016.07.008

Cited by 19 publications

(9 citation statements)

References 7 publications

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“…Let us recall the following basic properties of these sets. The proof of Proposition is essentially the same proof of [, Proposition 3.2] and [, Proposition 2.1]. We leave it to the reader to transpose to our context.…”

Section: Special Sets Associated To Isomorphisms Between Bold-italiccmentioning

confidence: 98%

Isomorphisms of spaces with large distortion

Galego

Silva

2018

Mathematische Nachrichten

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Letand be locally compact Hausdorff spaces and let be a strictly convex Banach space of finite dimension at least 2. In this paper, we prove that if there exists an isomorphism from 0 ( , ) onto 0 ( , ) satisfyingthen and are homeomorphic. Here ( ) denotes the Schäffer constant of . Even for the classical cases = , 1 < < ∞ and ≥ 2, this result is the -valued Banach-Stone theorem via isomorphism with the largest distortion that is known so far, namely ( ) = min { 2 1∕ , 2 1−1∕ } . On the other hand, it is well known that this result is not true for = , even though and are compact Hausdorff spaces. K E Y W O R D S 0 ( , ) spaces, spaces, Schäffer constant, strictly convex spaces, vector-valued Banach-Stone theorems M S C ( 2 0 1 0 ) Primary: 46B03, 46E15; Secondary: 46B25, 46E40 996

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Section: Special Sets Associated To Isomorphisms Between Bold-italiccmentioning

confidence: 98%

Isomorphisms of spaces with large distortion

Galego

Silva

2018

Mathematische Nachrichten

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“…The latest result in the direction of the Amir-Cambern theorem is due to E.M. Galego and A.L. Porto da Silva in [23] who proved the following theorem. If T is a function from C 0 (K, R) to C 0 (S, R), T (0) = 0, and both T and T −1 are bijective coarse (M, 1)-quasi-isometries with M < √ 2, then K and S are homeomorphic and there exists a homeomorphism φ from S to K and a continuous function λ : S → {−1, 1} such that for all s ∈ S and f ∈ C 0 (K, R) one has…”

Section: Introductionmentioning

confidence: 99%

Isomorphisms of spaces of affine continuous complex functions

Rondoš¹,

Spurný²

2019

Math. Scand.

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We deal with isomorphic Banach-Stone type theorems for closed subspaces of vector-valued continuous functions. Let F = R or C. For i = 1, 2, let E i be a reflexive Banach space over F with a certain parameter λ(E i ) > 1, which in the real case coincides with the Schaffer constant of E i , let K i be a locally compact (Hausdorff) topological space and let H i be a closed subspace of C 0 (K i , E i ) such that each point of the Choquet boundary Ch H i K i of H i is a weak peak point. We show that if there exists an isomorphism T : H 1 → H 2 with T · T −1 < min{λ(E 1 ), λ(E 2 )}, then Ch H 1 K 1 is homeomorphic to Ch H 2 K 2 . Next we provide an analogous version of the weak vector-valued Banach-Stone theorem for subspaces, where the target spaces do not contain an isomorphic copy of c 0 .

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“…On the other hand, as mentioned in Remark 9.2, it is possible to increase the number of digits of the decimal part of 2.054208 in such a way that the statement of Theorem 1.1 still holds. This raises the following conjecture about the exact value of BS(𝓁 The proof of Theorem 1.1 is based on a deeper understanding of the properties of some special subsets of locally compact spaces 𝐾 introduced in [7] in the study of the geometry of 𝐶 0 (𝐾, 𝑋) spaces, where 𝑋 = 𝐑. These special sets were studied in more detail in [10] in the case where 𝑋 is a finite-dimensional Hilbert space.…”

Section: Introductionmentioning

confidence: 99%

The strongest Banach–Stone theorem for C0(K,ℓ22)$C_{0}(K, \ell _2^2)$ spaces

Galego

2024

Mathematische Nachrichten

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As usual denote by the real two‐dimensional Hilbert space. We prove that if K and S are locally compact Hausdorff spaces and T is a linear isomorphism from onto satisfying then K and S are homeomorphic.This theorem is the strongest of all the other vector‐valued Banach–Stone theorems known so far in the sense that in none of them the distortion of the isomorphism T, denoted by , is as large as .Some remarks on the proof method developed here to prove our theorem suggest the conjecture that it is in fact very close to the optimal Banach–Stone theorem for spaces, or in more precise words, the exact value of the Banach–Stone constant of is between and .

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An optimal nonlinear extension of Banach–Stone theorem

Cited by 19 publications

References 7 publications

Isomorphisms of spaces with large distortion

Isomorphisms of spaces with large distortion

Isomorphisms of spaces of affine continuous complex functions

The strongest Banach–Stone theorem for C0(K,ℓ22)$C_{0}(K, \ell _2^2)$ spaces

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