Optimal extensions of the Banach–Stone theorem

Cidral, Fabiano C.; Galego, Elói Medina; Rincón‐Villamizar, Michael A.

doi:10.1016/j.jmaa.2015.04.060

Cited by 18 publications

(20 citation statements)

References 24 publications

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“…It is worth mentioning that in [14,Section 6], there is given a similar description of the second dual space of C 0 (K, E), where K is just locally compact. We will need to use only the case when K is compact, though.…”

Section: Notation and Auxiliary Resultsmentioning

confidence: 99%

“…Many of those results were recently unified and strengthened in [14], where it was showed that if E is a real or complex reflexive Banach space with λ(E) > 1, then for all locally compact spaces K 1 , K 2 , the existence of an isomorphism T : C 0 (K 1 , E) → C 0 (K 2 , E) with T · T −1 < λ(E) implies that the spaces K 1 , K 2 are homeomorphic. Here λ(E) = inf{max{ e 1 + λe 2 : λ ∈ F, |λ| = 1} : e 1 , e 2 ∈ S E } is a parameter introduced by Jarosz in [29].…”

Section: 2])mentioning

confidence: 94%

“…For H = C 0 (K, E), the Choquet boundary of H coincides with K and each point of K is a weak peak point, see Remarks 2.4. The proof of the above result combines the methods of [14] (which are in turn adapted from [9]) with the methods developed in [38]. The maximum principle for affine functions of the first Borel class (see Lemma 2.9) even allows some technical simplifications compared to [14] and [9], since we can construct the desired homeomorphism directly as a mapping from K 1 to K 2 (compare with the definition on pages 248 and 249 in [9]).…”

Section: 2])mentioning

confidence: 95%

“…It is easy to check that λ(F) = 2, thus this result recovers the theorem of Amir and Cambern. The authors of [14] also showed that the constant λ(E) = 2 1 p is the best possible for E = l p , where 2 ≤ p < ∞.…”

Section: 2])mentioning

confidence: 99%

“…The next lemma describes a property of the parameter λ that is crucial for the proof of the main theorem of [14]. Lemma 2.11.…”

Section: Thus (27)mentioning

confidence: 99%

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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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Section: Notation and Auxiliary Resultsmentioning

confidence: 99%

Section: 2])mentioning

confidence: 94%

Section: 2])mentioning

confidence: 95%

Section: 2])mentioning

confidence: 99%

“…The next lemma describes a property of the parameter λ that is crucial for the proof of the main theorem of [14]. Lemma 2.11.…”

Section: Thus (27)mentioning

confidence: 99%

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Isomorphisms of spaces of affine continuous complex functions

Rondoš¹,

Spurný²

2019

Math. Scand.

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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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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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Optimal extensions of the Banach–Stone theorem

Cited by 18 publications

References 24 publications

Isomorphisms of spaces of affine continuous complex functions

Isomorphisms of spaces of affine continuous complex functions

Isomorphisms of spaces with large distortion

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

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