Bulletin des Sciences Mathématiques

2015

DOI: 10.1016/j.bulsci.2015.04.002

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Weak forms of Banach–Stone theorem for C0(K,X) spaces via the αth derivatives of K

Elói Medina Galego

¹

,

Michael A. Rincón‐Villamizar

²

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A Vector‐valued Extension Of a Cengiz's Results On Bold-itali1

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Cited by 10 publications

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“…If α is an ordinal, we define the αth derivative S (α) of S as follows: S (0) � S, S (α+1) � (S (α) ) (1) , and S (α) � ∩ β<α S (β) , in the case where α is a limit ordinal. e following result is an extension of ( [19], eorem 5) for extremely regular subspaces. Theorem 4.…”

Section: Theoremmentioning

confidence: 76%

“…In order to prove eorem 2, we establish the following vector-valued result, which generalizes ( [19], eorem 2).…”

Section: Theoremmentioning

confidence: 99%

“…Paper is organized as follows: in the first section, we will establish some properties for extremely regular subspaces of C 0 (K). In the remaining sections, we will prove eorems 2, 3, 4, and 5. e idea of the proof of these theorems is inducing a set-valued map from isomorphisms defined on extremely regular subspaces and then improving the arguments given in [19] and [22] in order to obtain the wanted conclusions.…”

Section: Theoremmentioning

confidence: 99%

See 2 more Smart Citations

Isomorphisms from Extremely Regular Subspaces of C0K into C0
Cerpa-Torres
¹
,
Rincón‐Villamizar
²

2019
International Journal of Mathematics and Mathematical Sciences
Self Cite
1
0
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0
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For a locally compact Hausdorff space K and a Banach space X, let C0K,X be the Banach space of all X-valued continuous functions defined on K, which vanish at infinite provided with the sup norm. If X is ℝ, we denote C0K,X as C0K. If AK be an extremely regular subspace of C0K and T:AK⟶C0S,X is an into isomorphism, what can be said about the set-theoretical or topological properties of K and S? Answering the question, we will prove that if X contains no copy of c0, then the cardinality of K is less than that of S. Moreover, if TT−1<3 and AK is also a subalgebra of C0K, the cardinality of the αth derivative of K is less than that of the αth derivative of S, for each ordinal α. Finally, if λX>1 and TT−1<λX, then K is a continuous image of a subspace of S. Here, λX is the geometrical parameter introduced by Jarosz in 1989: λX=infmaxx+λy:λ=1:x=y=1. As a consequence, we improve classical results about into isomorphisms from extremely regular subspaces already obtained by Cengiz.

“…If α is an ordinal, we define the αth derivative S (α) of S as follows: S (0) � S, S (α+1) � (S (α) ) (1) , and S (α) � ∩ β<α S (β) , in the case where α is a limit ordinal. e following result is an extension of ( [19], eorem 5) for extremely regular subspaces. Theorem 4.…”

Section: Theoremmentioning

confidence: 76%

“…In order to prove eorem 2, we establish the following vector-valued result, which generalizes ( [19], eorem 2).…”

Section: Theoremmentioning

confidence: 99%

“…Paper is organized as follows: in the first section, we will establish some properties for extremely regular subspaces of C 0 (K). In the remaining sections, we will prove eorems 2, 3, 4, and 5. e idea of the proof of these theorems is inducing a set-valued map from isomorphisms defined on extremely regular subspaces and then improving the arguments given in [19] and [22] in order to obtain the wanted conclusions.…”

Section: Theoremmentioning

confidence: 99%

See 1 more Smart Citation

Isomorphisms from Extremely Regular Subspaces of C0K into C0
Cerpa-Torres
¹
,
Rincón‐Villamizar
²

2019
International Journal of Mathematics and Mathematical Sciences
Self Cite
1
0
1
0
View full text Add to dashboard Cite

For a locally compact Hausdorff space K and a Banach space X, let C0K,X be the Banach space of all X-valued continuous functions defined on K, which vanish at infinite provided with the sup norm. If X is ℝ, we denote C0K,X as C0K. If AK be an extremely regular subspace of C0K and T:AK⟶C0S,X is an into isomorphism, what can be said about the set-theoretical or topological properties of K and S? Answering the question, we will prove that if X contains no copy of c0, then the cardinality of K is less than that of S. Moreover, if TT−1<3 and AK is also a subalgebra of C0K, the cardinality of the αth derivative of K is less than that of the αth derivative of S, for each ordinal α. Finally, if λX>1 and TT−1<λX, then K is a continuous image of a subspace of S. Here, λX is the geometrical parameter introduced by Jarosz in 1989: λX=infmaxx+λy:λ=1:x=y=1. As a consequence, we improve classical results about into isomorphisms from extremely regular subspaces already obtained by Cengiz.

“…This result was improved by Galego and Rincn-Villamizar in [24], who showed that the same conclusion holds for Banach spaces not containing an isomorphic copy of c 0 . The way to this improvement was using a nice characterization of Banach spaces not containing an isomorphic copy of c 0 , see [35,Theorem 6.7], and a result of Plebanek, see [36,Theorem 3.3], which made it possible to remove the assumptions of separability and the Radon-Nikodym property.…”

Section: 2])mentioning

confidence: 83%

Isomorphisms of spaces of affine continuous complex functions

2019

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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 .

“…The following result is a lattice version of [, Proposition 3.1]. Lemma Let X be a Banach lattice containing no copy of c 0 and let K and S be locally compact Hausdorff spaces.…”

Section: A Vector‐valued Extension Of a Cengiz's Results On Bold-italimentioning

confidence: 99%

How do the positive embeddings of Banach lattices depend on the αth derivatives of K?

Rincón‐Villamizar

²

2016

Mathematische Nachrichten

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Let K and S be locally compact Hausdorff spaces and X be an abstract Lp space. Suppose that T is a positive Banach lattice isomorphism from COfalse(Kfalse) into COfalse(S,Xfalse). Then for each ordinal α the cardinalities of the αth derivatives Kfalse(αfalse) and Sfalse(αfalse) satisfy the following inequality ||K(α)1/p≤false∥Tfalse∥T−1||S(α)1/p.Moreover, if false∥Tfalse∥T−1<21/p,then Kfalse(αfalse) is a continuous image of a subset of Sfalse(αfalse) which can be taken closed when K is compact. The first statement of this result for p=1 is a vector‐valued extension of a Cengiz's theorem and the second one is vector‐valued version of a Holsztyński's theorem. A simple example shows that the number 21/p is sharp in these vector‐valued theorems.

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