On isomorphisms of Banach spaces of continuous functions

Plebanek, Grzegorz

doi:10.1007/s11856-015-1210-8

Cited by 15 publications

(19 citation statements)

References 17 publications

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“…In fact, as it is explained in the final section, what is really crucial here is the continuity of the mapping L ∋ y → ||T * δ y ||. In a recent preprint [16] we were able to extend some of the results presented here to the case of arbitrary isomorphisms between spaces of continuous functions.…”

Section: Introductionmentioning

confidence: 63%

On positive embeddings of C(K) spaces

Plebanek

2013

Studia Math.

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We investigate isomorphic embeddings T : C(K) → C(L) between Banach spaces of continuous functions. We show that if such an embedding T is a positive operator then K is an image of L under a upper semicontinuous set-function having finite values. Moreover we show that K has a π-base of sets which closures are continuous images of compact subspaces of L. Our results imply in particular that if C(K) can be positively embedded into C(L) then some topological properties of L, such as countable tightness of Frechetness, are inherited by the space K.We show that some arbitrary isomorphic embeddings C(K) → C(L) can be, in a sense, reduced to positive embeddings.

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Section: Introductionmentioning

confidence: 63%

On positive embeddings of C(K) spaces

Plebanek

2013

Studia Math.

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“…where quasi-open means that the image of every open set has nonempty interior (3.20). The second result is in fact a consequence of a study by G. Plebanek [39], however the proof of the first takes a considerable part of this paper. The possibility of obtaining these results is based on special properties of isomorphic embeddings and surjections.…”

Section: Introductionmentioning

confidence: 70%

“…13. An operator T : C(N * ) → C(N * ) is called fountainless or without fountains if, and only if, for every nowhere dense set F ⊆ N * the set In the following lemma we obtain a kind of left dual to an improvement of a theorem of Cengiz ("P" in [10]) obtained by Plebanek (Theorem 3.3. in [39]) which implies that if T is an isomorphic embedding then every x ∈ N * is in ϕ T (y) for some y ∈ N * .…”

Section: Local Behaviour Of Functions Associated With the Adjoint Opementioning

confidence: 99%

“…In general, for a linear bounded operator T acting on the Banach space C(K) for a compact K, the function which sends x ∈ K to T ⋆ (δ x ) is lower semicontinuous (e.g., Lemma 2.1. of [39]) and may be quite discontinuous. Then, there are an infinite B 1 ⊆ * B, a real number s, and partitions N = C n ∪ D n into infinite sets, such that for every y ∈ B * 1 we have:…”

Section: Local Behaviour Of Functions Associated With the Adjoint Opementioning

confidence: 99%

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On automorphisms of the Banach space $\ell _\infty /c_0$

Koszmider

Rodríguez-Porras

2016

Fund. Math.

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Abstract. We investigate Banach space automorphisms T : ℓ∞/c 0 → ℓ∞/c 0 focusing on the possibility of representing their fragments of the formfor A, B ⊆ N infinite by means of linear operators from ℓ∞(A) into ℓ∞(B), infinite A × B-matrices, continuous maps from B * = βB \ B into A * , or bijections from B to A. This leads to the analysis of general linear operators on ℓ∞/c 0 . We present many examples, introduce and investigate several classes of operators, for some of them we obtain satisfactory representations and for other give examples showing that it is impossible. In particular, we show that there are automorphisms of ℓ∞/c 0 which cannot be lifted to operators on ℓ∞ and assuming OCA+MA we show that every automorphism of ℓ∞/c 0 with no fountains or with no funnels is locally, i.e., for some infinite A, B ⊆ N as above, induced by a bijection from B to A. This additional set-theoretic assumption is necessary as we show that the continuum hypothesis implies the existence of counterexamples of diverse flavours. However, many basic problems, some of which are listed in the last section, remain open.

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“…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. We prove an analogous result for closed subspaces of vector-valued continuous functions, whose Choquet boundaries consist of weak peak points.…”

Section: 2])mentioning

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 isomorphisms of Banach spaces of continuous functions

Cited by 15 publications

References 17 publications

On positive embeddings of C(K) spaces

On positive embeddings of C(K) spaces

On automorphisms of the Banach space $\ell _\infty /c_0$

Isomorphisms of spaces of affine continuous complex functions

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