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1
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Banach spaces widely complemented in each other

100%
Galego E.
Colloquium Mathematicum
|
2013
|
tom 133
|
nr 2
283-291
EN
Suppose that X and Y are Banach spaces that embed complementably into each other. Are X and Y necessarily isomorphic? In this generality, the answer is no, as proved by W. T. Gowers in 1996. However, if X contains a complemented copy of its square X², then X is isomorphic to Y whenever there exists p ∈ ℕ such that $X^{p}$ can be decomposed into a direct sum of $X^{p-1}$ and Y. Motivated by this fact, we introduce the concept of (p,q,r) widely complemented subspaces in Banach spaces, where p,q and r ∈ ℕ. Then, we completely determine when X is isomorphic to Y whenever X is (p,q,r) widely complemented in Y and Y is (t,u,v) widely complemented in X. This new notion of complementability leads naturally to an extension of the Square-cube Problem for Banach spaces, the p-q-r Problem.
2
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The Schroeder-Bernstein index for Banach spaces

100%
Galego E.
Studia Mathematica
|
2004
|
tom 164
|
nr 1
29-38
EN
In relation to some Banach spaces recently constructed by W. T. Gowers and B. Maurey, we introduce the notion of Schroeder-Bernstein index SBi(X) for every Banach space X. This index is related to complemented subspaces of X which contain some complemented copy of X. Then we establish the existence of a Banach space E which is not isomorphic to Eⁿ for every n ∈ ℕ, n ≥ 2, but has a complemented subspace isomorphic to E². In particular, SBi(E) is uncountable. The construction of E follows the approach given in 1996 by W. T. Gowers to obtain the first solution to the Schroeder-Bernstein Problem for Banach spaces.
3
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An Isomorphic Classification of $C(2^{𝔪} × [0,α])$ Spaces

100%
Galego E.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2009
|
tom 57
|
nr 3
279-287
EN
We present an extension of the classical isomorphic classification of the Banach spaces C([0,α]) of all real continuous functions defined on the nondenumerable intervals of ordinals [0,α]. As an application, we establish the isomorphic classification of the Banach spaces $C(2^{𝔪} × [0,α])$ of all real continuous functions defined on the compact spaces $2^{𝔪} × [0,α]$, the topological product of the Cantor cubes $2^{𝔪}$ with 𝔪 smaller than the first sequential cardinal, and intervals of ordinal numbers [0,α]. Consequently, it is relatively consistent with ZFC that this yields a complete isomorphic classification of $C(2^{𝔪} × [0,α])$ spaces.
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On pairs of Banach spaces which are isomorphic to complemented subspaces of each other

100%
Galego E.
Colloquium Mathematicum
|
2004
|
tom 101
|
nr 2
279-287
EN
We establish the existence of Banach spaces E and F isomorphic to complemented subspaces of each other but with $E^m ⊕ Fⁿ$ isomorphic to $E^{p} ⊕ F^q$, m, n, p, q ∈ ℕ, if and only if m = p and n = q.
5
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Schroeder-Bernstein Quintuples for Banach Spaces

100%
Galego E.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2006
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tom 54
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nr 2
113-124
EN
Let X and Y be two Banach spaces, each isomorphic to a complemented subspace of the other. In 1996, W. T. Gowers solved the Schroeder-Bernstein Problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In this paper, we obtain necessary and sufficient conditions on the quintuples (p,q,r,s,t) in ℕ for X to be isomorphic to Y whenever ⎧$X ~ X^p ⊕ Y^q$, ⎨ ⎩ $Y^t ~ X^r ⊕ Y^s$. Such quintuples are called Schroeder-Bernstein quintuples for Banach spaces and they yield a unification of the known decomposition methods in Banach spaces involving finite sums of X and Y, similar to Pełczyński's decomposition method. Inspired by this result, we also introduce the notion of Schroeder-Bernstein sextuples for Banach spaces and pose a conjecture which would complete their characterization.
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An Arithmetic Characterization of Decomposition Methods in Banach Spaces Similar to Pełczyński's Decomposition Method

100%
Galego E.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2004
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tom 52
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nr 3
273-282
EN
Inspired by Pełczyński's decomposition method in Banach spaces, we introduce the notion of Schroeder-Bernstein quadruples for Banach spaces. Then we use some Banach spaces constructed by W. T. Gowers and B. Maurey in 1997 to characterize them.
7
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On isomorphism classes of $C(2^{𝔪} ⊕ [0,α])$ spaces

100%
Galego E.
Fundamenta Mathematicae
|
2009
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tom 204
|
nr 1
87-95
EN
We provide a complete isomorphic classification of the Banach spaces of continuous functions on the compact spaces $2^{𝔪} ⊕ [0,α]$, the topological sums of Cantor cubes $2^{𝔪}$, with 𝔪 smaller than the first sequential cardinal, and intervals of ordinal numbers [0,α]. In particular, we prove that it is relatively consistent with ZFC that the only isomorphism classes of $C(2^{𝔪} ⊕ [0,α])$ spaces with 𝔪 ≥ ℵ₀ and α ≥ ω₁ are the trivial ones. This result leads to some elementary questions on large cardinals.
8
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Cantor-Schroeder-Bernstein quadruples for Banach spaces

100%
Galego E.
Colloquium Mathematicum
|
2008
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tom 111
|
nr 1
105-115
EN
Two Banach spaces X and Y are symmetrically complemented in each other if there exists a supplement of Y in X which is isomorphic to some supplement of X in Y. In 1996, W. T. Gowers solved the Schroeder-Bernstein (or Cantor-Bernstein) Problem for Banach spaces by constructing two non-isomorphic Banach spaces which are symmetrically complemented in each other. In this paper, we show how to modify such a symmetry in order to ensure that X is isomorphic to Y. To do this, first we introduce the notion of Cantor-Schroeder-Bernstein Quadruples for Banach spaces. Then we characterize them by using some Banach spaces constructed by W. T. Gowers and B. Maurey in 1997. This new insight into the geometry of Banach spaces complemented in each other leads naturally to the Strong Square-hyperplane Problem which is closely related to the Schroeder-Bernstein Problem.
9
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A note on extensions of Pełczyński's decomposition method in Banach spaces

100%
Galego E.
Studia Mathematica
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2007
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tom 180
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nr 1
27-40
EN
Let X,Y,A and B be Banach spaces such that X is isomorphic to Y ⊕ A and Y is isomorphic to X ⊕ B. In 1996, W. T. Gowers solved the Schroeder-Bernstein problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In the present paper, we give a necessary and sufficient condition on sextuples (p,q,r,s,u,v) in ℕ with p + q ≥ 2, r + s ≥ 1 and u, v ∈ ℕ* for X to be isomorphic to Y whenever these spaces satisfy the following decomposition scheme: ⎧ $X^{u} ∼ X^{p} ⊕ Y^{q}$, ⎨ ⎩ $Y^{v} ∼ A^{r} ⊕ B^{s}$. Namely, Ω = (p-u)(s-r-v) - q(r-s) is different from zero and Ω divides p + q - u and v. In other words, we obtain an arithmetic characterization of some extensions of the classical Pełczyński decomposition method in Banach spaces. This result leads naturally to several problems closely related to the Schroeder-Bernstein problem.
10
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Spaces of compact operators on $C(2^{𝔪} × [0,α])$ spaces

100%
Galego E.
Colloquium Mathematicum
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2011
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tom 125
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nr 2
175-181
EN
We classify, up to isomorphism, the spaces of compact operators 𝒦(E,F), where E and F are the Banach spaces of all continuous functions defined on the compact spaces $2^{𝔪} × [0,α]$, the topological products of Cantor cubes $2^{𝔪}$ and intervals of ordinal numbers [0,α].
11
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The classical subspaces of the projective tensor products of $ℓ_{p}$ and C(α) spaces, α < ω₁

64%
Galego E., Samuel C.
Studia Mathematica
|
2013
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tom 214
|
nr 3
237-250
EN
We completely determine the $ℓ_{q}$ and C(K) spaces which are isomorphic to a subspace of $ℓ_{p} ⊗̂_{π} C(α)$, the projective tensor product of the classical $ℓ_{p}$ space, 1 ≤ p < ∞, and the space C(α) of all scalar valued continuous functions defined on the interval of ordinal numbers [1,α], α < ω₁. In order to do this, we extend a result of A. Tong concerning diagonal block matrices representing operators from $ℓ_{p}$ to ℓ₁, 1 ≤ p < ∞. The first main theorem is an extension of a result of E. Oja and states that the only $ℓ_{q}$ space which is isomorphic to a subspace of $ℓ_{p} ⊗̂_{π} C(α)$ with 1 ≤ p ≤ q < ∞ and ω ≤ α < ω₁ is $ℓ_{p}$. The second main theorem concerning C(K) spaces improves a result of Bessaga and Pełczyński which allows us to classify, up to isomorphism, the separable spaces 𝓝(X,Y) of nuclear operators, where X and Y are direct sums of $ℓ_{p}$ and C(K) spaces. More precisely, we prove the following cancellation law for separable Banach spaces. Suppose that K₁ and K₃ are finite or countable compact metric spaces of the same cardinality and 1 < p, q < ∞. Then, for any infinite compact metric spaces K₂ and K₄, the following statements are equivalent: (a) $𝓝(ℓ_{p}⊕ C(K₁),ℓ_{q}⊕ C(K₂))$ and $𝓝(ℓ_{p}⊕ C(K₃),ℓ_{q}⊕ C(K₄))$ are isomorphic. (b) C(K₂) is isomorphic to C(K₄).
12
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How far is C₀(Γ,X) with Γ discrete from C₀(K,X) spaces?

64%
Candido L., Galego E.
Fundamenta Mathematicae
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2012
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tom 218
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nr 2
151-163
EN
For a locally compact Hausdorff space K and a Banach space X we denote by C₀(K,X) the space of X-valued continuous functions on K which vanish at infinity, provided with the supremum norm. Let n be a positive integer, Γ an infinite set with the discrete topology, and X a Banach space having non-trivial cotype. We first prove that if the nth derived set of K is not empty, then the Banach-Mazur distance between C₀(Γ,X) and C₀(K,X) is greater than or equal to 2n + 1. We also show that the Banach-Mazur distance between C₀(ℕ,X) and C([1,ωⁿk],X) is exactly 2n + 1, for any positive integers n and k. These results extend and provide a vector-valued version of some 1970 Cambern theorems, concerning the cases where n = 1 and X is the scalar field.
13
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Embeddings of C(K) spaces into C(S,X) spaces with distortion strictly less than 3

64%
Candido L., Galego E.
Fundamenta Mathematicae
|
2013
|
tom 220
|
nr 1
83-92
EN
In the spirit of the classical Banach-Stone theorem, we prove that if K and S are intervals of ordinals and X is a Banach space having non-trivial cotype, then the existence of an isomorphism T from C(K, X) onto C(S,X) with distortion $||T|| ||T^{-1}||$ strictly less than 3 implies that some finite topological sum of K is homeomorphic to some finite topological sum of S. Moreover, if Xⁿ contains no subspace isomorphic to $X^{n+1}$ for every n ∈ ℕ, then K is homeomorphic to S. In other words, we obtain a vector-valued Banach-Stone theorem which is an extension of a Gordon theorem and at the same time an improvement of a Behrends and Cambern theorem. In order to prove this, we show that if there exists an embedding T of a C(K) space into a C(S,X) space, with distortion strictly less than 3, then the cardinality of the αth derivative of S is finite or greater than or equal to the cardinality of the αth derivative of K, for every ordinal α.
14
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How far is C(ω) from the other C(K) spaces?

64%
Candido L., Galego E.
Studia Mathematica
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2013
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tom 217
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nr 2
123-138
EN
Let us denote by C(α) the classical Banach space C(K) when K is the interval of ordinals [1,α] endowed with the order topology. In the present paper, we give an answer to a 1960 Bessaga and Pełczyński question by providing tight bounds for the Banach-Mazur distance between C(ω) and any other C(K) space which is isomorphic to it. More precisely, we obtain lower bounds L(n,k) and upper bounds U(n,k) on d(C(ω),C(ωⁿk)) such that U(n,k) - L(n,k) < 2 for all 1 ≤ n, k < ω.
15
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A quasi-dichotomy for C(α,X) spaces, α < ω₁

64%
Galego E., Zahn M.
Colloquium Mathematicum
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2015
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tom 141
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nr 1
51-59
EN
We prove the following quasi-dichotomy involving the Banach spaces C(α,X) of all X-valued continuous functions defined on the interval [0,α] of ordinals and endowed with the supremum norm. Suppose that X and Y are arbitrary Banach spaces of finite cotype. Then at least one of the following statements is true. (1) There exists a finite ordinal n such that either C(n,X) contains a copy of Y, or C(n,Y) contains a copy of X. (2) For any infinite countable ordinals α, β, ξ, η, the following are equivalent: (a) C(α,X) ⊕ C(ξ,Y) is isomorphic to C(β,X) ⊕ C(η,Y). (b) C(α) is isomorphic to C(β), and C(ξ) is isomorphic to C(η). This result is optimal in the sense that it cannot be extended to uncountable ordinals.
16
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Geometry of the Banach spaces C(βℕ × K,X) for compact metric spaces K

64%
Alspach D., Galego E.
Studia Mathematica
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2011
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tom 207
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nr 2
153-180
EN
A classical result of Cembranos and Freniche states that the C(K,X) space contains a complemented copy of c₀ whenever K is an infinite compact Hausdorff space and X is an infinite-dimensional Banach space. This paper takes this result as a starting point and begins a study of conditions under which the spaces C(α), α < ω₁, are quotients of or complemented in C(K,X). In contrast to the c₀ result, we prove that if C(βℕ ×[1,ω],X) contains a complemented copy of $C(ω^{ω})$ then X contains a copy of c₀. Moreover, we show that $C(ω^{ω})$ is not even a quotient of $C(βℕ ×[1,ω],ℓ_{p})$, 1 < p < ∞. We then completely determine the separable C(K) spaces which are isomorphic to a complemented subspace or a quotient of a $C(βℕ ×[1,α],ℓ_{p})$ space for countable ordinals α and 1 ≤ p < ∞. As a consequence, we obtain the isomorphic classification of the $C(βℕ ×K,ℓ_{p})$ spaces for infinite compact metric spaces K and 1 ≤ p < ∞. Indeed, we establish the following more general cancellation law. Suppose that the Banach space X contains no copy of c₀ and K₁ and K₂ are infinite compact metric spaces, then the following statements are equivalent: (1) C(βℕ ×K₁,X) is isomorphic to C(βℕ ×K₂,X). (2) C(K₁) is isomorphic to C(K₂). These results are applied to the isomorphic classification of some spaces of compact operators.
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