Wyniki wyszukiwania

1

Sobczyk's theorem and the Bounded Approximation Property

100%

Castillo J., Moreno Y.

Studia Mathematica

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2010

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tom 201

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nr 1

1-19

EN

Sobczyk's theorem asserts that every c₀-valued operator defined on a separable Banach space can be extended to every separable superspace. This paper is devoted to obtaining the most general vector valued version of the theorem, extending and completing previous results of Rosenthal, Johnson-Oikhberg and Cabello. Our approach is homological and nonlinear, transforming the problem of extension of operators into the problem of approximating z-linear maps by linear maps.

2

The structure of Lindenstrauss-Pełczyński spaces

81%

Castillo J., Moreno Y., Suárez J.

Studia Mathematica

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2009

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tom 194

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nr 2

105-115

EN

Lindenstrauss-Pełczyński (for short ℒ𝒫) spaces were introduced by these authors [Studia Math. 174 (2006)] as those Banach spaces X such that every operator from a subspace of c₀ into X can be extended to the whole c₀. Here we obtain the following structure theorem: a separable Banach space X is an ℒ𝒫-space if and only if every subspace of c₀ is placed in X in a unique position, up to automorphisms of X. This, in combination with a result of Kalton [New York J. Math. 13 (2007)], provides a negative answer to a problem posed by Lindenstrauss and Pełczyński [J. Funct. Anal. 8 (1971)]. We show that the class of ℒ𝒫-spaces does not have the 3-space property, which corrects a theorem in an earlier paper of the authors [Studia Math. 174 (2006)]. We then solve a problem in that paper showing that $ℒ_{∞}$ spaces not containing l₁ are not necessarily ℒ𝒫-spaces.

3

On Lindenstrauss-Pełczyński spaces

81%

Castillo J., Moreno Y., Suárez J.

Studia Mathematica

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2006

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tom 174

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nr 3

213-231

EN

We consider some stability aspects of the classical problem of extension of C(K)-valued operators. We introduce the class ℒ𝒫 of Banach spaces of Lindenstrauss-Pełczyński type as those such that every operator from a subspace of c₀ into them can be extended to c₀. We show that all ℒ𝒫-spaces are of type $ℒ_{∞}$ but not conversely. Moreover, $ℒ_{∞}$-spaces will be characterized as those spaces E such that E-valued operators from w*(l₁,c₀)-closed subspaces of l₁ extend to l₁. Regarding examples we will show that every separable $ℒ_{∞}$-space is a quotient of two ℒ𝒫-spaces; also, $ℒ_{∞}$-spaces not containing c₀ are ℒ𝒫-spaces; the complemented subspaces of C(K) and the separably injective spaces are subclasses of the ℒ𝒫-spaces and we show that the former does not contain the latter. Regarding stability properties, we prove that quotients of an ℒ𝒫-space by a separably injective space and twisted sums of ℒ𝒫-spaces are ℒ𝒫-spaces.

4

On ultrapowers of Banach spaces of type $ℒ_{∞}$

71%

Avilés A., Cabello Sánchez F., Castillo J., González M., Moreno Y.

Fundamenta Mathematicae

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2013

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tom 222

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nr 3

195-212

EN

We prove that no ultraproduct of Banach spaces via a countably incomplete ultrafilter can contain c₀ complemented. This shows that a "result" widely used in the theory of ultraproducts is wrong. We then amend a number of results whose proofs have been infected by that statement. In particular we provide proofs for the following statements: (i) All M-spaces, in particular all C(K)-spaces, have ultrapowers isomorphic to ultrapowers of c₀, as also do all their complemented subspaces isomorphic to their square. (ii) No ultrapower of the Gurariĭ space can be complemented in any M-space. (iii) There exist Banach spaces not complemented in any C(K)-space having ultrapowers isomorphic to a C(K)-space.

Ograniczanie wyników

3 Studia Mathematica

1 Fundamenta Mathematicae

4 Castillo J.

4 Moreno Y.

2 Suárez J.

1 Avilés A.

1 Cabello Sánchez F.

1 González M.

1 2013

1 2010

1 2009

1 2006

Sobczyk's theorem and the Bounded Approximation Property

The structure of Lindenstrauss-Pełczyński spaces

On Lindenstrauss-Pełczyński spaces

On ultrapowers of Banach spaces of type $ℒ_{∞}$