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The structure of Lindenstrauss-Pełczyński spaces

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Castillo J., Moreno Y., Suárez J.

Studia Mathematica

2009

tom 194

nr 2

105-115

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.

On Lindenstrauss-Pełczyński spaces

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Castillo J., Moreno Y., Suárez J.

Studia Mathematica

2006

tom 174

nr 3

213-231

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.

Ograniczanie wyników

2 Studia Mathematica

2 Castillo J.

2 Moreno Y.

2 Suárez J.

1 2009

1 2006

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The structure of Lindenstrauss-Pełczyński spaces

On Lindenstrauss-Pełczyński spaces