The structure of Lindenstrauss–Pełczyński spaces

Jesús M. F. Castillo ¹   ; Yolanda Moreno ¹   ; Jesús Suárez ¹

¹ Departamento de Matemáticas Universidad de Extremadura Avenida de Elvas 06071 Badajoz, España

Studia Mathematica, Tome 194 (2009) no. 2, pp. 105-115

Résumé

Lindenstrauss–Pełczyński (for short $\mathscr{LP}$) spaces were introduced by these authors [Studia Math. 174 (2006)] as those Banach spaces $X$ such that every operator from a subspace of $c_0$ into $X$ can be extended to the whole $c_0$. Here we obtain the following structure theorem: a separable Banach space $X$ is an $\mathscr{LP}$-space if and only if every subspace of $c_0$ 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 $\mathscr{LP}$-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 $\mathcal L_\infty$ spaces not containing $l_1$ are not necessarily $\mathscr{LP}$-spaces.

Détail

DOI : 10.4064/sm194-2-1

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