On 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, Spain

Studia Mathematica, Tome 174 (2006) no. 3, pp. 213-231

Résumé

We consider some stability aspects of the classical problem of extension of $C(K)$-valued operators. We introduce the class $\mathscr{LP}$ of Banach spaces of Lindenstrauss–Pełczyński type as those such that every operator from a subspace of $c_0$ into them can be extended to $c_0$. We show that all $\mathscr{LP}$-spaces are of type $\mathcal L_\infty$ but not conversely. Moreover, $\mathcal L_\infty$-spaces will be characterized as those spaces $E$ such that $E$-valued operators from $w^*(l_1,c_0)$-closed subspaces of $l_1$ extend to $l_1$. Regarding examples we will show that every separable $\mathcal L_\infty$-space is a quotient of two $\mathscr{LP}$-spaces; also, $\mathcal L_\infty$-spaces not containing $c_0$ are $\mathscr{LP}$-spaces; the complemented subspaces of $C(K)$ and the separably injective spaces are subclasses of the $\mathscr{LP}$-spaces and we show that the former does not contain the latter. Regarding stability properties, we prove that quotients of an $\mathscr{LP}$-space by a separably injective space and twisted sums of $\mathscr{LP}$-spaces are $\mathscr{LP}$-spaces.

Détail

DOI : 10.4064/sm174-3-1

Mots-clés : consider stability aspects classical problem extension valued operators introduce class mathscr banach spaces lindenstrauss czy ski type those every operator subspace extended mathscr spaces type mathcal infty conversely moreover mathcal infty spaces characterized those spaces e valued operators * closed subspaces extend regarding examples every separable mathcal infty space quotient mathscr spaces mathcal infty spaces containing mathscr spaces complemented subspaces separably injective spaces subclasses mathscr spaces former does contain latter regarding stability properties prove quotients mathscr space separably injective space twisted sums mathscr spaces mathscr spaces