On Lindenstrauss–Pełczyński spaces
Volume 174 / 2006
Abstract
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.
