On Lindenstrauss-Pełczyński spaces

Abstract

In this work we shall be concerned with some stability aspects of the classical problem of extension of $C(K)$-valued operators. We introduce the class $\mathcal{LP}$ 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 $\mathcal{LP}$-spaces are of type ${\mathcal{L}}_{\infty }$ but not the converse. Moreover, ${\mathcal{L}}_{\infty }$-spaces will be characterized as those spaces $E$ such that $E$-valued operators from $w^{\ast }(l_1,c_0)$-closed subspaces of $l_1$ extend to $l_1$. Complemented subspaces of $C(K)$ and separably injective spaces are subclasses of $\mathcal{LP}$-spaces and we show that the former does not contain the latter. It is established that ${\mathcal{L}}_{\infty }$-spaces not containing $l_1$ are quotients of $\mathcal{LP}$-spaces, while ${\mathcal{L}}_{\infty }$-spaces not containing $c_0$, quotients of an $\mathcal{LP}$-space by a separably injective space and twisted sums of $\mathcal{LP}$-spaces are $\mathcal{LP}$-spaces.