The structure of Lindenstrauss–Pełczyński spaces
Tom 194 / 2009
Streszczenie
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.
