The structure of Lindenstrauss–Pełczyński spaces

Abstract

Abstract. Lindenstrauss-Pełczyński (for short LP) spaces were introduced by these authors [Studia Math. 174 (2006)] as those Banach spaces X such that every operator from a subspace of c0 into X can be extended to the whole c0. Here we obtain the following structure theorem: a separable Banach space X is an LP-space if and only if every subspace of c0 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 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 L∞ spaces not containing l1 are not necessarily LP-spaces.1. LP-spaces have all subspaces of c 0 in a unique position. In [6] we introduced the class of Lindenstrauss-Pełczyński spaces (for short LP) as those Banach spaces E such that all operators from subspaces of c 0 into E can be extended to c 0 . The spaces are so named because Lindenstrauss and Pełczyński first proved in [9] that C(K)-spaces have this property. In [6] it was shown that every LP-space is an L ∞ -space, that not all L ∞ -spaces are LP-spaces, and that complemented subspaces of Lindenstrauss spaces (see also [9,7]), separably injective spaces and L ∞ -spaces not containing c 0 are LP-spaces.We now prove a fundamental structure theorem for this class; namely, separable LP-spaces are characterized as those L ∞ Banach spaces having all subspaces of c 0 placed in a unique position. Precisely, let Y, X be Banach spaces. Following [5] we say that X is Y -automorphic if any isomorphism between two subspaces of X isomorphic to Y can be extended to an automorphism of X. We agree that if X contains no copies of Y then it is Y -automorphic. Lindenstrauss and Pełczyński prove in

Banach spaces in various positions

“…Also, the Lindenstrauss-Pełczyński theorem [50], see also [24], yields that every subspace Y of c 0 has in a separable C(K)-space exactly one position. The paper [25] characterizes the Banach spaces with this property.…”

On isomorphically polyhedral L∞-spaces

“…Lindenstrauss spaces have also the property (see [17,6]) as well as L ∞ -spaces not containing c 0 [3] and, of course, all their complemented subspaces. The construction of the space Ω above has been modified in [4] to show that for every subspace H ⊂ c 0 there is an [17]; more precisely, there is an operator H → Ω H that cannot be extended to the whole c 0 . Proposition 1.…”