On spreading models in \(L^ 1(E)\). (English) Zbl 0579.46012

We construct a Banach space E which has the Schur property (hence \(\ell^ 1\) is its only spreading model) but such that for each family \((a_{n,k})\) with \(a_{n,k}\geq 1\), \(\lim_{n}a_{n,k}=+\infty\), there is a sequence \((f_ n)\) in \(L^ 1(E)\) for which \(\| \sum_{k\leq i\leq n}\pm f_ i\| \leq a_{n,k}\). In particular, \(L^ 1(E)\) has a spreading model isomorphic to \(c_ 0({\mathbb{N}})\).


46B25 Classical Banach spaces in the general theory
46B20 Geometry and structure of normed linear spaces
46E40 Spaces of vector- and operator-valued functions
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