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 $$c_ 0({\mathbb{N}})$$.

MSC:

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