On asymptotically symmetric Banach spaces

M. Junge ; D. Kutzarova ; E. Odell

Studia Mathematica, Tome 173 (2006), p. 203-231 / Harvested from The Polish Digital Mathematics Library

Résumé

A Banach space X is asymptotically symmetric (a.s.) if for some C < ∞, for all m ∈ ℕ, for all bounded sequences $(x_{j}^{i})_{j=1}^{∞} ⊆ X$, 1 ≤ i ≤ m, for all permutations σ of 1,...,m and all ultrafilters ₁,...,ₘ on ℕ, $lim_{n₁,₁} ... lim_{nₘ,ₘ} ||∑_{i=1}^{m} x_{n_i}^{i}|| ≤ C lim_{n_{σ(1)},_{σ(1)}} ... lim_{n_{σ(m)},_{σ(m)}} ||∑_{i=1}^{m} x_{n_{i}}^{i}||$. We investigate a.s. Banach spaces and several natural variations. X is weakly a.s. (w.a.s.) if the defining condition holds when restricted to weakly convergent sequences $(x_{j}^{i})_{j=1}^{∞}$. Moreover, X is w.n.a.s. if we restrict the condition further to normalized weakly null sequences. If X is a.s. then all spreading models of X are uniformly symmetric. We show that the converse fails. We also show that w.a.s. and w.n.a.s. are not equivalent properties and that Schlumprecht’s space S fails to be w.n.a.s. We show that if X is separable and has the property that every normalized weakly null sequence in X has a subsequence equivalent to the unit vector basis of c₀ then X is w.a.s. We obtain an analogous result if c₀ is replaced by ℓ₁ and also show it is false if c₀ is replaced by $ℓ_{p}$, 1 < p < ∞. We prove that if 1 ≤ p < ∞ and $||∑_{i=1}^{n} x_{i}|| ∼ n^{1/p}$ for all $(x_{i})_{i=1}^{n} ∈ {X}ₙ$, the nth asymptotic structure of X, then X contains an asymptotic $ℓ_{p}$, hence w.a.s. subspace.

Détail

Publié le : 2006-01-01

Zbl 1103.46005

EUDML-ID : urn:eudml:doc:284485