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## Studia Mathematica

2006 | 173 | 3 | 203-231
Tytuł artykułu

### On asymptotically symmetric Banach spaces

Autorzy
Treść / Zawartość
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Języki publikacji
EN
Abstrakty
EN
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 𝒰₁,...,𝒰ₘ on ℕ,
$lim_{n₁,𝒰₁} ... lim_{nₘ,𝒰ₘ} ||∑_{i=1}^{m} x_{n_i}^{i}|| ≤ C lim_{n_{σ(1)},𝒰_{σ(1)}} ... lim_{n_{σ(m)},𝒰_{σ(m)}} ||∑_{i=1}^{m} x_{n_{i}}^{i}||$.
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.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
203-231
Opis fizyczny
Daty
wydano
2006
Twórcy
autor
• Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.A.
autor
• Institute of Mathematics, Bulgarian Academy of Sciences, Sofia, Bulgaria
• Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.A.
autor
• Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, TX 78712-0257, U.S.A.
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