On asymptotically symmetric Banach spaces

M. Junge, D. Kutzarova, and E. Odell. "On asymptotically symmetric Banach spaces." Studia Mathematica 173.3 (2006): 203-231. <http://eudml.org/doc/284485>.

@article{M2006,
abstract = {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.},
author = {M. Junge, D. Kutzarova, E. Odell},
journal = {Studia Mathematica},
keywords = {asymptotically symmetric Banach space; spreading model; Schlumprecht's space},
language = {eng},
number = {3},
pages = {203-231},
title = {On asymptotically symmetric Banach spaces},
url = {http://eudml.org/doc/284485},
volume = {173},
year = {2006},
}

TY - JOUR
AU - M. Junge
AU - D. Kutzarova
AU - E. Odell
TI - On asymptotically symmetric Banach spaces
JO - Studia Mathematica
PY - 2006
VL - 173
IS - 3
SP - 203
EP - 231
AB - 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.
LA - eng
KW - asymptotically symmetric Banach space; spreading model; Schlumprecht's space
UR - http://eudml.org/doc/284485
ER -