## 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} ^{∞} ⊆C X, 1 ≤ i ≤ m, for all permutations a of {1,... ,m} and all ultrafilters u _{1},... ,u _{m} on ℕ, lim _{n1,u1}... lim _{nm,um} || Σ _{i=1} ^{m}x _{ni} ^{i}|| ≤ C lim _{nsigma;(1),uσ(1)}...lim _{nσ(m),uσ(m)}|| Σ _{i=1} ^{m}x _{ni} ^{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 c _{0} then X is w.a.s. We obtain an analogous result if c _{0} is replaced by ℓ _{1} and also show it is false if c _{0} is replaced by ℓ _{p}, 1 < p < ∞. We prove that if 1 ≤ p < ∞ and || Σ _{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.

Original language | English (US) |
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Pages (from-to) | 203-231 |

Number of pages | 29 |

Journal | Studia Mathematica |

Volume | 173 |

Issue number | 3 |

DOIs | |

State | Published - 2006 |

## Keywords

- Asymptotically symmetric Banach space
- Schlumprecht's space
- Spreading model

## ASJC Scopus subject areas

- General Mathematics