On asymptotically symmetric Banach spaces

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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 mx ni i|| ≤ C lim nsigma;(1),uσ(1)...lim nσ(m),uσ(m)|| Σ i=1 mx 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 languageEnglish (US)
Pages (from-to)203-231
Number of pages29
JournalStudia Mathematica
Issue number3
StatePublished - 2006


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

ASJC Scopus subject areas

  • General Mathematics


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