On asymptotically symmetric Banach spaces

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.