On asymptotically symmetric Banach spaces

A Banach space X is asymptotically symmetric (a.s.) if for some C < infinity,) for all m is an element of N, for all bounded sequences (x(j)(i))(j=1)(infinity) subset of X, 1 <= i <= m, for all permutations sigma of {2, ..., m} and all ultrafilters U-1, ..., U-m on N, [GRAPHICS] 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)(infinity) 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 CO 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 co is replaced by f(p), 1 < p < infinity. We prove that if 1 < p < infinity and parallel to Sigma(n)(i=1) x(i)parallel to similar to n(1/P) for all (x(i))(i=1)(n) is an element of {X}(n), the nth asymptotic structure of X, then X contains an asymptotic l(p), hence w.a.s. subspace.