On asymptotically symmetric Banach spaces
Volume 173 / 2006
Abstract
A Banach space $X$ is asymptotically symmetric (a.s.) if for some $C<\infty$, for all $m\in{\mathbb N}$, for all bounded sequences $(x_j^i)_{j=1}^\infty \subseteq X$, $1\le i\le m$, for all permutations $\sigma$ of $\{1,\ldots,m\}$ and all ultrafilters ${\cal U}_1,\ldots,{\cal U}_m$ on ${\mathbb N}$, $$\lim_{n_1,{\cal U}_1} \ldots \lim_{n_m,{\cal U}_m} \bigg\| \sum_{i=1}^m x_{n_i}^i\bigg\| \le C\lim_{n_{\sigma (1)},{\cal U}_{\sigma (1)}} \ldots \lim_{n_{\sigma(m)},{\cal U}_{\sigma(m)}} \bigg\|\sum_{i=1}^m x_{n_i}^i\bigg\| .$$ 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 $(x_j^i)_{j=1}^\infty$. 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 $\ell_1$ and also show it is false if $c_0$ is replaced by $\ell_p$, $1< p< \infty$.
We prove that if $1\le p< \infty$ and $\|\sum_{i=1}^n x_i\|\sim n^{1/p}$ for all $(x_i)_{i=1}^n\in \{X\}_n$, the $n${th} asymptotic structure of $X$, then $X$ contains an asymptotic $\ell_p$, hence w.a.s. subspace.
