## 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.