On asymptotically symmetric Banach spaces

M. Junge¹ ; D. Kutzarova² ; E. Odell³

¹Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL 61801, U.S.A.
²Institute of Mathematics Bulgarian Academy of Sciences Sofia, Bulgaria and Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL 61801, U.S.A.
³Department of Mathematics The University of Texas at Austin 1 University Station C1200 Austin, TX 78712-0257, U.S.A.

Studia Mathematica, Tome 173 (2006) no. 3, pp. 203-231

Résumé

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.

Détail

DOI : 10.4064/sm173-3-1

Keywords: banach space asymptotically symmetric infty mathbb bounded sequences infty subseteq permutations sigma ldots ultrafilters cal ldots cal mathbb lim cal ldots lim cal bigg sum bigg lim sigma cal sigma ldots lim sigma cal sigma bigg sum bigg investigate banach spaces several natural variations weakly defining condition holds restricted weakly convergent sequences infty moreover s restrict condition further normalized weakly null sequences spreading models uniformly symmetric converse fails s equivalent properties schlumprechts space fails s separable has property every normalized weakly null sequence has subsequence equivalent unit vector basis obtain analogous result replaced ell false replaced ell infty prove infty sum sim asymptotic structure contains asymptotic ell hence subspace