This is an announcement for the paper "On asymptotically symmetric Banach spaces" by M. Junge, D. Kutzarova and E. Odell.

Abstract: We define and study asymptotically symmetric Banach spaces (a.s.) and its variations: weakly a.s. (w.a.s.) and weakly normalized a.s. (w.n.a.s.). 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 < infinity. We prove that if 1 less than or equal p < infinity and the norm of the sum of (x_i)_1^n is of the order n^{1/p} for all (x_i)_1^n in the n^{th} asymptotic structure of $X$, then X contains an asymptotic ell_p, hence w.a.s. subspace.

Archive classification: Functional Analysis

Mathematics Subject Classification: 46B03; 46B20

Remarks: 22 pages, AMSLaTeX

The source file(s), jko31.tex: 68725 bytes, is(are) stored in gzipped form as 0508035.gz with size 21kb. The corresponding postcript file has gzipped size 107kb.

Submitted from: combs@mail.ma.utexas.edu

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