This is an announcement for the paper “On spreading sequences and asymptotic structures” by D. Freemanhttp://arxiv.org/find/math/1/au:+Freeman_D/0/1/0/all/0/1, E. Odellhttp://arxiv.org/find/math/1/au:+Odell_E/0/1/0/all/0/1, B. Sarihttp://arxiv.org/find/math/1/au:+Sari_B/0/1/0/all/0/1, B. Zhenghttp://arxiv.org/find/math/1/au:+Zheng_B/0/1/0/all/0/1.
Abstract: In the first part of the paper we study the structure of Banach spaces with a conditional spreading basis. The geometry of such spaces exhibit a striking resemblance to the geometry of James' space. Further, we show that the averaging projections onto subspaces spanned by constant coefficient blocks with no gaps between supports are bounded. As a consequence, every Banach space with a spreading basis contains a complemented subspace with an unconditional basis. This gives an affirmative answer to a question of H. Rosenthal. The second part contains two results on Banach spaces X whose asymptotic structures are closely related to $c_0$ and do not contain a copy of $\ell_1$: i) Suppose X has a normalized weakly null basis $(x_i)$ and every spreading model $(e_i)$ of a normalized weakly null block basis satisfies $|e_1-e_2|=1$. Then some subsequence of $(x_i)$ is equivalent to the unit vector basis of $c_0$. This generalizes a similar theorem of Odell and Schlumprecht, and yields a new proof of the Elton-Odell theorem on the existence of infinite $1+\epsilon$-separated sequences in the unit sphere of an arbitrary infinite dimensional Banach space. ii) Suppose that all asymptotic models of $X$ generated by weakly null arrays are equivalent to the unit vector basis of $c_0$. Then $X^*$ is separable and $X$ is asymptotic-$c_0$ with respect to a shrinking basis $(y_i)$ of $Y\supseteq X$.
The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1607.03587