On Spreading Sequences and Asymptotic Structures

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 c0 and do not contain a copy of `1: i) Suppose X has a normalized weakly null basis (xi) and every spreading model (ei) of a normalized weakly null block basis satisfies ‖e1 − e2‖ = 1. Then some subsequence of (xi) is equivalent to the unit vector basis of c0. 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 + ε)-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 c0. Then X ∗ is separable and X is asymptotic-c0 with respect to a shrinking basis (yi) of Y ⊇ X.