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 exhibits 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 l(1): i) Suppose X has a normalized weakly null basis (xi) and every spreading model (e(i)) of a normalized weakly null block basis satisfies vertical bar vertical bar e(1) - e(2)vertical bar vertical bar = 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 superset of X.