A Dichotomy for the Asymptotic Geometry of Sequences

José Iovino

A central concept in Banach space geometry is that of spreading model. (See [2, 3] and [1].) Every basic sequence in a Banach space has a subsequence that generates a Banach space called a spreading model of the sequence, and the spreading models generated by a sequence are the standard to analyze is asymptotic geometry.

A central question is to identify under what conditions a spreading contains either of the classical sequence spaces ℓ_{p} and c_{0}. To date, most of the results in this direction have been negative. (See, for example, [4] and [5]).

We will state the following dichotomy.

Theorem 1 Every basic sequence in a Banach space X has a normalized block basis (x_{n}) which satisfies one and only one of the following conditions.

1. (x_{n}) is cofinally isometric over X to either c_{0} or ℓ_{p}, for some p with 1 ≤ p < ∞.

2. (x_{n}) has the largest possible number of spreading models.

The proof is based on constructions from model theory. Specifically, we use the fact that there is an intimate connection between whether a spreading model contains ℓ_{p} or c_{0} and whether the spreading model has a type that is stable in the sense of model theory. The preceding theorem is a counterpart of Shelah’s landmark result that an unstable theory has the maximum number of models.

[1] B. Beauzamy and J.-T. Lapresté. Modèles étalés des espaces de Banach. Travaux en Cours. [Works in Progress]. Hermann, Paris, 1984.

[2] A. Brunel. Espaces associés à une suite bornée dans un espace de Banach. page 23, 1974.

[3] A. Brunel and L. Sucheston. On B-convex Banach spaces. Math. Systems Theory, 7(4):294–299, 1974.

[4] E. Odell and Th. Schlumprecht. On the richness of the set of p’s in Krivine’s theorem. In Geometric aspects of functional analysis (Israel, 1992–1994), volume 77 of Oper. Theory Adv. Appl., pages 177–198. Birkhäuser, Basel, 1995.

[5] E. Odell and Th. Schlumprecht. A problem on spreading models. J. Funct. Anal., 153(2):249–261, 1998.

Address: Division of Mathematics and Statistics, The University of Texas at San Antonio, San Antonio, TX 78249-0664, USA

E-mail: iovino@math.utsa.edu