On asymptotic models in Banach Spaces

Abstract

A well known application of Ramsey's Theorem to Banach Space Theory is the notion of a spreading model (ẽ i ) of a normalized basic sequence (x i ) in a Banach space X. We show how to generalize the construction to define a new creature (e i ), which we call an asymptotic model of X. Every spreading model of X is an asymptotic model of X and in most settings, such as if X is reflexive, every normalized block basis of an asymptotic model is itself an asymptotic model. We also show how to use the Hindman-Milliken Theorema strengthened form of Ramsey's Theorem-to generate asymptotic models with a stronger form of convergence.

On asymptotically symmetric Banach spaces

“…(f) If all asymptotic models of X are symmetric, then it follows easily from Krivine's theorem [K] that for some C < ∞, if (x i ) is an asymptotic model of X generated by either a weakly null array or a block basis array (if X has a basis), then (x i ) is C-equivalent to the unit vector basis of c 0 or ℓ p for some fixed p ∈ [1, ∞) (independent of (x i ); see [HO,4.7.4]). Thus X is w.a.s.…”

On the Structure of the Spreading Models of a Banach Space

“…Lemma 3.3 is actually a special case of a more general situation [11]. The results could also be phrased in terms of countably branching trees of order mn and proved much like the arguments in [13].…”

A New Coarsely Rigid Class of Banach Spaces

“…In order to prove the converse, the crucial step is to show that if a Banach space X satisfies the metric concentration inequality, then all its asymptotic models generated by weakly null arrays are isomorphic to c 0 . The notion of asymptotic models was introduced by Halbeisen and Odell in [11]. Then the conclusion follows from an unexpected link between the notions asymptotic structure and asymptotic models (see § 3).…”