A new metric invariant for Banach spaces

Abstract

We show that if the Szlenk index of a Banach space $X$ is larger than the first infinite ordinal $\omega$ or if the Szlenk index of its dual is larger than $\omega$, then the tree of all finite sequences of integers equipped with the hyperbolic distance metrically embeds into $X$. We show that the converse is true when $X$ is assumed to be reflexive. As an application, we exhibit new classes of Banach spaces that are stable under coarse-Lipschitz embeddings and therefore under uniform homeomorphisms.Comment: 22 page

(β)-distortion of some infinite graphs

“…We recall the main result from [4]. The same equivalences also hold without the separability assumption [8].…”

“…Since then other characterizations have been discovered [3], [15], [28]. The asymptotic analogue of Bourgain's characterization was proved by the first author, Kalton, and Lancien [4]. The definitions of the asymptotic versions of uniform convexity and uniform smoothness are briefly recalled.…”

“…If one uses a combination of Theorem 4.10 and Theorem 4.11 to prove that, for every Banach space Y admitting an equivalent norm with property (β) one has lim h→∞ c Y (T ω h ) = ∞, then one does not obtain an optimal estimate on the rate of growth of (c Y (T ω h )) h≥1 . Moreover the proof in [4] of the fact that lim h→∞ c Y (T ω h ) = ∞ for every reflexive Banach space Y that is a.u.s. renormable and a.u.c.…”

A New Coarsely Rigid Class of Banach Spaces

“…Recall that, as it was shown in [4], within the class of reflexive Banach spaces the subclass of reflexive spaces that admit an equivalent asymptotic uniformly smooth norm (i.e., they are AUS-able) and admit an equivalent asymptotic uniformly convex norm (i.e., they are AUC-able) is coarse Lipschitzly rigid. It was later proved in [3] that, within the class of reflexive spaces with an unconditional asymptotic structure, the subclass of such spaces that are additionally AUC-able is coarse Lipschitzly rigid.…”

Power type ξ-asymptotically uniformly smooth and ξ-asymptotically uniformly flat norms

“…In a number of recent results in the non-linear theory of Banach spaces, power type asymptotically uniformly smooth and asymptotically uniformly convex Banach spaces have played an important role. Given the recent, remarkable result of Motakis and Schlumprecht [16], which proves a transfinite version of results from [1], the existence of the transfinite notions of asymptotic uniform smoothness and asymptotic uniform convexity are potentially very useful in proving results akin to those in [16]. The notion of asymptotic uniform flatness was pivotal in [11] for the Lipschitz classification of C(K) spaces isomorphic to c 0 .…”