Property (β) and uniform quotient maps

Abstract

In 1999, Bates, Johnson, Lindenstrauss, Preiss and Schechtman asked whether a Banach space that is a uniform quotient of ℓp, 1 < p = 2 < ∞, must be isomorphic to a linear quotient of ℓp. We apply the geometric property (β) of Rolewicz to the study of uniform and Lipschitz quotient maps, and answer the above question positively for the case 1 < p < 2. We also give a necessary condition for a Banach space to have c0 as a uniform quotient.2010 Mathematics Subject Classification. 46B80, 46B25, 46T99.

(β)-distortion of some infinite graphs

“…Regarding the stability of the asymptotic structure of infinite-dimensional Banach spaces under nonlinear quotients, it is shown how this work unifies, and extends, a series of results from [21], [9], [31], and [10]. The quantitative approach devised in this article takes full advantage of the simple observation that the quantitative theories of biLipschitz embeddings and Lipschitz quotients coincide, in a precise sense, for trees.…”

“…Lipschitz subquotients have already been implicitly touched upon (e.g. in [25], [21], [10]). A "dual" notion was considered by Mendel and Naor in [24], where given α ∈ [1, ∞) they say that X has an α-Lipschitz quotient in Y if there is a subset S ⊂ Y and a Lipschitz quotient map f : X → S such that codist(f ) ≤ α.…”

“…Recently, the stability under nonlinear quotients of the asymptotic structure of infinite-dimensional Banach spaces has been investigated ( [21], [9], [31], [10]). The common feature of these articles is the implementation of a delicate and technical argument (or some slight variations of it) called "fork argument", which describes the behavior of a nonlinear lifting of points that are approximately in a fork configuration.…”

“…This behavior depends heavily on the asymptotic geometry of the spaces and can rule out the existence of certain nonlinear quotient maps. As explained in [21], the general idea is to built a collection of points approximately in a fork configuration in the target space whose set of pre-images contains a fork of comparable size, and then use the quantification of property (β) to get a contradiction. Our work unifies, and extends, a series of results from [21], [9], [31], and [10], which we now describe.…”

Asymptotic properties of Banach spaces and coarse quotient maps

“…Lima and Randrianarivony [13] showed that for q > p > 1, ℓ q is not a uniform quotient of ℓ p . Their proof relies on a technical argument called "fork argument".…”