Asymptotic structure and coarse Lipschitz geometry of Banach spaces

Abstract

Abstract. In this paper, we study the coarse Lipschitz geometry of Banach spaces with several asymptotic properties. Specifically, we look at asymptotic uniform smoothness and convexity, and several distinct Banach-Saks-like properties. Among other results, we characterize the Banach spaces which are either coarsely or uniformly homeomorphic to T p 1 ⊕ . . . ⊕ T pn , where each T p j denotes the p j -convexification of the Tsirelson space, for p 1 , . . . , pn ∈ (1, . . . , ∞), and 2 ∈ {p 1 , . . . , pn}. We obtain applications to the coarse Lipschitz geometry of the p-convexifications of the Schlumprecht space, and some hereditarily indecomposable Banach spaces. We also obtain some new results on the linear theory of Banach spaces.

Equivariant Geometry of Banach Spaces and Topological Groups

“…I.e., if X is an infinitedimensional Banach space uniformly embeddable into a super-reflexive space, does it follow that X contains an isomorphic copy of ℓ 1 or an infinite-dimensional super-reflexive subspace? However, as shown by B. Braga (Corollary 4.15 [14]), Tsirelson's space T uniformly embeds into the super-reflexive space (T 2 ) T 2 without, of course, containing ℓ 1 or a super-reflexive subspace. The best one might hope for is thus some asymptotic regularity property of X.…”

On Asymptotic Uniform Smoothness and Nonlinear Geometry of Banach Spaces

“…The last statement in the theorem follows from the fact that p-AUSable is analytic (see [Bra17a], page 82). 7.1.…”

“…3 A Banach space has the alternating p-Banach-Saks property if and only if it does not contain 1 and it has the weak p-Banach-Saks property (cf. [8,Proposition 3.1]).…”

On the weak maximizing properties

“…ρ X (t) ≤ (1 + t p ) 1/p − 1). For instance, the p-convexified Tsirelson space T p (see [8]) and the Lorentz sequence space d(w, p) satisfy an upper p -estimate with constant 1 [24, p. 177]. Moreover, d(w, p) contains almost isometric copies of p so ρ d(w,p) (t) = (1 + t p ) 1/p − 1 for all t > 0.…”