Asymptotic structure and coarse Lipschitz geometry of Banach spaces

TL;DR

This paper investigates the large-scale geometric structure of Banach spaces with specific asymptotic properties, providing characterizations and applications to various classical and complex Banach spaces.

Contribution

It characterizes Banach spaces coarsely or uniformly homeomorphic to sums of Tsirelson-type spaces and explores implications for the geometry of convexified Schlumprecht and indecomposable spaces.

Findings

Characterization of spaces coarsely or uniformly homeomorphic to sums of Tsirelson spaces.

Applications to the coarse Lipschitz geometry of convexified Schlumprecht spaces.

New results on the linear theory of Banach spaces.

Abstract

In this paper, we study the coarse Lipschitz geometry of Banach spaces with several asymptotic properties. Specifically, we look at asymptotically uniformly 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 $T^{p_1}\oplus \ldots \oplus T^{p_n}$, where each $T^{p_j}$ denotes the $p_j$-convexification of the Tsirelson space, for $p_1,\ldots,p_n\in (1,\ldots, \infty)$, and $2\not\in\{p_1,\ldots ,p_n\}$. 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.