Asymptotic structure and coarse Lipschitz geometry of Banach spaces

B. M. Braga¹

¹ Department of Mathematics, Statistics, and Computer Science (M/C 249) University of Illinois at Chicago 851 S. Morgan St. Chicago, IL 60607-7045, U.S.A.

Studia Mathematica, Tome 237 (2017) no. 1, pp. 71-97

Résumé

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. We characterize the Banach spaces which are either coarsely or uniformly homeomorphic to $T^{p_1}\oplus \cdots \oplus T^{p_n}$, 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 in the linear theory of Banach spaces.

Détail

DOI : 10.4064/sm8604-11-2016

Keywords: study coarse lipschitz geometry banach spaces several asymptotic properties specifically look asymptotic uniform smoothness convexity several distinct banach saks like properties characterize banach spaces which either coarsely uniformly homeomorphic oplus cdots oplus where each denotes j convexification tsirelson space ldots ldots infty ldots obtain applications coarse lipschitz geometry p convexifications schlumprecht space hereditarily indecomposable banach spaces obtain results linear theory banach spaces