Hello,

The next Banach spaces webinar is on Friday June 18 at 9AM Central time. Please join us at

https://unt.zoom.us/j/83807914306

Title: On the embeddability of the family of countably Branching trees into quasi-reflexive Banach spaces Speaker: Yoël Perreau (Besançon)

Abstract. This talk will be centered on an asymptotic analogue of Bourgain's metric characterization of superreflexivity due to F. Baudier, N. Kalton and G. Lancien which can be reformulated as follows: the family $(T_N)$ of hyperbolic countably branching trees is a test family for the property $(\beta)$ of Rolewickz (also known as asymptotic superreflexivity) inside the class of reflexive Banach spaces. In other words a reflexive Banach space $X$ admits an equivalent norm with property $(\beta)$ if and only if it does not contain equi-Lipschitz the family $(T_N)$. We will explain how this result can be reformulated in terms of the values of the Szlenk index of the Banach space $X$ and of its dual space $X^*$ as written in the original paper and we will show how this characterization can be extended to the larger class of superreflexive Banach spaces. As a consequence we will answer the question of the embeddability of the family $(T_N)$ into the James space $\mathcal{J}$ and show that the non-embeddability of the family $(T_N)$ is not a sufficient condition for reflexivity.

For more information about the past and future talks, and videos please visit the webinar website http://www.math.unt.edu/~bunyamin/banach