# On the embeddability of the family of countably branching trees into quasi-reflexive Banach spaces

## TL;DR

This paper extends non-embeddability results of countably branching trees to quasi-reflexive Banach spaces, showing they do not embed into certain classical quasi-reflexive spaces like James space or its dual.

## Contribution

It generalizes previous non-embeddability results from reflexive to quasi-reflexive Banach spaces, including James space and its dual.

## Key findings

- Countably branching trees do not embed into James space $\\mathcal{J}_p$ for $p\in(1,\infty)$.
- Countably branching trees do not embed into the dual of James space.
- Extension of non-embeddability results to quasi-reflexive Banach spaces.

## Abstract

In this note we extend to the quasi-reflexive setting the result of F. Baudier, N. Kalton and G. Lancien concerning the non-embeddability of the family of countably branching trees into reflexive Banach spaces whose Szlenk index and Szlenk index from the dual are both equal to the first infinite ordinal $\omega$. In particular we show that the family of countably branching trees does neither embed into the James space $\mathcal{J}_p$ nor into its dual space $\mathcal{J}_p^*$ for $p\in(1,\infty)$.