Non-asymptotic $\ell_1$ spaces with unique $\ell_1$ asymptotic model

TL;DR

This paper investigates the structure of Banach spaces with unique asymptotic models, showing that replacing $c_0$ with $ell_1$ invalidates previous results and constructing a reflexive space with a unique $ell_1$ asymptotic model.

Contribution

It demonstrates that the previous characterization for $c_0$ asymptotic models does not extend to $ell_1$, and constructs a reflexive space with a unique $ell_1$ asymptotic model.

Findings

Replacing $c_0$ with $ell_1$ invalidates the previous result.

Existence of a reflexive Banach space with a unique $ell_1$ asymptotic model.

Any subsequence of the basis generates a non-Asymptotic $ell_1$ subspace.

Abstract

A recent result of Freeman, Odell, Sari, and Zheng states that whenever a separable Banach space not containing $\ell_1$ has the property that all asymptotic models generated by weakly null sequences are equivalent to the unit vector basis of $c_0$ then the space is Asymptotic $c_0$. We show that if we replace $c_0$ with $\ell_1$ then this result is no longer true. Moreover, a stronger result of B. Maurey - H. P. Rosenthal type is presented, namely, there exists a reflexive Banach space with an unconditional basis admitting $\ell_1$ as a unique asymptotic model whereas any subsequence of the basis generates a non-Asymptotic $\ell_1$ subspace.