Non-asymptotic ${\ell }_1$ spaces with unique ${\ell }_1$ asymptotic model

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.

22 pages