This is an announcement for the paper “On the complete separation of asymptotic structures in Banach spaces” by Spiros A. Argyroshttps://arxiv.org/search/math?searchtype=author&query=Argyros%2C+S+A, Pavlos Motakishttps://arxiv.org/search/math?searchtype=author&query=Motakis%2C+P.

Abstract: Let $(e_i)_i$ denote the unit vector basis of $\ell_p$, $1\leq p< \infty$, or $c_0$. We construct a reflexive Banach space with an unconditional basis that admits $(e_i)_i$ as a uniformly unique spreading model while it has no subspace with a unique asymptotic model, and hence it has no asymptotic-$\ell_p$ or $c_0$ subspace. This solves a problem of E. Odell. We also construct a space with a unique $\ell_1$ spreading model and no subspace with a uniformly unique $\ell_1$ spreading model. These results are achieved with the utilization of a new version of the method of saturation under constraints that uses sequences of functionals with increasing weights.

https://arxiv.org/abs/1902.10092