The complete separation of the two finer asymptotic $\ell_{p}$ structures for $1\le p<\infty$

TL;DR

This paper constructs a reflexive Banach space with an unconditional basis that uniquely exhibits $ ext{ell}_p$ asymptotic structure without containing any asymptotic $ ext{ell}_p$ subspaces, advancing understanding of asymptotic structures in Banach spaces.

Contribution

It introduces a new Banach space with unique $ ext{ell}_p$ asymptotic behavior, using a novel norm definition based on asymptotically weakly incomparable constraints.

Findings

The space admits $ ext{ell}_p$ as a unique asymptotic model.

The space does not contain any asymptotic $ ext{ell}_p$ subspaces.

The construction answers a longstanding open problem.

Abstract

For $1\le p <\infty$, we present a reflexive Banach space $\mathfrak{X}^{(p)}_{\text{awi}}$, with an unconditional basis, that admits $\ell_p$ as a unique asymptotic model and does not contain any Asymptotic $\ell_p$ subspaces. D. Freeman, E. Odell, B. Sari and B. Zheng have shown that whenever a Banach space not containing $\ell_1$, in particular a reflexive Banach space, admits $c_0$ as a unique asymptotic model then it is Asymptotic $c_0$. These results provide a complete answer to a problem posed by L. Halbeisen and E. Odell and also complete a line of inquiry of the relation between specific asymptotic structures in Banach spaces, initiated in a previous paper by the first and fourth authors. For the definition of $\mathfrak{X}^{(p)}_{\text{awi}}$ we use saturation with asymptotically weakly incomparable constraints, a new method for defining a norm that remains small on a well-founded tree of vectors which penetrates any infinite dimensional closed subspace.