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

Spiros A. Argyros; Alexandros Georgiou; Antonis Manoussakis; Pavlos Motakis

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arXiv · 2106.12323

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

Spiros A. Argyros·Alexandros Georgiou·Antonis Manoussakis·Pavlos Motakis

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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.

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Spiros A. Argyros, Alexandros Georgiou, Antonis Manoussakis, Pavlos Motakis. 2021-06-23. The complete separation of the two finer asymptotic $\ell_{p}$ structures for $1\le p<\infty$. https://arxiv.org/abs/2106.12323

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