The complete separation of the two finer asymptotic $\ell_{p}$ structures for $1\le p<\infty$
| Autor: | Argyros, Spiros A., Georgiou, Alexandros, Manoussakis, Antonis, Motakis, Pavlos |
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| Rok vydání: | 2021 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 45 pages, 3 figures |
| Databáze: | arXiv |
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