The complete separation of the two finer asymptotic ℓp structures for 1≤p<∞The complete separation of the two finer asymptotic ℓp structures for 1≤p<∞ Peer-Reviewed Journal Publication Δημοσίευση σε Περιοδικό με Κριτές 2024-02-232022enFor 1 ≤ 𝑝 < ∞, we present a reflexive Banach space 𝔛( 𝑝)awi , with an unconditional basis, that admits ℓ𝑝 as a unique asymptotic model and does not contain any Asymptotic ℓ𝑝 subspaces. Freeman et al., Trans. AMS. 370 (2018), 6933–6953 have shown that whenever a Banach space not containing ℓ1, in particular a reflexive Banach space, admits 𝑐0 as a unique asymptotic model, then it is Asymptotic 𝑐0. These results provide a complete answer to a problem posed by Halbeisen and Odell [Isr. J. Math. 139 (2004), 253–291] 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 𝔛 ( 𝑝)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.http://creativecommons.org/licenses/by/4.0/Forum of Mathematics, Sigma10Argyros_et_al_Forum Math. Sigma_10_2022.pdfChania [Greece]Library of TUC2024-02-23application/pdf918.2 kBfree Argyros, Spiros, 1950- Georgiou Alexandros Manousakis Antonios Μανουσακης Αντωνιος Motakis, Pavlos Cambridge University Press Asymptotic structures