The complete separation of the two finer asymptotic ℓ_p structures for 1≤ p<∞

2021/06/23 by Spiros A. Argyros, Argyros, Spiros A., Alexandros Georgiou +5
Mathematics · Computer Science · #Advanced Banach Space Theory #Optimization and Variational Analysis #Advanced Operator Algebra Research

paper · pdf · doi:10.48550/arxiv.2106.12323

Abstract

For 1≤ p &lt;∞, we present a reflexive Banach space \mathfrakX^(p)_awi, with an unconditional basis, that admits ℓ_p as a unique asymptotic model and does not contain any Asymptotic ℓ_p subspaces. D. Freeman, E. Odell, B. Sari and B. Zheng 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 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 \mathfrakX^(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.

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