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Date
2024-03Type
- Report
ETH Bibliography
yes
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Abstract
We study the free Banach lattice FBL(p,∞)[E] with upper p-estimates generated by a Banach space E. Using a classical result of Pisier on factorization through Lp,∞(μ) together with a finite dimensional reduction, it is shown that the spaces ℓp,∞(n) witness the universal property of FBL(p,∞)[E] isomorphically. As a consequence, we obtain a functional representation for FBL(p,∞)[E], answering a previously open question. More generally, our proof allows us to identify the norm of any free Banach lattice over E associated with a rearrangement invariant function space. After obtaining the above functional representation, we take the first steps towards analyzing the fine structure of FBL(p,∞)[E]. Notably, we prove that the norm for FBL(p,∞)[E] cannot be isometrically witnessed by Lp,∞(μ) and settle the question of characterizing when an embedding between Banach spaces extends to a lattice embedding between the corresponding free Banach lattices with upper p-estimates. To prove this latter result, we introduce a novel push-out argument, which when combined with the injectivity of ℓp allows us to give an alternative proof of the subspace problem for free p-convex Banach lattices. On the other hand, we prove that ℓp,∞ is not injective in the class of Banach lattices with upper p-estimates, elucidating one of many difficulties arising in the study of FBL(p,∞)[E]. Show more
Publication status
publishedExternal links
Journal / series
SAM Research ReportVolume
Publisher
Seminar for Applied Mathematics, ETH ZurichOrganisational unit
09603 - Alaifari, Rima / Alaifari, Rima
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yes
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