Free quasi-Banach lattices
Alberto Salguero-Alarc\'on, Pedro Tradacete, Nazaret Trejo-Arroyo

TL;DR
This paper investigates free objects in quasi-Banach lattices, providing a functional representation for free p-convex p-Banach lattices and exploring their extension properties, structure, and density within free vector lattices.
Contribution
It introduces a functional representation for free p-convex p-Banach lattices generated by p-natural quasi-Banach spaces and analyzes their extension and structural properties.
Findings
Operators from Banach spaces extend to lattice homomorphisms with norm control
The space ℓ_p(Γ) is projective iff Γ is countable for 0<p<1
The free vector lattice embeds densely into the free p-convex p-Banach lattice
Abstract
We study different versions of \emph{free objects} in the setting of quasi-Banach spaces and quasi-Banach lattices. Special attention is devoted to the free -convex -Banach lattice generated by a -natural quasi-Banach space , for which we provide a functional representation by means of operators into . This representation yields, among other consequences: (1) Operators from a Banach space to any -convex quasi-Banach lattice can be extended to lattice homomorphisms with control of the norm. (2) The space is a projective -Banach lattice precisely when is countable. (3) The free vector lattice generated by sits inside as a dense sublattice.
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Taxonomy
TopicsAdvanced Banach Space Theory · Advanced Operator Algebra Research · Holomorphic and Operator Theory
