Modular functionals and perturbations of Nakano spaces

Abstract

We settle several questions regarding the model theory of Nakano spaces left open by the PhD thesis of Pedro Poitevin \cite{Poitevin:PhD}. We start by studying isometric Banach lattice embeddings of Nakano spaces, showing that in dimension two and above such embeddings have a particularly simple and rigid form. We use this to show show that in the Banach lattice language the modular functional is definable and that complete theories of atomless Nakano spaces are model complete. We also show that up to arbitrarily small perturbations of the exponent Nakano spaces are ${\aleph }_0$-categorical and ${\aleph }_0$-stable. In particular they are stable.

References in corpus (3)

• Topometric spaces and perturbations of metric structures
• On perturbations of continuous structures
• Model-theoretic independence in the Banach lattices $L_p(\mu )$

Cited by in corpus (9)

• On theories of random variables
• Topometric spaces and perturbations of metric structures
• Lipschitz functions on topometric spaces
• On perturbations of Hilbert spaces and probability algebras with a generic automorphism
• ${\aleph }_0$-categorical Banach spaces contain ${\ell }_p$ or $c_0$
• Model Theory of R-trees
• On uniform canonical bases in $L_p$ lattices and other metric structures
• Model theory of a non-degenerate representation of a unital C*-algebra
• Forking and stability in the representations of a C*-algebra