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.
Highlights
Nakano spaces are a generalisation of Lp function spaces in which the exponent p is allowed to vary as a measurable function of the underlying measure space
The PhD thesis of Pedro Poitevin [Poi06] studies Nakano spaces as Banach lattices from a model theoretic standpoint
He viewed Nakano spaces as continuous metric structures in the language of Banach lattices, possibly augmented by a predicate symbol Θ for the modular functional, showed that natural classes of such structures are elementary in the sense of continuous first order logic, and studied properties of their theories
Summary
Nakano spaces are a generalisation of Lp function spaces in which the exponent p is allowed to vary as a measurable function of the underlying measure space. The PhD thesis of Pedro Poitevin [Poi06] studies Nakano spaces as Banach lattices from a model theoretic standpoint. Poitevin studies Nakano spaces in two natural languages: that of Banach lattices, and the same augmented with an additional predicate symbol for the modular functional. It is natural to ask whether, up to small perturbations of the exponent, a complete theory of atomless Nakano spaces is א0-categorical and א0-stable. The theory of atomless Nakano spaces with a fixed essential range for the exponent function is model complete in the Banach lattice language. Up to such perturbations the theory of atomless Nakano spaces is א0-stable, and every completion thereof is א0-categorical. Appendix B contains some approximation results for the modular functional which were used in earlier versions of this paper to be superseded later by Theorem 1.10, but which might be useful
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