Modular functionals and perturbations of Nakano spaces

We settle several questions regarding the model theory of Nakano spaces left open by the PhD thesis of Poitevin [Poi06]. 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 א0-categorical and א0-stable. In particular they are stable. Introduction 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. More specifically, he viewed Nakano spaces as continuous metric structures (in the sense of continuous logic, see [BU]) 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. In the present paper we propose to answer a few questions left open by Poitevin. • First, 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 these languages are truly distinct, i.e., whether adding the modular functional adds new structure. • Even if the naming of the modular functional does not add structure, it does give quantifier elimination in atomless Nakano spaces. While it is clear that Date: June 19, 2009. 2000 Mathematics Subject Classification. 03C98,46E30. Research initiated during the workshop “Model theory of metric structures”, American Institute of Mathematics Research Conference Centre, 18 to 22 September 2006. Research supported by ANR chaire d’excellence junior (projet THEMODMET) and by the European Commission Marie Curie Research Network ModNet. 1