Model theory of Nakano spaces Ita Ben Yaacov February 2007 Nakano - PDF document

Model theory of Nakano spaces Ita¨ ı Ben Yaacov February 2007 Nakano spaces • Let ( X, B , µ ) be a measure space, p ∈ [1 , ∞ ) constant. Then: � | f ( x ) | p dµ < ∞} . L p ( X ) = { f : X → R : • Now let p : X → [1 , r ] be measurable. We still define: � | f ( x ) | p ( x ) dµ < ∞} . L p ( · ) ( X ) = { f : X → R : � | f ( x ) | p ( x ) dµ is the modular . The mapping Θ p ( f ) = • We define the essential range of p : ess rng( p ) = { t ∈ R : ( ∀ open U ∋ t )( µ ( { p ∈ U } ) > 0) } . It is a compact subset of [1 , r ]. • Why p ( x ) ≤ r < ∞ ? Allowing p ( x ) to be unbounded would have been just as bad as allowing p = ∞ . The norm in Nakano spaces • It does not make sense to define � f � = Θ p ( f ) 1 /p . • But: for f � = 0 there is a unique constant c > 0 such that Θ p ( f/c ) = 1, and we define � f � = c . • This is a norm on L p ( · ) ( X, B , µ ), making it a Banach space. For constant p it agrees with the classical L p norm. • Since we consider real-valued functions: L p ( · ) ( X, B , µ ) is a Banach lattice. It is order-complete and order-continuous . 1

Model theory for Nakano spaces - Poitevin’s thesis Nakano spaces were first studied from a model theoretic standpoint in the PhD thesis of Pedro Poitevin [Poi06] (UIUC, 2006, under Ward Henson). Classes of Nakano spaces Let K ⊆ [1 , ∞ ) be compact (so: K ⊆ [1 , r ] for some r ). N K = { Nakano Banach lattices with ess rng p = K } N ⊆ K = { . . . ⊆ K . . . } Proposition (Poitevin) . If L p ( · ) ( X, B , µ ) ∼ = L p ′ ( · ) ( X ′ , B ′ , µ ′ ) (isometrically!) as Banach lattices, and ( X, B , µ ) is not reduced to a single atom, then ess rng p = ess rng p ′ . In other words: L p ( · ) ( X, B , µ ) ∈ N K ⇐ ⇒ ess rng p = K. L p ( · ) ( X, B , µ ) ∈ N ⊆ K ⇐ ⇒ ess rng p ⊆ K. Theories of Nakano spaces Let K ⊆ [1 , ∞ ) be compact (so: K ⊆ [1 , r ] for some r ). N K = { Nakano Banach lattices with ess rng p = K } N ⊆ K = { . . . ⊆ K . . . } N Θ K , N Θ ⊆ K = { Same structures, augmented by a symbol for Θ } AN K = { atomless members of N K } , etc. Theorem (Poitevin) . • The classes above are elementary. • Th( AN Θ K ) admits quantifier elimination. ∈ K then Th( AN Θ • If 1 / K ) is superstable. It follows that Th( AN K ) is superstable, as a reduct. Further questions In case p is constant (B., Berenstein, Henson [BBH]): • Θ is definable in the lattice structure: Θ( f ) = � f � p . in particular, naming Θ does not add structure. • The theory is ℵ 0 -stable (even when p = 1) and ℵ 0 -categorical. 2

Question. In the general case: • Does naming Θ add structure? I.e., is Θ definable in the Banach lattice language? • Even when Th( AN K ) does not eliminate quantifiers, is it at least model complete? • What about stability when 1 ∈ K ? • What about ℵ 0 -stability or ℵ 0 -categoricity? Continuous logic Continuous logic [BU] generalises first order logic. Classical logic Continuous logic { T, F } truth values [0 , 1] (0 = T ) M n → M M n → M functions M n → { T, F } M n → [0 , 1] predicates { T, F } n → { T, F } [0 , 1] n → [0 , 1] connectives quantifiers ∀ x , ∃ x sup x , inf x x = y ∈ { T, F } d ( x, y ) ∈ [0 , 1] is a complete metric, with respect “equality” is true quality to which symbols are uniformly continuous Describes classes of complete (bounded) metric spaces. Banach spaces Banach spaces are unbounded metric structures. We have several options: • Consider a multi-sorted structure: the n th sort consists of ¯ B (0 , n ). • Work inside the closed unit ball ¯ B (0 , 1). • “Add a point at infinity.” We prefer to work in the unit ball. The appropriate languages are: L Bs = { 0 , − , x + y � x − y � � 2 ; � · �} , d ( x, y ) = � 2 L Bl = L Bs ∪ { x ∨ y 2 , x ∧ y 2 } , L Θ Bl = L Bl ∪ { Θ } Note that in a Nakano space: Θ( f/ � f � ) = 1, so � f � ≤ 1 ⇐ ⇒ Θ( f ) ≤ 1. 3

Axiomatisability • N ⊆ K is the class of unit balls of Nakano spaces with ess rng p ⊆ K , as L Bl - structures. • AN Θ K is the class of unit balls of Nakano spaces with ess rng p = K , as L Θ Bl - structures. • Etc. A theory is a set of conditions of the form ϕ = 0 (or ϕ ≤ s , ϕ ≥ s . . . ) where ϕ is a sentence. Theorem (Poitevin) . The classes N K , N ⊆ K , AN ⊆ K , etc., are elementary, i.e., axioma- tised by continuous first order theories. Example: expressing atomlessness 2 | x | = x ∨ ( − x ) Note that | x | = x ∨ − x , which is not in our language, but 1 is. Then we 2 can express atomlessness by: � � � 1 1 � 1 � 2 ( 1 2 | x | − 1 � sup x inf 2 � y � − 2 y ) � � � 2 � y � = 0 � � 1 2 ( 1 1 2 y ∧ 1 2 ( 1 2 | x | − 1 � ∨ 2 y )) 2 I.e.: � ∨ � = 0 � � � � sup x inf � � y � − �| x | − y � � y ∧ ( | x | − y ) y Or: � � � � � � ∀ x “ ∃ y ” � y � = � | x | − y & y ∧ ( | x | − y ) = 0 � Notions of definability 4

Classical logic Continuous logic definable predicate “definable predicate” M n → [0 , 1] M n → { T, F } defined by a uniform limit defined by a formula. of formulae. definable set D “True set”: – A definable structure A set such that d (¯ x, D ) is (e.g., a group). a definable predicate. – A set over which one can quantify. Quantifier elimination Definition. A theory T eliminates quantifiers if every formula is a quantifier-free defin- able predicate in model of T : ϕ (¯ x ) = lim ψ i (¯ x ) uniformly, ψ i quantifier free. Theorem (Poitevin) . Th( AN Θ K ) eliminates quantifiers. Definability of symbols Definition. Let L ′ = L ∪ { P } , T a L ′ -theory. • T admits an explicit L -definition for P if P coincides with a L 0 -definable predicate in models of T . • T admits an implicit L -definition for P if a L -structure admits at most one expan- sion into a L ′ -structure modelling T . Theorem (Beth’s Theorem for continuous logic) . A theory T admits an explicit L - definition for P if and only if it admits an implicit one. Definability of the modular � | f ( x ) | p ( x ) dµ ( x ). Recall: Θ( f ) = Theorem. Th( N Θ [ ⊆ K ] ) admits an explicit L Bl -definition for Θ . • Let N = L p ( · ) ( X, B , µ ), fix n . Then N = � Proof. k N k where N k ∈ N ⊆ [ n + k ] . n , n + k +1 n 5

� � � � � � • Let f ∈ N , f = � f k : n + k +1 � � Θ( f ) = Θ( f k ) ≈ � f k � . n • The decomposition is respected by isomorphisms of Banach lattices. It follows that an isomorphism of Banach lattices respects Θ. Model completeness – understanding mappings Theorem. L p ′ ( · ) ( X ′ , B ′ , µ ′ ) L p ( · ) ( X, B , µ ) � � � χ A �→ χ θ ( A ) � � � � � � � ∼ � � = density change � � � � � � � � � � � � � L p ′ ( · ) ( X ′ , B ′ , µ ′′ ) ∼ = � � � ⊆ � � � � � � ⊆ � � � � � � � � � � � � � � L θ ∗ p ( · ) ( X ′′ , B ′′ , θ ∗ µ ) L p ′ ( · ) ↾ X ′′ ( X ′′ , B ′′ , µ ′′ ↾ B ′′ ) Where ( X ′′ , B ′′ ) = θ ( X, B ) ⊆ ( X ′ , B ′ ) . Lemma (long and technical. . . ) . Under the conditions above (and ( X, B , µ ) atomless): θ ∗ p = p ′ ↾ X ′′ and θ ∗ µ = µ ′′ ↾ B ′′ . All these applications respect Θ . Lots of nice consequences Corollary. Th( AN K ) is model complete. Corollary. Th( AN ⊆ K ) is inductive. Corollary. Alternative proof for definability of Θ in atomless Nakano spaces. Question. Extend the technical lemma to spaces with atoms. Pertubations of the exponent Let p, p ′ : ( X, B , µ ) → [1 , r ]. ρ : L p ( · ) ( X ) → L p ′ ( · ) ( X ) sgn( f ( x )) | f ( x ) | p ( x ) /p ′ ( x ) f ( x ) �→ 6

• ρ respects: Θ, 0, − x , x ∨ y , x ∧ y , x ≤ y . In particular: a bijection of the unit balls. • Does not respect: +, x + y 2 , x ∧ y 2 , � x � , d ( x, y ). Theorem. For every ε > 0 there is δ > 0 such that if e − δ ≤ p ( x ) /p ′ ( x ) ≤ e δ a.e. then ρ is an “ ε -perturbation”: � ≤ ε, � � � � f � − � ρf � � ≤ ε, � � d ( f, g + h 2 ) − d ( ρf, ρg + ρh � ) 2 . . . e − εe ε d ( f, g ) e ε ≤ d ( ρf, ρg ) ≤ e ε d ( f, g ) e − ε . Note: A composition of an ε -perturbation and an ε ′ -perturbation is an ( ε + ε ′ )- perturbation. We obtain a notion of perturbation distance between types: d p ( p, q ) ≤ ε ⇐ ⇒∃ M, N, ρ ∈ Pert ε ( M, N ) , ¯ a ∈ M ¯ a � p , ρ (¯ a ) � q Properties up to perturbation Theorem. Up to an arbitrarily small perturbation of the exponent, Th( AN K ) is: • ℵ 0 -categorical: For every two separable M, N ∈ AN K and every ε > 0 there exists an ε -perturbation ρ : M → N . • Superstable, and in fact, ℵ 0 -stable (even if 1 ∈ K ): In the metric of moving and/or perturbing realisations there are separably many types over separable sets. Equiva- lently: metric Cantor-Bendixson ranks (i.e., Morley ranks) exist on type spaces. Corollary. Th( AN K ) is complete and stable. Further questions • Generalisation to Musielak-Orlicz lattices: We extend the family of functions x �→ | x | p to include other convex functions. • Extend the “big technical lemma” to spaces with atoms. 7