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:

Lp(X) = {f : X → R:

• |f(x)|pdµ < ∞}.
• Now let p: X → [1, r] be measurable. We still define:

Lp(·)(X) = {f : X → R:

• |f(x)|p(x)dµ < ∞}.

The mapping Θp(f) =

• |f(x)|p(x)dµ is the modular.
• 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 = Θp(f)1/p.
• But: for f = 0 there is a unique constant c > 0 such that Θp(f/c) = 1, and we

define f = c.

• This is a norm on Lp(·)(X, B, µ), making it a Banach space. For constant p it agrees

with the classical Lp norm.

• Since we consider real-valued functions: Lp(·)(X, B, µ) is a Banach lattice. It is
• rder-complete and order-continuous.

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Model theory for Nakano spaces - Poitevin’s thesis Nakano spaces were first studied from a model theoretic standpoint in the PhD thesis

• f Pedro Poitevin [Poi06] (UIUC, 2006, under Ward Henson).

Classes of Nakano spaces Let K ⊆ [1, ∞) be compact (so: K ⊆ [1, r] for some r). NK = {Nakano Banach lattices with ess rng p = K} N⊆K = { . . . ⊆ K . . . } Proposition (Poitevin). If Lp(·)(X, B, µ) ∼ = Lp′(·)(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: Lp(·)(X, B, µ) ∈ NK ⇐ ⇒ ess rng p = K. Lp(·)(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). NK = {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}

ANK = {atomless members of NK}, etc. Theorem (Poitevin).

• The classes above are elementary.
• Th(AN Θ

K) admits quantifier elimination.

• If 1 /

∈ K then Th(AN Θ

K) is superstable.

It follows that Th(ANK) 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) = fp. in particular, naming Θ does

not add structure.

• The theory is ℵ0-stable (even when p = 1) and ℵ0-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(ANK) does not eliminate quantifiers, is it at least model complete?
• What about stability when 1 ∈ K?
• What about ℵ0-stability or ℵ0-categoricity?

Continuous logic Continuous logic [BU] generalises first order logic. Classical logic Continuous logic truth values {T, F} [0, 1] (0 = T) functions M n → M M n → M predicates M n → {T, F} M n → [0, 1] connectives {T, F}n → {T, F} [0, 1]n → [0, 1] quantifiers ∀x, ∃x supx, infx x = y ∈ {T, F} d(x, y) ∈ [0, 1] “equality” is true quality is a complete metric, with respect 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 nth 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: LBs = {0, −, x+y

2 ; · },

d(x, y) =

• x−y

2

• LBl = LBs ∪ {x∨y

2 , x∧y 2 },

LΘ

Bl = LBl ∪ {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 LBl-

structures.

• AN Θ

K is the class of unit balls of Nakano spaces with ess rng p = K, as LΘ 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 NK, N⊆K, AN⊆K, etc., are elementary, i.e., axioma- tised by continuous first order theories. Example: expressing atomlessness Note that |x| = x ∨ −x, which is not in our language, but 1

2|x| = x∨(−x) 2

• is. Then we

can express atomlessness by: sup

x inf y

• 1

2 1 2y −

• 1

2(1 2|x| − 1 2y)

• ∨
• 1

2(1 2 1 2y ∧ 1 2(1 2|x| − 1 2y))

• = 0

I.e.: sup

x inf y

• y − |x| − y
• ∨
• y ∧ (|x| − y)
• = 0

Or: ∀x“∃y”

• y =
• |x| − y
• &
• y ∧ (|x| − y) = 0
• Notions of definability

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Classical logic Continuous logic “definable predicate” M n → {T, F} defined by a formula. definable predicate M n → [0, 1] defined by a uniform limit

• f formulae.

definable set D “True set”: – A definable structure (e.g., a group). – A set over which one can quantify. A set such that d(¯ x, D) is a definable predicate. 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 ψi(¯ x) uniformly, ψi 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 L0-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 Recall: Θ(f) =

• |f(x)|p(x)dµ(x).
• Theorem. Th(N Θ

[⊆K]) admits an explicit LBl-definition for Θ.

Proof.

• Let N = Lp(·)(X, B, µ), fix n. Then N =

k Nk where Nk ∈ N⊆[ n+k

n , n+k+1 n

].

5

• Let f ∈ N, f = fk:

Θ(f) =

• Θ(fk) ≈
• fk

n+k+1 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. Lp(·)(X, B, µ)

• χA→χθ(A)
• ∼

=

• Lp′(·)(X′, B′, µ′)

Lp′(·)(X′, B′, µ′′)

∼ = density change

• Lθ∗p(·)(X′′, B′′, θ∗µ)

⊆

• Lp′(·)↾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(ANK) 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: Lp(·)(X) → Lp′(·)(X) f(x) → sgn(f(x))|f(x)|p(x)/p′(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) ≤ eδ a.e. then ρ

is an “ε-perturbation”:

• f − ρf
• ≤ ε,
• d(f, g+h

2 ) − d(ρf, ρg+ρh 2

)

• ≤ ε,

. . . e−εeεd(f, g)eε ≤ d(ρf, ρg) ≤ eεd(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: dp(p, q) ≤ ε ⇐ ⇒∃M, N, ρ ∈ Pertε(M, N), ¯ a ∈ M ¯ a p , ρ(¯ a) q Properties up to perturbation

• Theorem. Up to an arbitrarily small perturbation of the exponent, Th(ANK) is:
• ℵ0-categorical: For every two separable M, N ∈ ANK and every ε > 0 there exists

an ε-perturbation ρ: M → N.

• Superstable, and in fact, ℵ0-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(ANK) 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.

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References

[BBH] Ita¨ ı Ben Yaacov, Alexander Berenstein, and C. Ward Henson, Model-theoretic independence in the Banach lattices Lp(µ), submitted. [BU] Ita¨ ı Ben Yaacov and Alexander Usvyatsov, Continuous first order logic and local stability, submitted. [Poi06] Luis P. Poitevin, Model theory of Nakano spaces, Ph.D. thesis, University of Illinois at Urbana-Champaign, 2006. 8