Banach algebras of infinitary formulas

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Basic model theoretical concepts

In this subsection, we define several basic concepts of metric model theory that we will need throughout the thesis. These concepts are metric generalisations of concepts from classical first-order logic.
The first concept we define is satisfaction of formulas in a structure.
Definition 2.2.1. Let L be a signature and let M be an L-structure. We say that M satisfies the formula ‘(v) with the tuple a 2Mjvj if ‘M(a) = 0. This is denoted M ‘(a).
In particular, if we are given a sentence , then M satisfies if M = 0. We will denote this by M . The notions of satisfiable and complete theories are then defined as follows:
Definition 2.2.2. We say that a theory T is satisfiable if there is a structure M satisfying all sentences of T. In this case, we say that M is a model of T and write M T.
IfMis a structure, then the theory ofMis the collection of all sentences that are satisfied by M. The theory of M is denoted by Th(M). A theory T is said to be complete if there is a structure M such that T = Th(M). We can also define the concept of elementarity as follows:
Definition 2.2.3. Let L be a signature and let M and N be two L-structures.
We say that M and N are elementarily equivalent, denoted M N, if Th(M) = Th(N). If M€ N, i.e. if M is a substructure of N, and for all formulas ‘(x) and all tuples a of elements of M, we have ‘M(a) = ‘N(a), we say that M is an elementary substructure of N and that N is an elementary extension of M. We denote this by M N.
An elementary embedding of M into N is a map : M ! N such that for all formulas ‘(x) and all tuples a of elements of M, we have ‘M(a) = ‘N((a)).

The Banach algebras of formulas

In this section we introduce notions from functional analysis that will be helpful when dealing with formulas and, more importantly, uniform limits of formulas.
The collection of all formulas L(L) generated by a signature L carries the structure of a commutative unital algebra in a natural way. If we fix some enumeration v! = (v0; : : 🙂 of the variables of V (recall we assumed V to be countable), then we may consider the formulas of L(L) as uniformly continuous maps M! ! R for any L-structure M. Since multiplication and addition are continuous maps, they are connectives, and thus we can add and multiply formulas and multiply with real scalars as well. Therefore, L(L) becomes an algebra. Moreover, given any complete theory T, we can define a natural semi-norm on L(L) as follows:
Definition 2.3.1. Let L be a metric signature and let L(L) denote the formulas it generates. Let moreover T be a complete theory. Given ‘(v!) 2 L(L), we define the map k kT : L(L) ! [0;1) by k'(v!)kT = supfj’M(a)j :M T & a 2M!g: (2.3).
We will omit the variables from time to time and simply write k’kT for a formula ‘. If the theory is empty, or if it is clear from context which theory we are considering, we will just write k’k. Since any formula has bounded image, it follows that k kT only attains finite values. Moreover, it is easy to see that this map defines a semi-norm. However, there are several formulas such that k’kT = 0, so k kT is not an actual norm. Therefore, we make the following definition: Definition 2.3.2. Two formulas ‘; 2 L(L) are said to be logically equivalent with respect to T if k’ 􀀀 kT = 0.

Atomic & homogeneous structures

Two important concepts of classical model theory, that generalise nicely to the metric setting, are atomic and homogeneous structures. In this section, we study these concepts and show that an atomic structure knows the type distance, i.e. that the infimum defining the type distance between two types realised by the structure may be taken over realisations in the structure itself instead of over all models of the given theory. The concept of knowledge of @ will be important to us in Chapter 4, when we study the Roelcke completion of a Polish group as a set of types.

Elementary substructures

We move on to show generalisations of two classical theorems of model theory. These are the downwards Löwenheim-Skolem Theorem and the Tarski-Vaught Test. In order to state them properly, we need to generalise some of the concepts of finitary metric model theory to the infinitary case. What this means more precisely is that we need to make some of the definitions relative to a fragment.
First of all, embeddings, substructures, isomorphisms and automorphisms are all defined as in the finitary case (cf. Definition 2.1.6). The notion of elementarity is then defined with respect to some fragment. Suppose therefore that F is a fragment of L!1!(L).
Definition 3.2.1. We say that two structuresMand N are F-elementarily equivalent if they have the same F-theory, i.e. if it holds that for any sentence 2 F, we have M = N. We denote this by MF N. If M€ N and it holds that for all formulas ‘ 2 F we have ‘M(a) = ‘N(a).

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Banach algebras of infinitary formulas

The construction of the Banach algebras of infinitary formulas or infinitary predicates is more or less identical to what we did in the previous chapter, but we do need to fix some notation.
The semi-norm on the commutative algebra (over Q) of L!1!-formulas is defined as in Definition 2.3.1 above. Logical equivalence is then defined as in Definition 2.3.2. The associated Banach algebra is denoted F!1! or, if we consider equivalence with respect to a theory T, F!1!(T). We will still write ‘ if we want to emphasise that we are considering the class of the formula ‘. Elements of F!1!(T) are called L!1!-definable prediates or, by an abuse of language, simply formulas. If we make this construction starting with a given fragment F, the resulting algebra is denoted FF(T) or, if T is empty, simply FF.
Similarly, if we restrict ourselves to formulas with free variables among some set I V, we obtain Banach algebras denoted F!1!;I (T) and FFI (T). To simplify our notation, these algebras will also be denoted FI (T) whenever it is clear whether we consider a fragment or the full algebra. We recall that since all these algebras are commutative Banach algebras over R, it follows from [1, Theorem 2.3] that they are isometrically isomorphic to the space of continuous functions on some compact Hausdorff space. It is these spaces that we will use as our type spaces in infinitary metric model theory (cf. Definition 3.3.2 below). However, before moving on, we have the following convenient proposition, saying that the formulas with finitely many free variables are dense in L!1!(L) and therefore give us the same Banach algebras.
Proposition 3.3.1. Let L be a signature and T a complete theory. The formulas with finitely many free variables are dense inside L!1!(L) with respect to the semi-norm kkT . Hence, they give rise to the same Banach algebra over R.

Table of contents :

1 Polish Group Theory 
1.1 Polish spaces
1.2 Baire category
1.3 Polish groups
1.4 Uniform spaces
1.4.1 Uniformities on Polish groups
2 Model Theory for Metric Structures 
2.1 Languages and structures
2.1.1 Signatures
2.1.2 Metric structures
2.1.3 Syntax
2.1.4 Semantics
2.2 Basic model theoretical concepts
2.3 The Banach algebras of formulas
2.4 Type spaces
2.4.1 The logic topology
2.4.2 The type distance
2.5 Atomic & homogeneous structures
2.5.1 Knowledge of @
2.5.2 The exact homogeneous case
3 Infinitary Metric Model Theory 
3.1 Infinitary languages
3.1.1 Infinitary syntax
3.1.2 Infinitary semantics
3.2 Elementary substructures
3.3 Type spaces in infinitary logic
3.3.1 Banach algebras of infinitary formulas
3.3.2 Infinitary types
3.3.3 Connection to the usual definition
3.3.4 The infinitary type distance
3.3.5 Principality of infinitary types
3.4 Omitting types
3.5 The infinitary Ryll-Nardzewski Theorem
4 Polish Groups & Metric Model Theory 
4.1 The canonical metric structure
4.2 The Roelcke completion & model theory
4.3 The exact homogeneous case
5 The Urysohn Metric Space 
5.1 Preliminaries on U
5.2 The theory of metric spaces
5.3 Model theory of U
6 The Urysohn Diversity 
6.1 Introduction to diversties
6.2 Model theory of the Urysohn diversity
6.3 Universality of Aut(U)
6.4 A dense conjugacy class in Aut(U)
6.4.1 Fraïssé theory
6.4.2 A dense conjugacy class
6.5 Ample generics of Aut(UQ)
A Open problems 


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