# 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).

### 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; : : đź™‚ 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’ ô€€€ 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 inÂ  of a Polish group as a set of types. Homogeneity of metric structures is defined as follows:
Definition 2.5.1. A metric structure M is approximately homogeneous or simply homogeneous if for any n 2 N and any two n-tuples a;b 2Mn of the same type, we have Aut(M) a = Aut(M) b.

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Elementary substructures

We move on to show generalisations of two classical theorems of model theory. These are the downwards LĂ¶wenheim-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 MÂ€ N and it holds that for all formulas ‘ 2 F we have ‘M(a) = ‘N(a).

IntroductionÂ
1 Polish Group TheoryÂ
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 StructuresÂ
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 TheoryÂ
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 TheoryÂ
4.1 The canonical metric structure
4.2 The Roelcke completion & model theory
4.3 The exact homogeneous case
5 The Urysohn Metric SpaceÂ
5.1 Preliminaries on U
5.2 The theory of metric spaces
5.3 Model theory of U
6 The Urysohn DiversityÂ
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)