Bradd Hart

Coordinatisation and Canonical Bases in Simple Theories

In this paper we discuss several generalization of theorems from stability theory to simple theories. Cherlin and Hrushovski, in [2] develop a substitute for canonical bases in finite rank, ω-categorical supersimple theories. Motivated by methods there, we prove the existence of canonical bases (in a suitable sense) for types in any simple theory. This is done in Section 2. In general these canonical bases will (as far as we know) exist only as “hyperimaginaries”, namely objects of the form a / E where a is a possibly infinite tuple and E a type-definable equivalence relation. (In the supersimple, ω-categorical case, these reduce to ordinary imaginaries.) So in Section 1 we develop the general theory of hyperimaginaries and show how first order model theory (including the theory of forking) generalises to hyperimaginaries. We go on, in Section 3 to show the existence and ubiquity of regular types in supersimple theories, ω-categorical simple structures and modularity is discussed in Section 4. It is also shown here how the general machinery of simplicity simplifies some of the general theory of smoothly approximable (or Lie-coordinatizable) structures from [2]. Throughout this paper we will work in a large, saturated model M of a complete theory T . All types, sets and sequences will have size smaller than the size of M . We will assume that the reader is familiar with the basics of forking in simple theories as laid out in [4] and [6]. For basic stability-theoretic results concerning regular types, orthogonality etc., see [1] or [9].

Model theory of operator algebras I: stability

Several authors have considered whether the ultrapower and the relative com- mutant of a C*-algebra or II1 factor depend on the choice of the ultralter. We show that the negative answer to each of these questions is equivalent to the Continuum Hypothesis, extending results of Ge{Hadwin and the rst author.

Model theory of operator algebras III: elementary equivalence and II1 factors

We use continuous model theory to obtain several results concerning isomorphisms and embeddings between II_1 factors and their ultrapowers. Among other things, we show that for any II_1 factor M, there are continuum many nonisomorphic separable II_1 factors that have an ultrapower isomorphic to an ultrapower of M. We also give a poor mans resolution of the Connes Embedding Problem: there exists a separable II_1 factor such that all II_1 factors embed into one of its ultrapowers.

Quasiminimal structures and excellence

We show that the excellence axiom in the definition of Zilbers quasiminimal excellent classes is redundant, in that it follows from the other axioms. This substantially simplifies a number of categoricity proofs.

Models with second order properties V: A general principle

Abstract Shelah, S., C. Laflamme and B. Hart, Models with second order properties V: A general principle, Annals of Pure and Applied Logic 64 (1993) 169–194. We present a general framework for carrying out the construction in [2-10] and others of the same type. The unifying factor is a combinatorial principle which we present in terms of a game in which the first player challenges the second player to carry out constructions which would be much easier in a generic extension of the universe, and the second player cheats with the aid of ♦. Section 1 contains an axiomatic framework suitable for the description of a number of related constructions, and the statement of the main theorem 1.9 in terms of this framework. In Section 2 we illustrate the use of our combinatorial principle. The proof of the main result is then carried out in Sections 3–5.

Omitting types and AF algebras

We prove that the classes of UHF algebras and AF algebras, while not axiomatizable, can be characterized as those C*-algebras that omit certain types in the logic of metric structures.

The uncountable spectra of countable theories

Let T be a complete, flrst-order theory in a flnite or countable language having inflnite models. Let I(T;•) be the number of isomorphism types of models of T of cardinality •. We denote by „ (respectively ^ „) the number of cardinals (respectively inflnite cardinals) less than or equal to •.

Algebraic model theory

Preface. An Introduction to Independence and Local Modularity E. Bouscaren. Groups Definable in ACFA Z. Chatzidakis. Large Finite Structures with Few Types G. Cherlin. A Survey of the Uncountable Spectra of Countable Theories B. Hart, M.C. Laskowski. An Introduction to Tame Congruence Theory E.W. Kiss. Stable Finitely Homogeneous Structures: A Survey A.H. Lachlan. Homogeneous and Smoothly Approximated Structures D. Macpherson. Khovanskiis Theorem D. Marker. ACFA and the Manin-Mumford Conjecture A. Pillay. Decidable Equational Classes M.A. Valeriote. Schanuels Conjecture and the Decidability of the Real Exponential Field A.J. Wilkie. Three Lectures on the RS Problem R. Willard. Decidable Modules M. Ziegler. Index.

Fraïssé limits of C*-algebras

We realize the Jiang-Su algebra, all UHF algebras, and the hyperfinite II_1 factor as Fraisse limits of suitable classes of structures. Moreover by means of Fraisse theory we provide new examples of AF algebras with strong homogeneity properties. As a consequence of our analysis we deduce Ramsey-theoretic results about the class of full-matrix algebras.

The classification of excellent classes

In [9] and [12], Shelah defined a certain type of Scott sentence which he called excellent. He proved, among other things, that if a Scott sentence is excellent and categorical in some uncountable power then it is categorical in all uncountable powers: the analog of the Morley categoricity theorem. Proving such an analog is often the starting point in the classification of a family of classes. Before beginning this classification in the case of excellent Scott sentences, let us say a few words about what this paper is and what it is not. It is not the beginning of a classification theory for complete sentences in where is countable. Although excellence arises in the study of the model theory of Scott sentences, it is not a dividing line in a classification of them. In particular, the assumption of nonexcellence does not yield much information. In fact, in [3] there is an example of a nonexcellent Scott sentence, categorical in ℵ 1 which is. not fully categorical. It seems to the second author that a classification of sentences analogous to the classification of first order theories is a long way off and may not be accomplishable in ZFC. This is not to say that the study of excellent Scott sentences (or the class of models of such which we will call excellent classes) is unproductive. Besides its extreme usefulness in [12], Mekler and Shelah have shown that excellence plays a decisive role in the study of almost free algebras (see [7]). Moreover, as the class of ω -saturated models of an ω -stable theory is an example of an excellent class, the study of excellent classes is at least as difficult as the study of first order ω -stable theories.