Abstract We show that the following operator algebras have hyperarithmetic theory: the hyperfinite II$_1$ factor $\mathcal R$, $L(\varGamma )$ for $\varGamma $ a finitely generated group with solvable word problem, $C^*(\varGamma )$ for $\varGamma $ a finitely presented group, $C^*_\lambda (\varGamma )$ for $\varGamma $ a finitely generated group with solvable word problem, $C(2^\omega )$ and $C(\mathbb P)$ (where $\mathbb P$ is the pseudoarc). We also show that the Cuntz algebra $\mathcal O_2$ has a hyperarithmetic theory provided that the Kirchberg embedding problems have affirmative answers. Finally, we prove that if there is an existentially closed (e.c.) II$_1$ factor (resp. $\textrm{C}^*$-algebra) that does not have hyperarithmetic theory, then there are continuum many theories of e.c. II$_1$ factors (resp. e.c. $\textrm{C}^*$-algebras).

Abstract We realize the Jiang-Su algebra, all UHF algebras, and the hyperfinite II1factor as Fraïssé limits of suitable classes of structures. Moreover by means of Fraïssé 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.

Abstract In this note, we show that the theory of tracial von Neumann algebras does not have a model companion. This will follow from the fact that the theory of any locally universal, McDuff II1 factor does not have quantifier elimination. We also show how a positive solution to the Connes Embedding Problem implies that there can be no model-complete theory of II1 factors.

Abstract Let T be simple, work in Ceq over a boundedly closed set. Let p Є S(∅) be internal in a quasi-stably-embedded type-definable set Q (e.g., Q is definable or stably-embedded) and suppose (p, Q) is ACL-embedded in Q (see definitions below). Then Aut(p/Q) with its action on pc is type-definable in Ceq over ∅. In particular, if p Є S(∅) is internal in a stably-embedded type-definable set Q, and pc ⋃ Q is stably-embedded, then Aut(p/Q) is type-definable with its action on pc.

By a classifiable theory we shall mean a theory which is superstable, without the dimensional order property, which has prime models over pairs. In order to define what we mean by unique decomposition, we remind the reader of several definitions and results. We adopt the usual conventions of stability theory and work inside a large saturated model of a fixed classifiable theory T; for instance, if we write M ⊆ N for models of T, M and N we are thinking of these models as elementary submodels of this fixed saturated models; so, in particular, M is an elementary submodel of N. Although the results will not depend on it, we will assume that T is countable to ease notation. We do adopt one piece of notation which is not completely standard: if T is classifiable, M0 ⊆ Mi for i = 1, 2 are models of T and M1 is independent from M2 over M0 then we write M1 M2 for the prime model over M1 ∪ M2.

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 forma/Ewhereais a possibly infinite tuple andEa 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 modelMof a complete theoryT. All types, sets and sequences will have size smaller than the size ofM. 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].

Abstract We answer a question of Cassidy and Kolchin about the universality of the constrained closure of a differential field by working in a larger category of models.

By a variety we mean a class of algebras in a language , containing only function symbols, which is closed under homomorphisms, submodels, and products. A variety is said to be strongly abelian if for any term in , the quasi-identity holds in . In [1] it was proved that if a strongly abelian variety has less than the maximal possible uncountable spectrum, then it is equivalent to a multisorted unary variety. Using Shelah's Main Gap theorem one can conclude that if is a classifiable (superstable without DOP or OTOP and shallow) strongly abelian variety then is a multisorted unary variety. In fact, it was known that this conclusion followed from the assumption of superstable without DOP alone. This paper is devoted to the proof that the superstability assumption is enough to obtain the same structure result. This fulfills a promise made in [2]. Namely, we will prove the following Theorem 0.1. If is a superstable strongly abelian variety, then it is multisorted unary.

By a variety, we mean a class of structures in some language containing only function symbols which is equationally defined or equivalently is closed under homomorphisms, submodels and products. If K is a class of -structures then I ( K , λ) denotes the number of nonisomorphic models in K of cardinality λ. When we say that K has few models, we mean that I ( K ,λ) < 2 λ for some λ > ∣ ∣. If I ( K ,λ) = 2 λ for all λ > ∣ ∣, then we say K has many models. In [9] and [10], Shelah has shown that for an elementary class K , having few models is a strong structural condition.

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 ℵ1 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.

In the 1960's, it was conjectured that a complete first order theory in a countable language would have a nondecreasing spectrum on uncountable cardinals. This conjecture became known as Morley's conjecture. Shelah has proved this in [10]. The intent of this paper is to give a different proof which resembles a more naive way of approaching this theorem. Let I(T, λ) = the number of nonisomorphic models of T in cardinality λ. We prove: Theorem 0.1. If T is a complete countable first order theory then for ℵ0 < κ < λ, I(T,K) ≤ I(T, λ). In some sense, one can view Shelah's work on the classification of first order theories as an attack on Morley's conjecture. Over the years, he has shown that certain assumptions on a first order theory would lead to its having maximal spectrum in powers larger than the cardinality of its language (see §6 for precise references). At some point it must have seemed that Morley's conjecture would be a corollary to an exact calculation of all possible spectrums. In the end, this did not occur and, in fact, the exact spectrum functions are still not known (see [10]). Let us consider a naive approach to the proof. If we have two nonisomorphic models of the same cardinality and their cardinality is “large enough” then there should be some reason, irrespective of their cardinalities, which causes this nonisomorphism. If we could isolate this property and extend these models to a larger cardinality preserving this property, then the larger models would also be nonisomorphic. The notion of extendibility introduced in §2 is such a property which allows a version of this naive proof to work. Let us preview the sections.