We prove an analogue of Voiculescu’s theorem: Relative bicommutant of a separable unital subalgebra A of an ultraproduct of simple unital C*-algebras is equal to A. Ultrapowers1 AU of separable algebras are, being subject to well-developed model-theoretic methods, reasonably well-understood (see e.g. [9, Theorem 1.2] and §2). Since the early 1970s and the influential work of McDuff and Connes c...

The Connes embedding problem (CEP) is a in the theory of tracial von Neumann algebras and asks whether or not every algebra embeds into an ultrapower hyperfinite II 1 <mml:annotation encoding="application/x-...

In the first edition of Classification Theory, the second author characterized the stable theories in terms of saturation of ultrapowers. Prior to this theorem, stability had already been defined in terms of counting types, and the unstable formula theorem was known. A contribution of the ultrapower characterization was that it involved sorting out the global theory, and introducing nonforking,...

We investigate the structure of rank-to-rank elementary embeddings at successor rank, working in $\mathrm{ZF}$ set theory without Axiom Choice. Recall that set-theoretic universe is naturally stratified by cumulative hierarchy, whose levels $V\_\alpha$ are defined via iterated application power operation, starting from $V\_0=\emptyset$, setting $V\_{\alpha+1}=\mathcal{P}(V\_\alpha)$, and taking...

Among the results of this paper are the following: 1. Every Boolean (u1tra)power is the union of an updirected elementary family of direct ultrapowers. 2. Under certain conditions, a finitely iterated Boolean ultrapower is isomorphic to a single Boolean ultrapower. 3. A w-bounded filtral power is an elementary substructure of a filtral power. 4. Let . X be an elementary class closed under updir...

We study solutions to the free stochastic differential equation dXt = dSt − 1 2DV (Xt)dt, where V is a locally convex polynomial potential in m non-commuting variables. We show that for self-adjoint V , the law μV of a stationary solution is the limit law of a random matrix model, in which an m-tuple of self-adjoint matrices are chosen according to the law exp(−NTr(V (A1, . . . , Am)))dA1 · · ·...

Fine’s influential Canonicity Theorem states that if a modal logic is determined by a first-order definable class of Kripke frames, then it is valid in its canonical frames. This article reviews the background and context of this result, and the history of its impact on further research. It then develops a new characterisation of when a logic is canonically valid, providing a precise point of d...

Historically one of the great successes of model theory has been Shelah’s stability theory: a program, described in [17], of showing that the arrangement of first-order theories into complexity classes according to a priori set-theoretic criteria (e.g. counting types over sets) in fact pushes down to reveal a very rich and entirely model-theoretic structure theory for the classes involved: what...

We present unified proofs of several properties of the corona of σ-unital C*-algebras such as AA-CRISP, SAW*, being sub-σ-Stonean in the sense of Kirchberg, and the conclusion of Kasparov’s Technical Theorem. Although our results were obtained by considering C*algebras as models of the logic for metric structures, the reader is not required to have any knowledge of model theory of metric struct...

This paper, a continuation of [2]–[8], aims to carry on the project of establishing model-theoretic concepts and methods within the topological context; namely that of compacta (i.e., compact Hausdorff spaces). Since there is a precise duality between the categories of compacta (plus continuous maps) and commutative B-algebras (plus nonexpansive linear maps) (the Gel’fand-Năımark theorem [21]),...