Ilijas Farah

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.

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.

A DICHOTOMY FOR THE NUMBER OF ULTRAPOWERS

We prove a strong dichotomy for the number of ultrapowers of a given countable model associated with nonprincipal ultrafilters on N. They are either all isomorphic, or else there are

RIGIDITY OF CONTINUOUS QUOTIENTS

2^{2^{\aleph_0}}

Automorphisms of corona algebras, and group cohomology

many nonisomorphic ultrapowers. We prove the analogous result for metric structures, including C*-algebras and II

A Dichotomy for the Mackey Borel Structure

_1

Four and more

factors, as well as their relative commutants and include several applications. We also show that the C*-algebra B(H) always has nonisomorphic relative commutants in its ultrapowers associated with nonprincipal ultrafilters on N.

A NONSEPARABLE AMENABLE OPERATOR ALGEBRA WHICH IS NOT ISOMORPHIC TO A C -ALGEBRA

We study countable saturation of the metric reduced products and introduce continuous fields of metric models indexed by locally compact, separable, completely metrizable spaces. Saturation of the reduced product depends both on the underlying index space and the model. By using the Gelfand--Naimark duality we conclude that the assertion that the \vCech--Stone remainder of the half-line has only trivial automorphisms is independent from ZFC. The consistency of this statement follows from Proper Forcing Axiom and this is the first known example of a connected space with this property.

Analytic Hausdorff gaps II: The density zero ideal

In 2007 Phillips and Weaver showed that, assuming the Continuum Hypothesis, there exists an outer automorphism of the Calkin algebra. (The Calkin algebra is the algebra of bounded operators on a separable complex Hilbert space, modulo the compact operators.) In this paper we establish that the analogous conclusion holds for a broad family of quotient algebras. Specifically, we will show that assuming the Continuum Hypothesis, if