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Ilijas Farah

Publications and source records attributed to Ilijas Farah.

At least 55 records · Page 3Linked to original sources
arXiv2018

The Calkin algebra is $\aleph_1$-universal

We discuss the existence of (injectively) universal C*-algebras and prove that all C*-algebras of density character $\aleph_1$ embed into the Calkin algebra, $Q(H)$. Together with other results, this shows that each of the following assertions is relatively consistent with ZFC: (i) $Q(H)$ is a $2^{\aleph_0}$-universal C*-algebra. (ii) There exists a $2^{\aleph_0}$-universal C*-algebra, but $Q(H)$ is not $2^{\aleph_0}$-universal. (iii) A $2^{\aleph_0}$-universal C*-algebra does not exist. We also prove that it is relatively consistent with ZFC that (iv) there is no $\aleph_1$-universal nuclear C*-algebra, and that (v) there is no $\aleph_1$-universal simple nuclear C*-algebra.

math.OA↗
arXiv2017

Omitting types in logic of metric structures

This paper is about omitting types in logic of metric structures introduced by Ben Yaacov, Berenstein, Henson and Usvyatsov. While a complete type is omissible in some model of a countable complete theory if and only if it is not principal, this is not true for the incomplete types by a result of Ben Yaacov. We prove that there is no simple test for determining whether a type is omissible in a model of a theory $T$ in a countable language. More precisely, we find a theory in a countable language such that the set of types omissible in some of its models is a complete $Σ^1_2$ set and a complete theory in a countable language such that the set of types omissible in some of its models is a complete $Π^1_1$ set. Two more unexpected examples are given: (i) a complete theory $T$ and a countable set of types such that each of its finite sets is jointly omissible in a model of $T$, but the whole set is not and (ii) a complete theory and two types that are separately omissible, but not jointly omissible, in its models.

math.LO↗
arXiv2017

A simple AF algebra not isomorphic to its opposite

We show that it is consistent with ZFC that there is a simple nuclear non-separable C*-algebra which is not isomorphic to its opposite algebra. We can furthermore guarantee that this example is an inductive limit of unital copies of the Cuntz algebra O_2, or of the CAR algebra.

math.OA↗
arXiv2017

A new bicommutant theorem

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 $A$.

math.OA↗
arXiv2017

Axiomatizability of the stable rank of C*-algebras

We show that the class of C*-algebras with stable rank greater than a given positive integer is axiomatizable in logic of metric structures. As a consequence we show that the stable rank is continuous with respect to forming ultrapowers of C*-algebras, and that stable rank is Kadison--Kastler stable.

math.OA↗
arXiv2016

Fraïssé limits of C*-algebras

We realize the Jiang-Su algebra, all UHF algebras, and the hyperfinite II$_{1}$ factor 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.

math.LO↗
arXiv2016

Quantifier elimination in C*-algebras

The only C*-algebras that admit elimination of quantifiers in continuous logic are $\mathbb{C}, \mathbb{C}^2$, $C($Cantor space$)$ and $M_2(\mathbb{C})$. We also prove that the theory of C*-algebras does not have model companion and show that the theory of $M_n(\mathcal {O_{n+1}})$ is not $\forall\exists$-axiomatizable for any $n\geq 2$.

math.LO↗
arXiv2016

The extender algebra and $Σ^2_1$-absoluteness

We present a self-contained account of Woodin's extender algebra and its use in proving absoluteness results, including a proof of the $Σ^2_1$-absoluteness theorem. We also include a proof that the existence of an inner model with Woodin limit of Woodin cardinals implies the existence of divergent models of $\AD^+$.

math.LO↗
arXiv2016

Relative commutants of strongly self-absorbing C*-algebras

The relative commutant $A'\cap A^{\mathcal{U}}$ of a strongly self-absorbing algebra $A$ is indistinguishable from its ultrapower $A^{\mathcal{U}}$. This applies both to the case when $A$ is the hyperfinite II$_1$ factor and to the case when it is a strongly self-absorbing C*-algebra. In the latter case we prove analogous results for $\ell_\infty(A)/c_0(A)$ and reduced powers corresponding to other filters on $\bf N$. Examples of algebras with approximately inner flip and approximately inner half-flip are provided, showing the optimality of our results. We also prove that strongly self-absorbing algebras are smoothly classifiable, unlike the algebras with approximately inner half-flip.

math.LO↗
arXiv2016

The Calkin algebra is not countably homogeneous

We show that the Calkin algebra is not countably homogeneous, in the sense of continuous model theory. We furthermore show that the connected component of the unitary group of the Calkin algebra is not countably homogeneous.

math.OA↗
arXiv2014

Rigidity of continuous quotients

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.

math.LO↗
arXiv2014

Logic and operator algebras

The most recent wave of applications of logic to operator algebras is a young and rapidly developing field. This is a snapshot of the current state of the art.

math.LO↗
arXiv2014

Automorphisms of corona algebras, and group cohomology

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 $A$ is a separable algebra which is either simple or stable, then the corona of $A$ has nontrivial automorphisms. We also discuss a connection with cohomology theory, namely, that our proof can be viewed as a computation of the cardinality of a particular derived inverse limit.

math.OA↗
arXiv2013

A nonseparable amenable operator algebra which is not isomorphic to a C*-algebra

It has been a longstanding problem whether every amenable operator algebra is isomorphic to a (necessarily nuclear) C*-algebra. In this note, we give a nonseparable counterexample. The existence of a separable counterexample remains an open problem. We also initiate a general study of unitarizability of representations of amenable groups in C*-algebras and show that our method cannot produce a separable counterexample.

math.OA↗
arXiv2013

Existentially closed II_1 factors

We examine the properties of existentially closed (R^omega-embeddable) II_1 factors. In particular, we use the fact that every automorphism of an existentially closed (R^omega-embeddable) II_1 factor is approximately inner to prove that Th(R) is not model-complete. We also show that Th(R) is complete for both finite and infinite forcing and use the latter result to prove that there exist continuum many nonisomorphic existentially closed models of Th(R).

math.OA↗
arXiv2013

Model theory of operator algebras II: Model theory

We introduce a version of logic for metric structures suitable for applications to C*-algebras and tracial von Neumann algebras. We also prove a purely model-theoretic result to the effect that the theory of a separable metric structure is stable if and only if all of its ultrapowers associated with nonprincipal ultrafilters on N are isomorphic even when the Continuum Hypothesis fails.

math.LO↗
arXiv2013

Traces, Ultrapowers and the Pedersen-Petersen C*-Algebras

Our motivating question was whether all traces on a U-ultrapower of a C*-algebra A, where U is a non-principal ultrafilter on N, are necessarily U-limits of traces on A. We show that this is false so long as A has infinitely many extremal traces, and even exhibit a 2^c size family of such traces on the ultrapower. For this to fail even when A has finitely many traces implies that A contains operators that can be expressed as sums of n+1 but not n *-commutators, for arbitrarily large n. We show that this happens for a direct sum of Pedersen-Petersen C*-algebras, and analyze some other interesting properties of these C*-algebras.

math.OA↗