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Ilijas Farah

  1. All automorphisms of all Calkin algebras.

    Пнд, 07/26/2010 - 15:01 — SomNambul
    Authors: Ilijas Farah
    Subjects: Logic
    Abstract

    The Proper Forcing Axiom implies all automorphisms of every Calkin algebra
    associated with an infinite-dimensional complex Hilbert space and the ideal of
    compact operators are inner. As a means of the proof we introduce the notion of
    Polish $\omega_1$-trees and cohrerent families of Polish spaces and prove some
    uniformization results.

  2. The extender algebra and $\Sigma^2_1$-absoluteness.

    Втр, 05/25/2010 - 15:01 — SomNambul
    Authors: Ilijas Farah
    Subjects: Logic
    Abstract

    We present a self-contained account of Woodin's extender algebra and its use
    in proving absoluteness results, including a proof of the
    $\Sigma^2_1$-absoluteness theorem.

  3. Model theory of operator algebras II: Model theory.

    Ср, 04/07/2010 - 15:01 — SomNambul
    Authors: Ilijas Farah, Bradd Hart, David Sherman
    Subjects: Logic
    Abstract

    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 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.

  4. A dichotomy for the number of ultrapowers.

    Чт, 12/03/2009 - 16:01 — SomNambul
    Authors: Ilijas Farah, Saharon Shelah
    Subjects: Logic
    Abstract

    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 $2^{2^{\aleph_0}}$ many nonisomorphic
    ultrapowers. We prove the analogous result for metric structures, including
    C*-algebras and II$_1$ 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.

  5. Graphs and CCR algebras.

    Пт, 08/28/2009 - 23:56 — SomNambul
    Authors: Ilijas Farah
    Subjects: Operator Algebras
    Abstract

    I introduce yet another way to associate a C*-algebra to a graph and
    construct a simple nuclear C*-algebra that has irreducible representations both
    on a separable and a nonseparable Hilbert space.

  6. Model theory of operator algebras I: Stability.

    Чт, 08/20/2009 - 23:06 — SomNambul
    Authors: Ilijas Farah, Bradd Hart, David Sherman
    Subjects: Operator Algebras
    Abstract

    Several authors have considered whether the ultrapower and the relative
    commutant of a C*-algebra or II_1 factor depend on the choice of the
    ultrafilter. We show that the negative answer to each of these questions is
    equivalent to the Continuum Hypothesis, extending results of Ge-Hadwin and the
    first author.

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