Ilijas Farah
All automorphisms of all Calkin algebras.
Mon, 07/26/2010 - 15:01 — SomNambulAuthors: Ilijas Farah
Subjects: Logic
AbstractThe 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.The extender algebra and $\Sigma^2_1$-absoluteness.
Tue, 05/25/2010 - 15:01 — SomNambulAuthors: Ilijas Farah
Subjects: Logic
AbstractWe 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.Model theory of operator algebras II: Model theory.
Wed, 04/07/2010 - 15:01 — SomNambulAuthors: Ilijas Farah, Bradd Hart, David Sherman
Subjects: Logic
AbstractWe 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.A dichotomy for the number of ultrapowers.
Thu, 12/03/2009 - 16:01 — SomNambulAuthors: Ilijas Farah, Saharon Shelah
Subjects: Logic
AbstractWe 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.Graphs and CCR algebras.
Fri, 08/28/2009 - 23:56 — SomNambulAuthors: Ilijas Farah
Subjects: Operator Algebras
AbstractI 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.Model theory of operator algebras I: Stability.
Thu, 08/20/2009 - 23:06 — SomNambulAuthors: Ilijas Farah, Bradd Hart, David Sherman
Subjects: Operator Algebras
AbstractSeveral 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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