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Rigidity of Corona Algebras

Rigidity of Corona Algebras

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Title: Rigidity of Corona Algebras
Author: Ghasemi, Saeed
Abstract: In this thesis we use techniques from set theory and model theory to study the isomorphisms between certain classes of C*-algebras. In particular we look at the isomorphisms between corona algebras of the form $\prod\mathbb{M}_{k(n)}(\mathbb{C})/\bigoplus \mathbb{M}_{k(n)}(\mathbb{C})$ for sequences of natural numbers $\{k(n): n\in\mathbb{N}\}$. We will show that the question ``whether any isomorphism between these C*-algebras is trivial", is independent from the usual axioms of set theory (ZFC).
We extend the classical Feferman-Vaught theorem to reduced products of metric structures. This implies that the reduced powers of elementarily equivalent structures are elementarily equivalent. We also use this to find examples of corona algebras of the form $\prod\mathbb{M}_{k(n)}(\mathbb{C}) / \bigoplus \mathbb{M}_{k(n)}(\mathbb{C})$ which are non-trivially isomorphic under the Continuum Hypothesis. This gives the first example of genuinely non-commutative structures with this property.
In chapter 6 we show that $SAW^{*}$-algebras are not isomorphic to $\nu$-tensor products of two infinite dimensional C*-algebras, for any C*-norm $\nu$. This answers a question of S. Wassermann who asked whether the Calkin algebra has this property.
Subject: Mathematics
Theoretical mathematics
Keywords: Corona algebra
Rigidity
C*-algebra
Reduced products
FDD-algebras
Isomorphism
Automorphism
Metric Feferman-Vaught theorem
SAW*-algebras
Type: Electronic Thesis or Dissertation
Rights: Author owns copyright, except where explicitly noted. Please contact the author directly with licensing requests.
URI: http://hdl.handle.net/10315/30112
Supervisor: Farah, Ilijas
Degree: PhD - Doctor of Philosophy
Program: Mathematics & Statistics
Exam date: 2015-04-13
Publish on: 2015-08-28

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