Rigidity of Corona Algebras

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dc.contributor.advisorFarah, Ilijas
dc.creatorGhasemi, Saeed
dc.date.accessioned2015-08-28T15:44:37Z
dc.date.available2015-08-28T15:44:37Z
dc.date.copyright2015-04-13
dc.date.issued2015-08-28
dc.date.updated2015-08-28T15:44:37Z
dc.degree.disciplineMathematics & Statistics
dc.degree.levelDoctoral
dc.degree.namePhD - Doctor of Philosophy
dc.description.abstractIn 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.
dc.identifier.urihttp://hdl.handle.net/10315/30112
dc.language.isoen
dc.rightsAuthor owns copyright, except where explicitly noted. Please contact the author directly with licensing requests.
dc.subjectMathematics
dc.subjectTheoretical mathematics
dc.subject.keywordsCorona algebra
dc.subject.keywordsRigidity
dc.subject.keywordsC*-algebra
dc.subject.keywordsReduced products
dc.subject.keywordsFDD-algebras
dc.subject.keywordsIsomorphism
dc.subject.keywordsAutomorphism
dc.subject.keywordsMetric Feferman-Vaught theorem
dc.subject.keywordsSAW*-algebras
dc.titleRigidity of Corona Algebras
dc.typeElectronic Thesis or Dissertationen_US

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