dc.contributor.author Kim, Se-Jin dc.date.accessioned 2016-09-15 15:26:46 (GMT) dc.date.available 2016-09-15 15:26:46 (GMT) dc.date.issued 2016-09-15 dc.date.submitted 2016-09-07 dc.identifier.uri http://hdl.handle.net/10012/10854 dc.description.abstract The purpose of this thesis is to give an exposition of two topics, mostly following the books \cite{R & W} and \cite{Wil}. First, we wish to investigate crossed product $C^*$-algebras in its most general form. Crossed product $C^*$-algebras are $C^*$-algebras which encode information about the action of a locally compact Hausdorff group $G$ as automorphisms on a $C^*$-algebra $A$. One of the prettiest example of such a dynamical system that I have observed in the wild arises in the gauge-invariant uniqueness theorem \cite{Rae}, which assigns to every $C^*$-algebra $C^*(E)$ associated with a graph $E$ a \emph{gauge action} of the unit circle $\T$ to automorphisms on $C^*(E)$. Group $C^*$-algebras also arise as a crossed product of a dynamical system. I found crossed products in its most general form very abstract and much of its constructions motivated by phenomena in a simpler case. Because of this, much of the initial portion of this exposition is dedicated to the action of a discrete group on a unital $C^*$-algebra, where most of the examples are given. en I must admit that I find calculations of crossed products when one has an indiscrete group $G$ acting on our $C^*$-algebra daunting except under very simple cases. This leads to our second topic, on imprimitivity theorems of crossed product $C^*$-algebras. Imprimitivity theorems are machines that output (strong) Morita equivalences between crossed products. Morita equivalence is an invariant on $C^*$-algebras which preserve properties like the ideal structure and the associated $K$-groups. For example, no two commutative $C^*$-algebras are Morita equivalent, but $C(X) \otimes M_n$ is Morita equivalent to $C(X)$ whenever $n$ is a positive integer and $X$ is a compact Hausdorff space. Notice that Morita equivalence can be used to prove that a given $C^*$-algebra is simple. All this leads to our concluding application: Takai duality. The set-up is as follows: we have an action $\alpha$ of an abelian group $G$ on a $C^*$-algebra $A$. On the associated crossed product $A \rtimes_\alpha G$, there is a dual action $\Hat{\alpha}$ from the Pontryagin dual $\Hat{G}$. Takai duality states that the iterated crossed product $(A \rtimes_\alpha G) \rtimes \Hat{G}$ is isomorphic to $A \otimes \calK(L^2(G))$ in a canonical way. This theorem is used to show for example that all graph $C^*$-algebras are nuclear or to establish theorems on the $K$-theory on crossed product $C^*$-algebras. dc.language.iso en en dc.publisher University of Waterloo en dc.subject Harmonic Analysis en dc.subject Dynamical Systems en dc.subject Operator Theory en dc.title Strong Morita Equivalence and Imprimitivity Theorems en dc.type Master Thesis en dc.pending false uws-etd.degree.department Pure Mathematics en uws-etd.degree.discipline Pure Mathematics en uws-etd.degree.grantor University of Waterloo en uws-etd.degree Master of Mathematics en uws.contributor.advisor Davidson, Kenneth uws.contributor.affiliation1 Faculty of Mathematics en uws.published.city Waterloo en uws.published.country Canada en uws.published.province Ontario en uws.typeOfResource Text en uws.peerReviewStatus Unreviewed en uws.scholarLevel Graduate en
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