Strong Morita Equivalence and Imprimitivity Theorems

Harmonic Analysis;Dynamical Systems;Operator Theory

Kim, Se-Jin

University of Waterloo

关键词: Harmonic Analysis;    Dynamical Systems;    Operator Theory;   

Others  :  https://uwspace.uwaterloo.ca/bitstream/10012/10854/1/Kim_Se-Jin.pdf

瑞士|英语

来源: UWSPACE Waterloo Institutional Repository

【 摘 要 】

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.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 $A times_alpha G$, there is a dual action $Hat{alpha}$ from the Pontryagin dual $Hat{G}$. Takai duality states that the iterated crossed product $(A times_alpha G) times Hat{G}$ 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.