20 research outputs found
Strong Morita Equivalence and Imprimitivity Theorems
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 -algebras in its most general form. Crossed product -algebras are -algebras which encode information about the action of a locally compact Hausdorff group as automorphisms on a -algebra . 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 -algebra associated with a graph a \emph{gauge action} of the unit circle \T to automorphisms on . Group -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 -algebra, where most of the examples are given.
I must admit that I find calculations of crossed products when one has an indiscrete group acting on our -algebra daunting except under very simple cases. This leads to our second topic, on imprimitivity theorems of crossed product -algebras. Imprimitivity theorems are machines that output (strong) Morita equivalences between crossed products. Morita equivalence is an invariant on -algebras which preserve properties like the ideal structure and the associated -groups. For example, no two commutative -algebras are Morita equivalent, but is Morita equivalent to whenever is a positive integer and is a compact Hausdorff space. Notice that Morita equivalence can be used to prove that a given -algebra is simple.
All this leads to our concluding application: Takai duality. The set-up is as follows: we have an action of an abelian group on a -algebra . On the associated crossed product , 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 -algebras are nuclear or to establish theorems on the -theory on crossed product -algebras