Bratteli Diagrams - Modeling AF-algebras and Cantor Minimal Systems Using Infinite Graphs
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This thesis gives a thorough introduction to the infinite graphs known as Bratteli diagrams and their two most common uses - namely as a means of representing AF-algebras in a combinatorial way and as a way of modeling Cantor minimal systems. The thesis is therefore naturally divided into two parts. In the first part, the machinery needed study AF-algebras is introduced, in particular the structure theorem for finite-dimensional C*-algebras is proved from first principles, and direct limits of finite-dimensional C*-algebras are constructed. We show how an AF-algebra can be represented by a Bratteli diagram and how information about the AF-algebra may be extracted from its Bratteli diagram. In particular we demonstrate how the ideals of an AF-algebra may be read off its Bratteli diagram and also how the Bratteli diagrams of isomorphic AF-algebras are related. Some classic examples of AF-algebras are given, and their Bratteli diagrams are computed and used to illustrate the general theory. In the second part, ordered Bratteli diagrams are introduced and we construct the associated Cantor minimal systems, known as Bratteli-Vershik systems. The associated dimension groups are also briefly introduced. We give the full proof of the model theorem for Cantor minimal systems.