Motivation: Leavitt and Cuntz Algebras

Exploring the Classification of Leavitt and Cuntz Algebras

Abstract

This essay is meant to be an exposition of the theory of Leavitt path algebras and graph C*-algebras, with an aim to discuss some current classification questions. These two classes of algebras sit on opposite sides of a mirror, each reacting aspects of the other. The majority of these notes is taken to describe the basic properties of Leavitt path algebras and graph C*-algebras, the main theme being the translation of graph-theoretic properties into exclusively (C*-)algebraic properties.A pair of well-known results in the classification of C*-algebras, due to Elliott and Kirchberg {Phillips, state that the classes of approximately ønite-dimensional (af) C*- algebras and purely infinite simple C*-algebras can be classified, up to isomorphism or Morita equivalence, by a pair of functors K0;K1 from the category of C*-algebras to category of abelian groups. Since simple graph C*-algebras must either be AF or purely infinite, combining the Elliott and Kirchberg {Phillips theorems yields a full classification of simple graph C*-algebras