Motivation: Leavitt and Cuntz Algebras: Exploring the Classification of Leavitt and Cuntz Algebras (Journal of Advances and Scholarly Researches in Allied Education, 2017)
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