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