Classification of graph algebras: A selective survey

This survey reports on current progress of programs to classify graph C*-algebras and Leavitt path algebras up to Morita equivalence using K-theory. Beginning with an overview and some history, we trace the development of the classification of simple and nonsimple graph C*-algebras and state theorems summarizing the current status of these efforts. We then discuss the much more nascent efforts to classify Leavitt path algebras, and we describe the current status of these efforts as well as outline current impediments that must be solved for this classification program to progress. In particular, we give two specific open problems that must be addressed in order to identify the correct K-theoretic invariant for classification of simple Leavitt path algebras, and we discuss the significance of various possible outcomes to these open problems.