Non-Stable K-Theory of an Arbitrary Graph Algebra

In 2005, Gene Abrams and Gonzalo Aranda Pino showed a way to associate an algebra to a row-finite, directed graph, which they generalized in 2008 to arbitrary countable, directed graphs. In 2007, Ara, Moreno, and Pardo computed the non-stable K-theory of a Leavitt path algebra and they also showed that this monoid satisfies the refinement property and separative cancellation. They achieved this result by showing that the non-stable K-theory of the Leavitt path algebra can be completely described by generators and relations. Another consequence of their result is that they described the ideal structure of the Leavitt path algebra. We discuss our recent results which generalize the work of Ara, Moreno, and Pardo to arbitrary countable directed graphs.  To obtain this result we defined a similar monoid that accounts for the presence of infinite emitters. We then showed that our monoid is naturally isomorphic to the monoid generated by the disingularization of an arbitrary directed graph. This allowed us to use the results of Ara, Moreno, and Pardo to obtain the desired natural isomorphism.