Non-stable $K$-theory for Leavitt path algebras

Abstract

We compute the monoid $\mathcal{V} [ L_{K} (E) ]$ of isomorphism classes of finitely generated projective modules of a Leavitt path algebra over an arbitrary directed graph. Our result generalizes the result of Ara, Moreno and Pardo in which they computed the monoid $\mathcal{V}[ L_{K} (E) ]$ of a Leavitt path algebra over a countable row-finite directed graph.