Nonstable \(K\)-theory for graph algebras.
In this paper the authors compute the monoid \(V(L_K(E))\) of isomorphism classes of finitely generated projective modules over certain graph algebras \(L_K(E)\), and the authors show that this monoid satisfies the refinement property and separative cancellation. The authors also show that there is a natural isomorphism between the lattice of graded ideals of \(L_K(E)\) and the lattice of order-ideals of \(V(L_K(E))\). When \(K\) is the field \(\mathbb{C}\) of complex numbers, the algebra \(L_\mathbb{C}(E)\) is a dense subalgebra of the graph \(C^*\)-algebra \(C^*(E)\), and the authors show that the inclusion map induces an isomorphism between the corresponding monoids. As a consequence, the graph \(C^*\)-algebra of any row-finite graph turns out to satisfy the stable weak cancellation property.
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