Nonstable $K$-theory for graph algebras

Abstract

We compute the monoid $V(L_K(E))$ of isomorphism classes of finitely generated projective modules over certain graph algebras $L_K(E)$, and we show that this monoid satisfies the refinement property and separative cancellation. We 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 $C^{\ast }$-algebra $C^{\ast }(E)$, and we 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.

Final version, to appear in "Algebra and Representation Theory"

References in corpus (1)

• The Leavitt path algebra of a graph

Cited by in corpus (1)

• Purely infinite simple Leavitt path algebras