Nonstable $K$-theory for graph algebras

Details Nonstable $K$-theory for graph algebras Nonstable $K$-theory for graph algebras

2004-12-13

arxiv.org/abs/math/0412243v3

Mathematics

Rings and Algebras

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

Scientific paper

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 $C^*$-algebra $C^*(E)$, 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.

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