K-theory of locally finite graph $C^*$-algebras

We calculate the K-theory of the Cuntz-Krieger algebra ${\cal O}_E$ associated with an infinite, locally finite graph, via the Bass-Hashimoto operator. The formulae we get express the Grothendieck group and the Whitehead group in purely graph theoretic terms. We consider the category of finite (black-and-white, bi-directed) subgraphs with certain graph homomorphisms and construct a continuous functor to abelian groups. In this category $K_0$ is an inductive limit of $K$-groups of finite graphs, which were calculated in \cite{MM}. In the case of an infinite graph with the finite Betti number we obtain the formula for the Grothendieck group $K_0({\cal O}_E)= {\mathbb Z}^{β(E)+γ(E)},\,$ where $β(E)$ is the first Betti number and $γ(E)$ is the valency number of the graph $E$. We note, that in the infinite case the torsion part of $K_0$, which is present in the case of a finite graph, vanishes. The Whitehead group depends only on the first Betti number: $K_1({\cal O}_E)= {\mathbb Z}^{β(E)}$. These allow us to provide a counterexample to the fact, which holds for finite graphs, that $K_1({\cal O}_E)$ is the torsion free part of $K_0({\cal O}_E)$.

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