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

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

2010-07-22

arxiv.org/abs/1007.3952v3

Mathematics

Operator Algebras

V3: calculations of Whitehead group added

Scientific paper

We calculate the K-theory of Cuntz-Krieger algebras associated to locally finite infinite graphs via the Bass-Hashimoto operator. The formula we get express the Grothendieck group and Whitehead group in purely graph theoretic terms. We consider the category of finite (black-and-white, double) subgraphs with certain graph homomorphisms and construct a continuous functor to abelian groups. This allows to present $K_0$ as an inductive limit of $K$-groups of finite graphs, which are calculated in \cite{MM}. Then we specify this construction in the case of an infinite graph with finite Betti number and obtain a formula $K_0({\cal O}_E)= {\mathbb Z}^{\beta(E)+\gamma(E)},\,$ where $\beta(E)$ is the first Betti number and $\gamma(E)$ is the branching number of the graph $E$. We note, that in the infinite case the torsion part of the group, which is present in the case of finite graph, vanishes. The formula for the Whitehead group expresses the group only via the first Betti number: $K_1({\cal O}_E)= {\mathbb Z}^{\beta(E)}$. This allows to provide a contrexample to the fact, that $K_1({\cal O}_E)$ is a torsion free part of $K_0({\cal O}_E)$, which holds for finite graphs.

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