[formula omitted]-theory of locally finite graph [formula omitted]-algebras

Abstract

We calculate the K-theory of the Cuntz–Krieger algebra OE 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 K0 is an inductive limit of K-groups of finite graphs, which were calculated in Cornelissen et al. (2008) [3].In the case of an infinite graph with the finite Betti number we obtain the formula for the Grothendieck group K0(OE)=Zβ(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 K0, which is present in the case of a finite graph, vanishes. The Whitehead group depends only on the first Betti number: K1(OE)=Zβ(E). These allow us to provide a counterexample to the fact, which holds for finite graphs, that K1(OE) is the torsion free part of K0(OE).