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 $K_0({\cal O}_E)= {\mathbb Z}^{\beta(E)+\gamma(E)},,$ where $\beta(E)$ is the first Betti number and $\gamma(E)$ 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: $K_1({\cal O}_E)= {\mathbb Z}^{\beta(E)}$. 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 $K_0({\cal O}_E)$.

💡 Research Summary

The paper addresses the K‑theory of Cuntz‑Krieger algebras 𝒪_E associated with infinite, locally finite directed graphs E. The authors employ the Bass‑Hashimoto operator T_E, a linear map on the space of oriented edges that captures the adjacency structure of the graph, as the central analytical tool. By considering the category 𝔊 of finite black‑and‑white bi‑directed subgraphs of E together with inclusion morphisms that preserve the black‑white coloring, they construct a continuous functor K: 𝔊 → Ab that assigns to each finite subgraph F the already known K‑groups K₀(𝒪_F) and K₁(𝒪_F) (computed in earlier work by the same authors).

The key methodological step is to take the inductive limit of these groups over the directed system of subgraphs. For an inclusion F₁ ⊂ F₂, the induced map K(F₁) → K(F₂) is realized by a block‑extension of the matrices describing the Bass‑Hashimoto operator on the smaller subgraph. Passing to the limit yields \