Complete Characterization of K-Theory for C*-algebras Associated to Locally Finite Unoriented Graphs

TL;DR

This paper provides a complete characterization of the K-theory groups for C*-algebras associated with locally finite graphs, extending previous results to infinite graphs using graph-theoretical methods.

Contribution

It generalizes the K-theory description of Cuntz-Krieger algebras to infinite, locally finite graphs, employing a graph-theoretical approach.

Findings

K_0(O_E) = Z^{β(E)} ⊕ Z^{γ(E)}

K_1(O_E) = Z^{γ(E)}

Extension of previous finite graph results to infinite graphs

Abstract

In this paper we give a complete description of K-theory groups for Cuntz-Krieger C*-algebras associated to general locally-finite (topologically connected) graphs via Bass-Hashimoto operator. Our result generalizes the one obtained by the second author for the case of graphs with not necessarily finite first Betti numbers. On the basis of purely graph-theoretical method introduced by G. Cornelissen, O. Lorscheid, M. Marcolli and developed further by N.Iyudu, we prove that for the algebra O_E associated to an infinite graph E of the above form holds K_0(O_E)=Z^{\beta(E)} \oplus Z^{\gamma(E)} and K_1(O_E) = Z^{\gamma(E)}, where \beta(E)=\dim H_1(E) and \gamma(E) stands for the cardinality of the valency set of E, defined in the paper.