The $C^{\ast }$-algebras of locally finite undirected graphs: A complete description of their K-theory

Abstract

We study the $C^{\ast }$-algebra $C^{\ast }(\Gamma )$ of a locally finite undirected (Serre) graph $\Gamma$ and compute its K-theory. The algebra is defined intrinsically, as the graph-of-groups algebra with all groups trivial, and is shown to be independent of the choice of orientation. Its structure is accessed through a passage to directed graphs: for every locally finite $\Gamma$ there is a row-finite directed graph $E_{\Gamma }$ with $C^{\ast }(\Gamma )\cong C^{\ast }(E_{\Gamma })$, and for essential $\Gamma$ this identifies $C^{\ast }(\Gamma )$ with the Cuntz--Krieger algebra of the Bass--Hashimoto (non-backtracking) matrix $T_{\Gamma }$. This makes the whole directed-graph K-theory machinery available and, unlike the directed case, produces K-groups that read off the geometry of $\Gamma$: for a finite essential graph of genus $g\ge 2$ one has $K_0(C^{\ast }(\Gamma ))\cong {\mathbb{Z}}^g\oplus \mathbb{Z}/(g-1)\mathbb{Z}$ and $K_1(C^{\ast }(\Gamma ))\cong H_1(\Gamma )\cong {\mathbb{Z}}^g$, and in general $K_{\ast }$ is governed by the genus, the number of ends and the number of dead-ends. We record the resulting classification by genus: the algebras are finite-dimensional or AF at genus $0$, AT precisely for finite graphs at genus $1$, and unital Kirchberg algebras for finite essential graphs of genus $\ge 2$.