The $C^*$-algebras of locally finite undirected graphs: A complete description of their K-theory

Abstract

We study the $C^*$-algebra $C^*(\Gamma)$ 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 $C^*(\Gamma)\cong C^*(E_\Gamma)$, and for essential $\Gamma$ this identifies $C^*(\Gamma)$ 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 $K_0(C^*(\Gamma))\cong\mathbb{Z}^{g}\oplus\mathbb{Z}/(g-1)\mathbb{Z}$ and $K_1(C^*(\Gamma))\cong H_1(\Gamma)\cong\mathbb{Z}^{g}$, 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 $\ge 2$.