The K-Theory of Toeplitz C*-Algebras of Right-Angled Artin Groups

Abstract

To a graph $\Gamma$ one can associate a C^*-algebra $C^{\ast }(\Gamma )$ generated by isometries. Such $C^{\ast }$-algebras were studied recently by Crisp and Laca. They are a special case of the Toeplitz C^*-algebras $\mathcal{T}(G,P)$ associated to quasi-latice ordered groups (G, P) introduced by Nica. Crisp and Laca proved that the so called "boundary quotients" $C_q^{\ast }(\Gamma )$ of $C^{\ast }(\Gamma )$ are simple and purely infinite. For a certain class of finite graphs $\Gamma$ we show that $C_q^{\ast }(\Gamma )$ can be represented as a full corner of a crossed product of an appropriate C^*-subalgebra of $C_q^{\ast }(\Gamma )$ built by using $C^{\ast }({\Gamma }^{\prime })$, where ${\Gamma }^{\prime }$ is a subgraph of $\Gamma$ with one less vertex, by the group $\mathbb{Z}$. Using induction on the number of the vertices of $\Gamma$ we show that $C_q^{\ast }(\Gamma )$ are nuclear and belong to the small bootstrap class. This also enables us to use the Pimsner-Voiculescu exact sequence to find their K-theory. Finally we use the Kirchberg-Phillips classification theorem to show that those C^*-algebras are isomorphic to tensor products of ${\mathcal{O}}_n$ for $1\le n\le \infty$.

26 pages