The $K$-theory of Toeplitz $C^*$-algebras of right-angled Artin groups

Toeplitz $C^*$-algebras of right-angled Artin groups were studied by Crisp and Laca. They are a special case of the Toeplitz $C^*$-algebras $\mathcal{T}(G, P)$ associated with quasi-lattice ordered groups $(G, P)$ introduced by Nica. Crisp and Laca proved that the so-called ``boundary quotients'' $C^*_Q(\Gamma)$ of $C^*(\Gamma)$ are simple and purely infinite. For a certain class of finite graphs $\Gamma$ we show that $C^*_Q(\Gamma)$ can be represented as a full corner of a crossed product of an appropriate $C^*$-subalgebra of $C^*_Q(\Gamma)$ built by using $C^*(\Gamma')$, where $\Gamma'$ 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 $C^*_Q(\Gamma)$ are nuclear and moreover belong to the small bootstrap class. We also 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 $\mathcal{O}_n$ with $1 \leq n \leq \infty$.