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

To a graph $\Gamma$ one can associate a C^-algebra $C^(\Gamma)$ generated by isometries. Such $C^$-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” $C^_q(\Gamma)$ 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 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 $1 \leq n \leq \infty$.

💡 Research Summary

The paper investigates the C*-algebras associated with right‑angled Artin groups (RAAGs) via the graph‑theoretic construction introduced by Crisp and Laca. Starting from a finite simplicial graph Γ, one defines a Toeplitz‑type C*-algebra C*(Γ) generated by isometries subject to commutation relations dictated by the edges of Γ. This construction fits into Nica’s framework of Toeplitz C*-algebras 𝒯(G,P) for quasi‑lattice ordered groups (G,P). Crisp and Laca showed that the “boundary quotient’’ C*_q(Γ) = C*(Γ)/I, where I is the canonical ideal generated by the Cuntz–Krieger relations, is simple and purely infinite.

The central achievement of the article is a structural decomposition of C*_q(Γ) for a broad class of finite graphs. By removing a single vertex v from Γ one obtains a subgraph Γ′. Using C*(Γ′) one builds a natural C*-subalgebra A ⊂ C*_q(Γ) that is isomorphic to the boundary quotient of the smaller graph, C*_q(Γ′). The authors then define a canonical action of the integers ℤ on A, and prove that the crossed product A ⋊ ℤ contains a full corner p(A ⋊ ℤ)p which is ‑isomorphic to C_q(Γ). In other words, C*_q(Γ) can be realized as a full corner of a ℤ‑crossed product of a smaller boundary quotient.

This corner description enables an induction on the number of vertices. The base case (a single vertex) is already known: the boundary quotient is a Cuntz algebra O_n (or O_∞) depending on the valence of the vertex. Assuming that for Γ′ the algebra C*_q(Γ′) is nuclear and belongs to the bootstrap class, the crossed product A ⋊ ℤ inherits nuclearity, and the Pimsner–Voiculescu six‑term exact sequence applies. The exact sequence \