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

TL;DR

This paper computes the K-theory of boundary quotients of Toeplitz C*-algebras associated with right-angled Artin groups, showing they are nuclear, classifiable, and isomorphic to tensor products of Cuntz algebras.

Contribution

It introduces a new inductive approach to analyze boundary quotients, establishing their nuclearity, bootstrap class membership, and explicit classification as tensor products of Cuntz algebras.

Findings

Boundary quotients are nuclear and in the bootstrap class.

K-theory computed via Pimsner-Voiculescu exact sequence.

Boundary quotients are isomorphic to tensor products of Cuntz algebras.

Abstract

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 $1 \leq n \leq \infty$.