Boundary quotients and ideals of Toeplitz C∗-algebras of Artin groups

We study the quotients of the Toeplitz C∗-algebra of a quasi-lattice ordered group (G,P), which we view as crossed products by a partial actions of G on closed invariant subsets of a totally disconnected compact Hausdorff space, the Nica spectrum of (G,P). Our original motivation and our main examples are drawn from right-angled Artin groups, but many of our results are valid for more general quasi-lattice ordered groups. We show that the Nica spectrum has a unique minimal closed invariant subset, which we call the boundary spectrum, and we define the boundary quotient to be the crossed product of the corresponding restricted partial action. The main technical tools used are the results of Exel, Laca, and Quigg on simplicity and ideal structure of partial crossed products, which depend on amenability and topological freeness of the partial action and its restriction to closed invariant subsets. When there exists a generalised length function, or controlled map, defined on G and taking values in an amenable group, we prove that the partial action is amenable on arbitrary closed invariant subsets. The topological freeness of the boundary action depends on topological freeness of the restriction to a certain lattice subgroup of G, the “core” of (G,P), which often turns out to be trivial. Our main results are obtained for right-angled Artin groups with trivial centre, that is, those with no cyclic direct factor; they include a presentation of the boundary quotient in terms of generators and relations that generalises Cuntz’s presentation of On, a proof that the boundary quotient is purely infinite and simple, and a parametrisation of the ideals of the Toeplitz C∗-algebra in terms of subsets of the standard generators of the Artin group. © 2006 Elsevier Inc. All rights reserved.