  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. 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 O_n,
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.
