83 research outputs found
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. 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.Comment: 26 page
Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers
We show that the group of
orientation-preserving affine transformations of the rational numbers is
quasi-lattice ordered by its subsemigroup . The associated Toeplitz -algebra is universal for isometric representations which
are covariant in the sense of Nica. We give a presentation of this Toeplitz
algebra in terms of generators and relations, and use this to show that the
-algebra recently introduced by Cuntz is the
boundary quotient of in the sense of Crisp and Laca. The Toeplitz algebra
carries a natural dynamics
, which induces the one considered by Cuntz on the quotient , and our main result is the computation of the KMS
(equilibrium) states of the dynamical system for all values of the inverse
temperature . For there is a unique KMS
state, and the KMS state factors through the quotient map onto , giving the unique KMS state discovered by Cuntz. At
there is a phase transition, and for the KMS states are
indexed by probability measures on the circle. There is a further phase
transition at , where the KMS states are indexed by the
probability measures on the circle, but the ground states are indexed by the
states on the classical Toeplitz algebra .Comment: 38 page
KMS states of quasi-free dynamics on Pimsner algebras
A continuous one-parameter group of unitary isometries of a right Hilbert
C*-bimodule induces a quasi-free dynamics on the Cuntz-Pimsner C*-algebra of
the bimodule and on its Toeplitz extension. The restriction of such a dynamics
to the algebra of coefficients of the bimodule is trivial, and the
corresponding KMS states of the Toeplitz-Cuntz-Pimsner and Cuntz-Pimsner
C*-algebras are characterized in terms of traces on the algebra of
coefficients. This generalizes and sheds light onto various earlier results
about KMS states of the gauge actions on Cuntz algebras, Cuntz-Krieger
algebras, and crossed products by endomorphisms. We also obtain a more general
characterization, in terms of KMS weights, for the case in which the inducing
isometries are not unitary, and accordingly, the restriction of the quasi-free
dynamics to the algebra of coefficients is nontrivial
Type III_1 equilibrium states of the Toeplitz algebra of the affine semigroup over the natural numbers
We complete the analysis of KMS-states of the Toeplitz algebra of the affine
semigroup over the natural numbers, recently studied by Raeburn and the first
author, by showing that for every inverse temperature beta in the critical
interval [1,2], the unique KMS_beta-state is of type III_1. We prove this by
reducing the type classification from the Toeplitz algebra to that of the
symmetric part of the Bost-Connes system, with a shift in inverse temperature.
To carry out this reduction we first obtain a parametrization of the Nica
spectrum of the Toeplitz algebra in terms of an adelic space. Combining a
characterization of traces on crossed products due to the second author with an
analysis of the action of the affine semigroup on the Nica spectrum, we can
also recover all the KMS-states originally computed by Raeburn and the first
author. Our computation sheds light on why there is a free transitive circle
action on the extremal KMS_beta-states for beta>2 that does not ostensibly come
from an action on the C*-algebra.Comment: 15 pages, AMS-LaTe
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