44 research outputs found
Amenability and exactness for dynamical systems and their C*-algebras
In this survey, we study the relations between amenability (resp. amenability
at infinity) of C*-dynamical systems and equality or nuclearity (resp.
exactness) of the corresponding crossed products.Comment: 16 pages, Ams-Tex, minor grammatical change
Measuring processes and the Heisenberg picture
In this paper, we attempt to establish quantum measurement theory in the
Heisenberg picture. First, we review foundations of quantum measurement theory,
that is usually based on the Schr\"{o}dinger picture. The concept of instrument
is introduced there. Next, we define the concept of system of measurement
correlations and that of measuring process. The former is the exact counterpart
of instrument in the (generalized) Heisenberg picture. In quantum mechanical
systems, we then show a one-to-one correspondence between systems of
measurement correlations and measuring processes up to complete equivalence.
This is nothing but a unitary dilation theorem of systems of measurement
correlations. Furthermore, from the viewpoint of the statistical approach to
quantum measurement theory, we focus on the extendability of instruments to
systems of measurement correlations. It is shown that all completely positive
(CP) instruments are extended into systems of measurement correlations. Lastly,
we study the approximate realizability of CP instruments by measuring processes
within arbitrarily given error limits.Comment: v
KMS states and conformal measures
From a non-constant holomorphic map on a connected Riemann surface we
construct an 'etale second countable locally compact Hausdorff groupoid whose
associated groupoid C*-algebra admits a one-parameter group of automorphisms
with the property that its KMS states corresponds to conformal measures in the
sense of Sullivan. In this way certain quadratic polynomials give rise to
quantum statistical models with a phase transition arising from spontaneous
symmetry breaking.Comment: The last section revised. This version will appear in Comm. Math.
Phy
Radon--Nikodym representations of Cuntz--Krieger algebras and Lyapunov spectra for KMS states
We study relations between --KMS states on Cuntz--Krieger algebras
and the dual of the Perron--Frobenius operator .
Generalising the well--studied purely hyperbolic situation, we obtain under
mild conditions that for an expansive dynamical system there is a one--one
correspondence between --KMS states and eigenmeasures of
for the eigenvalue 1. We then consider
representations of Cuntz--Krieger algebras which are induced by Markov fibred
systems, and show that if the associated incidence matrix is irreducible then
these are --isomorphic to the given Cuntz--Krieger algebra. Finally, we
apply these general results to study multifractal decompositions of limit sets
of essentially free Kleinian groups which may have parabolic elements. We
show that for the Cuntz--Krieger algebra arising from there exists an
analytic family of KMS states induced by the Lyapunov spectrum of the analogue
of the Bowen--Series map associated with . Furthermore, we obtain a formula
for the Hausdorff dimensions of the restrictions of these KMS states to the set
of continuous functions on the limit set of . If has no parabolic
elements, then this formula can be interpreted as the singularity spectrum of
the measure of maximal entropy associated with .Comment: 30 pages, minor changes in the proofs of Theorem 3.9 and Fact
The tight groupoid of an inverse semigroup
In this work we present algebraic conditions on an inverse semigroup S (with zero) which imply that its associated tight groupoid G_tight(S) is: Hausdorff, essentially principal, minimal and contracting, respectively. In some cases these conditions are in fact necessary and sufficient.The first-named author was partially supported by CNPq. The second-named author was partially supported by PAI III grants FQM-298 and P11-FQM-7156 of the Junta de Andalucía and by the DGI- MICINN and European Regional Development Fund, jointly, through Project MTM2011-28992-C02-02
Egorov's theorem for transversally elliptic operators on foliated manifolds and noncommutative geodesic flow
The main result of the paper is Egorov's theorem for transversally elliptic
operators on compact foliated manifolds. This theorem is applied to describe
the noncommutative geodesic flow in noncommutative geometry of Riemannian
foliations.Comment: 23 pages, no figures. Completely revised and improved version of
dg-ga/970301
Progress in noncommutative function theory
In this expository paper we describe the study of certain non-self-adjoint
operator algebras, the Hardy algebras, and their representation theory. We view
these algebras as algebras of (operator valued) functions on their spaces of
representations. We will show that these spaces of representations can be
parameterized as unit balls of certain -correspondences and the
functions can be viewed as Schur class operator functions on these balls. We
will provide evidence to show that the elements in these (non commutative)
Hardy algebras behave very much like bounded analytic functions and the study
of these algebras should be viewed as noncommutative function theory
Group measure space decomposition of II_1 factors and W*-superrigidity
We prove a "unique crossed product decomposition" result for group measure
space II_1 factors arising from arbitrary free ergodic probability measure
preserving (p.m.p.) actions of groups \Gamma in a fairly large family G, which
contains all free products of a Kazhdan group and a non-trivial group, as well
as certain amalgamated free products over an amenable subgroup. We deduce that
if T_n denotes the group of upper triangular matrices in PSL(n,Z), then any
free, mixing p.m.p. action of the amalgamated free product of PSL(n,Z) with
itself over T_n, is W*-superrigid, i.e. any isomorphism between L^\infty(X)
\rtimes \Gamma and an arbitrary group measure space factor L^\infty(Y) \rtimes
\Lambda, comes from a conjugacy of the actions. We also prove that for many
groups \Gamma in the family G, the Bernoulli actions of \Gamma are
W*-superrigid.Comment: Final version. Some extra details have been added to improve the
expositio
C*-simplicity and the unique trace property for discrete groups
In this paper, we introduce new methods for working with group and crossed product C*-algebras that allow us to settle the longstanding open problem of characterizing groups with the unique trace property