47 research outputs found
Geodesic laminations revisited
The Bratteli diagram is an infinite graph which reflects the structure of
projections in a C*-algebra. We prove that every strictly ergodic unimodular
Bratteli diagram of rank 2g+m-1 gives rise to a minimal geodesic lamination
with the m-component principal region on a surface of genus g greater or equal
to 1. The proof is based on the Morse theory of the recurrent geodesics on the
hyperbolic surfaces.Comment: 13 pages, 2 figures, revised versio
On the Grothendieck Theorem for jointly completely bounded bilinear forms
We show how the proof of the Grothendieck Theorem for jointly completely
bounded bilinear forms on C*-algebras by Haagerup and Musat can be modified in
such a way that the method of proof is essentially C*-algebraic. To this
purpose, we use Cuntz algebras rather than type III factors. Furthermore, we
show that the best constant in Blecher's inequality is strictly greater than
one.Comment: 9 pages, minor change
Structure, classifcation, and conformal symmetry, of elementary particles over non-archimedean space-time
It is known that no length or time measurements are possible in sub-Planckian
regions of spacetime. The Volovich hypothesis postulates that the
micro-geometry of spacetime may therefore be assumed to be non-archimedean. In
this letter, the consequences of this hypothesis for the structure,
classification, and conformal symmetry of elementary particles, when spacetime
is a flat space over a non-archimedean field such as the -adic numbers, is
explored. Both the Poincar\'e and Galilean groups are treated. The results are
based on a new variant of the Mackey machine for projective unitary
representations of semidirect product groups which are locally compact and
second countable. Conformal spacetime is constructed over -adic fields and
the impossibility of conformal symmetry of massive and eventually massive
particles is proved
Multiplicativity of completely bounded p-norms implies a new additivity result
We prove additivity of the minimal conditional entropy associated with a
quantum channel Phi, represented by a completely positive (CP),
trace-preserving map, when the infimum of S(gamma_{12}) - S(gamma_1) is
restricted to states of the form gamma_{12} = (I \ot Phi)(| psi >< psi |). We
show that this follows from multiplicativity of the completely bounded norm of
Phi considered as a map from L_1 -> L_p for L_p spaces defined by the Schatten
p-norm on matrices; we also give an independent proof based on entropy
inequalities. Several related multiplicativity results are discussed and
proved. In particular, we show that both the usual L_1 -> L_p norm of a CP map
and the corresponding completely bounded norm are achieved for positive
semi-definite matrices. Physical interpretations are considered, and a new
proof of strong subadditivity is presented.Comment: Final version for Commun. Math. Physics. Section 5.2 of previous
version deleted in view of the results in quant-ph/0601071 Other changes
mino
Extensions and degenerations of spectral triples
For a unital C*-algebra A, which is equipped with a spectral triple and an
extension T of A by the compacts, we construct a family of spectral triples
associated to T and depending on the two positive parameters (s,t).
Using Rieffel's notation of quantum Gromov-Hausdorff distance between compact
quantum metric spaces it is possible to define a metric on this family of
spectral triples, and we show that the distance between a pair of spectral
triples varies continuously with respect to the parameters. It turns out that a
spectral triple associated to the unitarization of the algebra of compact
operators is obtained under the limit - in this metric - for (s,1) -> (0, 1),
while the basic spectral triple, associated to A, is obtained from this family
under a sort of a dual limiting process for (1, t) -> (1, 0).
We show that our constructions will provide families of spectral triples for
the unitarized compacts and for the Podles sphere. In the case of the compacts
we investigate to which extent our proposed spectral triple satisfies Connes' 7
axioms for noncommutative geometry.Comment: 40 pages. Addedd in ver. 2: Examples for the compacts and the Podle`s
sphere plus comments on the relations to matricial quantum metrics. In ver.3
the word "deformations" in the original title has changed to "degenerations"
and some illustrative remarks on this aspect are adde
Metric adjusted skew information: Convexity and restricted forms of superadditivity
We give a truly elementary proof of the convexity of metric adjusted skew
information following an idea of Effros. We extend earlier results of weak
forms of superadditivity to general metric adjusted skew informations.
Recently, Luo and Zhang introduced the notion of semi-quantum states on a
bipartite system and proved superadditivity of the Wigner-Yanase-Dyson skew
informations for such states. We extend this result to general metric adjusted
skew informations. We finally show that a recently introduced extension to
parameter values of the WYD-information is a special case of
(unbounded) metric adjusted skew information.Comment: An error in the literature is pointed ou
Semilattices of groups and nonstable K-theory of extended Cuntz limits
We give an elementary characterization of those abelian monoidsM
that are direct limits of countable sequences of finite direct sums of monoids
of the form either (Z/nZ) ⊔ {0} or Z ⊔ {0}. This characterization involves the
Riesz refinement property together with lattice-theoretical properties of the
collection of all subgroups of M (viewed as a semigroup), and it makes it pos-
sible to express M as a certain submonoid of a direct product ×G, where
is a distributive semilattice with zero and G is an abelian group. When applied
to the monoids V (A) appearing in the nonstable K-theory of C*-algebras, our
results yield a full description of V (A) for C*-inductive limits A of finite sums
of full matrix algebras over either Cuntz algebras On, where 2 ≤ n < ∞, or
corners of O1 by projections, thus extending to the case including O1 earlier
work by the authors together with K.R. Goodearl
Twisted k-graph algebras associated to Bratteli diagrams
Given a system of coverings of k-graphs, we show that the cohomology of the
resulting (k+1)-graph is isomorphic to that of any one of the k-graphs in the
system. We then consider Bratteli diagrams of 2-graphs whose twisted
C*-algebras are matrix algebras over noncommutative tori. For such systems we
calculate the ordered K-theory and the gauge-invariant semifinite traces of the
resulting 3-graph C*-algebras. We deduce that every simple C*-algebra of this
form is Morita equivalent to the C*-algebra of a rank-2 Bratteli diagram in the
sense of Pask-Raeburn-R{\o}rdam-Sims.Comment: 28 pages, pictures prepared using tik
Meixner class of non-commutative generalized stochastic processes with freely independent values I. A characterization
Let be an underlying space with a non-atomic measure on it (e.g.
and is the Lebesgue measure). We introduce and study a
class of non-commutative generalized stochastic processes, indexed by points of
, with freely independent values. Such a process (field),
, , is given a rigorous meaning through smearing out
with test functions on , with being a
(bounded) linear operator in a full Fock space. We define a set
of all continuous polynomials of , and then define a con-commutative
-space by taking the closure of in the norm
, where is the vacuum in the Fock
space. Through procedure of orthogonalization of polynomials, we construct a
unitary isomorphism between and a (Fock-space-type) Hilbert space
, with
explicitly given measures . We identify the Meixner class as those
processes for which the procedure of orthogonalization leaves the set invariant. (Note that, in the general case, the projection of a
continuous monomial of oder onto the -th chaos need not remain a
continuous polynomial.) Each element of the Meixner class is characterized by
two continuous functions and on , such that, in the
space, has representation
\omega(t)=\di_t^\dag+\lambda(t)\di_t^\dag\di_t+\di_t+\eta(t)\di_t^\dag\di^2_t,
where \di_t^\dag and \di_t are the usual creation and annihilation
operators at point
Large violation of Bell inequalities with low entanglement
In this paper we obtain violations of general bipartite Bell inequalities of
order with inputs, outputs and
-dimensional Hilbert spaces. Moreover, we construct explicitly, up to a
random choice of signs, all the elements involved in such violations: the
coefficients of the Bell inequalities, POVMs measurements and quantum states.
Analyzing this construction we find that, even though entanglement is necessary
to obtain violation of Bell inequalities, the Entropy of entanglement of the
underlying state is essentially irrelevant in obtaining large violation. We
also indicate why the maximally entangled state is a rather poor candidate in
producing large violations with arbitrary coefficients. However, we also show
that for Bell inequalities with positive coefficients (in particular, games)
the maximally entangled state achieves the largest violation up to a
logarithmic factor.Comment: Reference [16] added. Some typos correcte