95 research outputs found
Fermi Markov states
We investigate the structure of the Markov states on general Fermion
algebras. The situation treated in the present paper covers, beyond the
d--Markov states on the CAR algebra on Z (i.e. when there are d--annihilators
and creators on each site), also the non homogeneous case (i.e. when the
numbers of generators depends on the localization). The present analysis
provides the first necessary step for the study of the general properties, and
the construction of nontrivial examples of Fermi Markov states on the
d--standard lattice, that is the Fermi Markov fields. Natural connections with
the KMS boundary condition and entropy of Fermi Markov states are studied in
detail. Apart from a class of Markov states quite similar to those arising in
the tensor product algebras (called "strongly even" in the sequel), other
interesting examples of Fermi Markov states naturally appear. Contrarily to the
strongly even examples, the latter are highly entangled and it is expected that
they describe interactions which are not "commuting nearest neighbor".
Therefore, the non strongly even Markov states, in addition to the natural
applications to quantum statistical mechanics, might be of interest for the
information theory as well.Comment: 32 pages. Journal of Operator Theory, to appea
Harmonic analysis on Cayley Trees II: the Bose Einstein condensation
We investigate the Bose-Einstein Condensation on non homogeneous non amenable
networks for the model describing arrays of Josephson junctions on perturbed
Cayley Trees. The resulting topological model has also a mathematical interest
in itself. The present paper is then the application to the Bose-Einstein
Condensation phenomena, of the harmonic analysis aspects arising from additive
and density zero perturbations, previously investigated by the author in a
separate work. Concerning the appearance of the Bose-Einstein Condensation, the
results are surprisingly in accordance with the previous ones, despite the lack
of amenability. We indeed first show the following fact. Even when the critical
density is finite (which is implied in all the models under consideration,
thanks to the appearance of the hidden spectrum), if the adjacency operator of
the graph is recurrent, it is impossible to exhibit temperature locally normal
states (i.e. states for which the local particle density is finite) describing
the condensation at all. The same occurs in the transient cases for which it is
impossible to exhibit locally normal states describing the Bose--Einstein
Condensation at mean particle density strictly greater than the critical
density . In addition, for the transient cases, in order to construct locally
normal temperature states through infinite volume limits of finite volume Gibbs
states, a careful choice of the the sequence of the finite volume chemical
potential should be done. For all such states, the condensate is essentially
allocated on the base--point supporting the perturbation. This leads that the
particle density always coincide with the critical one. It is shown that all
such temperature states are Kubo-Martin-Schwinger states for a natural
dynamics. The construction of such a dynamics, which is a very delicate issue,
is also done.Comment: 28 pages, 6 figures, 1 tabl
A note on Boolean stochastic processes
For the quantum stochastic processes generated by the Boolean Commutation
Relations, we prove the following version of De Finetti Theorem: each of such
Boolean process is exchangeable if and only if it is independent and
identically distributed with respect to the tail algebra.Comment: 9 page
Some operator ideals in non-commutative functional analysis
We characterize classes of linear maps between operator spaces , which
factorize through maps arising in a natural manner via the Pisier vector-valued
non-commutative spaces based on the Schatten classes on the
separable Hilbert space . These classes of maps can be viewed as
quasi-normed operator ideals in the category of operator spaces, that is in
non-commutative (quantized) functional analysis. The case provides a
Banach operator ideal and allows us to characterize the split property for
inclusions of -algebras by the 2-factorable maps. The various
characterizations of the split property have interesting applications in
Quantum Field Theory.Comment: 23 pages, LaTe
Infinite dimensional entangled Markov chains
We continue the analysis of nontrivial examples of quantum Markov processes.
This is done by applying the construction of entangled Markov chains obtained
from classical Markov chains with infinite state--space. The formula giving the
joint correlations arises from the corresponding classical formula by replacing
the usual matrix multiplication by the Schur multiplication. In this way, we
provide nontrivial examples of entangled Markov chains on , being any infinite dimensional type
factor, a finite interval of , and the bar the von Neumann tensor
product between von Neumann algebras. We then have new nontrivial examples of
quantum random walks which could play a r\^ole in quantum information theory.
In view of applications to quantum statistical mechanics too, we see that the
ergodic type of an entangled Markov chain is completely determined by the
corresponding ergodic type of the underlying classical chain, provided that the
latter admits an invariant probability distribution. This result parallels the
corresponding one relative to the finite dimensional case.
Finally, starting from random walks on discrete ICC groups, we exhibit
examples of quantum Markov processes based on type von Neumann factors.Comment: 16 page
New topics in ergodic theory
The entangled ergodic theorem concerns the study of the convergence in the
strong, or merely weak operator topology, of the multiple Cesaro mean
\frac{1}{N^{k}}\sum_{n_{1},...,n_{k}=0}^{N-1}
U^{n_{\a(1)}}A_{1}U^{n_{\a(2)}}... U^{n_{\a(2k-1)}}A_{2k-1}U^{n_{\a(2k)}} ,
where is a unitary operator acting on the Hilbert space , \a:\{1,...,
m\}\mapsto\{1,..., k\} is a partition of the set made of elements in
parts, and finally are bounded operators acting on .
While reviewing recent results about the entangled ergodic theorem, we provide
some natural applications to dynamical systems based on compact operators.
Namely, let be a --dynamical system, where
, and is an automorphism implemented by the
unitary . We show that
pointwise in
the weak topology of \K(H). Here, is a conditional expectation projecting
onto the --subalgebra If in addition is weakly
mixing with the unique up to a phase, invariant vector under
and , we have the following recurrence result. If
fulfils , and are natural
numbers kept fixed, then there exists an such that
for each .Comment: 18 page
- β¦