1,721,036 research outputs found
Fermi-Markov states
No full textWe investigate the structure of the Markov states on general
Fermi 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 Fermionic annihilators
and creators on each site), also the nonhomogeneous 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 Zn, 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 quantum information
theory as well
Operator space structures and the split property
No full textA characterization of the split property for an inclusion N⊂M of W∗-factors with separable predual is established in terms of the canonical non-commutative L2 embedding considered in \cite{B1,B2}
\F_2:a\in N\to
\D_{M,\Om}^{1/4}a\Om\in L^2(M,\Om)
associated with an arbitrary fixed standard vector \Om for M. This characterization follows an analogous characterization related to the canonical non-commutative L1 embedding
\F_1:a\in N\to (\cdot\Om,J_{M,\Om}a\Om)\in L^1(M,\Om)
also considered in \cite{B1,B2} and studied in \cite{F}. The split property for a Quantum Field Theory is characterized by equivalent conditions relative to the non-commutative embeddings \F_i, i=1,2, constructed by the modular Hamiltonian of a privileged faithful state such as e.g. the vacuum state. The above characterization would be also useful for theories on a curved space-time where there exists no a-priori privileged state
HARMONIC ANALYSIS ON CAYLEY TREES II: THE BOSE-EINSTEIN CONDENSATION
No full textWe investigate the Bose-Einstein Condensation on non-homogeneous non-amenable networks for the model describing arrays of Josephson junctions. The graphs under investigation are obtained by adding density zero perturbations to the homogeneous Cayley Trees. The resulting topological model, whose Hamiltonian is the pure hopping one given by the opposite of the adjacency operator, 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, previously investigated in a separate work, for such non-amenable graphs. Concerning the appearance of the Bose-Einstein Condensation, the results are surprisingly in accordance with the previous ones, despite the lack of amenability. The appearance of the hidden spectrum for low energies always implies that the critical density is finite for all the models under consideration. We also show that, even when the critical density is finite, if the adjacency operator of the graph is recurrent, it is impossible to exhibit temperature states which are locally normal (i.e. states for which the local particle density is finite) describing the condensation at all. A similar situation seems to occur in the transient cases for which it is impossible to exhibit locally normal states omega describing the Bose-Einstein Condensation with mean particle density rho(omega) strictly greater than the critical density rho(c). Indeed, it is shown that the transient cases admit locally normal states exhibiting Bose-Einstein Condensation phenomena. In order to construct such locally normal temperature states by infinite volume limits of finite volume Gibbs states, a careful choice of the sequence of the chemical potentials should be done. For all such states, the condensate is essentially allocated on the base point supporting the perturbation. This leads to rho(omega) = rho(c) as the perturbation is negligible with respect to the whole network. We prove that all such temperature states are Kubo-Martin-Schwinger states for the natural dynamics associated to the (formal) pure hopping Hamiltonian. The construction of such a dynamics, which is a delicate issue, is also provided in detail
De Finetti Theorem on the CAR Algebra
No full textThe symmetric states on a quasi local C*-algebra on the infinite set of indices J are those invariant under the action of the group of the permutations moving only a finite, but arbitrary, number of elements of J. The celebrated De Finetti Theorem describes the structure of the symmetric states (i.e. exchangeable probability measures) in classical probability. In the present paper we extend the De Finetti Theorem to the case of the CAR algebra, that is for physical systems describing Fermions. Namely, after showing that a symmetric state is automatically even under the natural action of the parity automorphism, we prove that the compact convex set of such states is a Choquet simplex, whose extremal (i.e. ergodic w.r.t. the action of the group of permutations previously described) are precisely the product states in the sense of Araki-Moriya. In order to do that, we also prove some ergodic properties naturally enjoyed by the symmetric states which have a self-containing interest
Corrigendum to "Harmonic analysis on perturbed Cayley Trees" [J.funct.anal. 261 (3), (2011) 604-634]
No full textDue to the boundary effects, the standard definition of the integrated density of the states (i.d.s. for short) used in [F. Fidaleo, Harmonic analysis on perturbed Cayley Trees, J. Funct. Anal. 261 (3) (2011) 604-634], does not work for nonamenable graphs like Cayley Trees and density zero perturbations of those. On the other hand, Proposition 2.3 in the previous mentioned paper works under the right definition we are going to describe, and which is useful for all the applications. For the sake of completeness and the convenience of the reader, we also show that both the definitions coincide in the amenable cas
Canonical operator space structures in non-commutative L^p spaces
No full textWe analyze canonical operator space structures on the non-commutative Lp spaces Lpη(M; ϕ, ω) constructed by interpolation a la Stein–Weiss based on two normal semifinite faithful weights ϕ, ω on a W*-algebra M. We show that there is only one canonical (i.e. arising by interpolation operator space structure on Lp(M) when M and p are kept fixed. Namely, for any n.s.f. weights ϕ, ω on M and η∈[0, 1], the spaces Lpη(M; ϕ, ω) are all completely isomorphic when they are canonically considered as operator spaces. Finally, we also describe the norms on all matrix spaces n(Lp(M)) which determine such a canonical quantized structure
Weak and strong martingale convergences of generalized conditional expectations in non-commutative L^{p} spaces
No full textWhile briefly reviewing results about (nets of) generalized conditional expectations and martingale-type convergences, we present an unified version of the topic and prove martingale-type convergence results of monotone nets of generalized conditional expectations in the full generality. In so doing, we fill some gaps
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