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Francesco Fidaleo

Publications and source records attributed to Francesco Fidaleo.

At least 37 records · Page 2Linked to original sources
arXiv2018

type III representations and modular spectral triples for the noncommutative torus

It is well known that for any irrational rotation number $\a$, the noncommutative torus $\ba_\a$ must have representations $π$ such that the generated von Neumann algebra $π(\ba_\a)"$ is of type $\ty{III}$. Therefore, it could be of interest to exhibit and investigate such kind of representations, together with the associated spectral triples whose twist of the Dirac operator and the corresponding derivation arises from the Tomita modular operator. In the present paper, we show that this program can be carried out, at least when $\a$ is a Liouville number satisfying a faster approximation property by rationals. In this case, we exhibit several type $\ty{II_\infty}$ and $\ty{III_ł}$, $ł\in[0,1]$, factor representations and modular spectral triples. The method developed in the present paper can be generalised to CCR algebras based on a locally compact abelian group equipped with a symplectic form.

math.OA↗
arXiv2016

From discrete to continuous monotone $C^*$-algebras via quantum central limit theorems

We prove that all finite joint distributions of creation and annihilation operators in Monotone and anti-Monotone Fock spaces can be realized as Quantum Central Limit of certain operators on a $C^*$-algebra, at least when the test functions are Riemann integrable. Namely, the approximation is given by weighted sequences of creators and annihilators in discrete monotone $C^*$-algebras, the weight being the above cited test functions. The construction is then generalized to processes by an invariance principle.

math.PR↗
arXiv2016

A proposal for the thermodynamics of certain open systems

Motivated by the fact that the (inverse) temperature might be a function of the energy levels in the Planck distribution $n_ε=\frac1{ζ^{-1}e^{β(ε)ε}-1}$ for the occupation number $n_ε$ of the level $ε$, we show that it can be naturally achieved by imposing the constraint concerning the conservation of a weighted sum $\sum_εf(ε)εn_ε$, with a fixed positive weight function $f$, of the contributions of the single energy levels occupation in the Microcanonical Ensemble scheme, obtaining $β(ε)\propto f(ε)$. This immediately addresses the possibility that also a weighted sum $\sum_εg(ε)n_ε$ of the particles occupation number is conserved, having as a consequence that the chemical potential might be a function of the energy levels of the system as well. This scheme leads to a thermodynamics of open systems in the following way: the equilibrium is reached when the entropy function is maximised under the constraints that some weighed sums of occupation of the energy levels and the occupation numbers are conserved. The standard case of isolated systems corresponds to the weight functions being trivial (i.e. $f, g$ are identically 1). For such open systems, new and unexpected phenomena which might happen in nature can appear, like the Bose Einstein Condensation in excited levels. The ideas outlined in the present paper may provide a new approach for the treatment of the irreversible thermodynamics.

cond-mat.stat-mech↗
arXiv2016

Symmetries and ergodic properties in quantum probability

We deal with the general structure of (noncommutative) stochastic processes by using the standard techniques of Operator Algebras. Any stochastic process is associated to a state on a universal object, i.e. the free product $C^*$-algebra in a natural way. In this setting one recovers the classical (i.e. commutative) probability scheme and many others, like those associated to the Monotone, Boolean and the $q$-deformed canonical commutation relations including the Bose/Fermi and Boltzmann cases. Natural symmetries like stationarity and exchangeability, as well as the ergodic properties of the stochastic processes are reviewed in detail for many interesting cases arising from Quantum Physics and Probability.

math.PR↗
arXiv2016

Ergodic theorems in quantum probability: an application to the monotone stochastic processes

We give sufficient conditions ensuring the strong ergodic property of unique mixing for $C^*$-dynamical systems arising from Yang-Baxter-Hecke quantisation. We discuss whether they can be applied to some important cases including monotone, Boson, Fermion and Boolean $C^*$-algebras in a unified version. The monotone and the Boolean cases are treated in full generality, the Bose/Fermi cases being already widely investigated. In fact, on one hand we show that the set of stationary stochastic processes are isomorphic to a segment in both the situations, on the other hand the Boolean processes enjoy the very strong property of unique mixing with respect to the fixed point subalgebra and the monotone ones do not

math.OA↗
arXiv2015

Quantitative BT-Theorem and automatic continuity for standard von Neumann algebras

We prove a general criterion for a von Neumann algebra $M$ in order to be in standard form. It is formulated in terms of an everywhere defined, invertible, antilinear, a priori not necessarily bounded operator, intertwining $M$ with its commutant $M'$ and acting as the $*$-operation on the centre. We also prove a generalized version of the BT-Theorem which enables us to see that such an intertwiner must be necessarily bounded. It is shown that this extension of the BT-Theorem leads to the automatic boundedness of quite general operators which intertwine the identity map of a von Neumann algebra with a general bounded, real linear, operator valued map. We apply the last result to the automatic boundedness of linear operators implementing algebraic morphisms of a von Neumann algebra onto some Banach algebra, and to the structure of a $W^*$-algebra $M$ endowed with a normal, semi-finite, faithful weight $φ\,$, whose left ideal $\mathfrak N_φ$ admits an algebraic complement in the GNS representation space $H_φ\,$, invariant under the canonical action of $M$.

math.OA↗
arXiv2015

Bose-Einstein condensation and condensation of $q$-particles in equilibrium and non equilibrium thermodynamics: a new approach

In the setting of the principle of local equilibrium which asserts that the temperature is a function of the energy levels of the system, we exhibit plenty of steady states describing the condensation of free Bosons which are not in thermal equilibrium. The surprising facts are that the condensation can occur both in dimension less than 3 in configuration space, and even in excited energy levels. The investigation relative to non equilibrium suggests a new approach to the condensation, which allows an unified analysis involving also the condensation of $q$-particles, $-1\leq q\leq 1$, where $q=\pm1$ corresponds to the Bose/Fermi alternative. For such $q$-particles, the condensation can occur only if $0<q\leq1$, the case 1 corresponding to the standard Bose-Einstein condensation. In this more general approach, completely new and unexpected states exhibiting condensation phenomena naturally occur also in the usual situation of equilibrium thermodynamics. The new approach proposed in the present paper for the situation of $2^\text{nd}$ quantisation of free particles, is naturally based on the theory of the Distributions, which might hopefully be extended to more general cases

math-ph↗
arXiv2013

Exchangeable stochastic processes and symmetric states in quantum probability

We analyze general aspects of exchangeable quantum stochastic processes, as well as some concrete cases relevant for several applications to Quantum Physics and Probability. We establish that there is a one-to-one correspondence between quantum stochastic processes, either preserving or not the identity, and states on free product C*-algebras, unital or not unital respectively, where the exchangeable ones correspond precisely to the symmetric states. We also connect some algebraic properties of exchangeable processes, that is the fact that they satisfy the product state or the block-singleton conditions, to some natural ergodic ones. We then specialize the investigation for the q-deformed Commutation Relations, q in(-1,1) (the case q=0 corresponding to the reduced group C*-algebra of the free group on infinitely many generators), and the Boolean ones. We also provide a generalization of De Finetti Theorem to the Fermi CAR algebra (corresponding to the q-deformed Commutation Relations with q=-1), by showing that any state is symmetric if and only if it is conditionally independent and identically distributed with respect to the tail algebra. The Boolean stochastic processes provide examples for which the condition to be independent and identically distributed w.r.t. the tail algebra, without mentioning the {\it a-priori} existence of a preserving conditional expectation, is in general meaningless in the quantum setting. Finally, we study the ergodic properties of a class of remarkable states on the full group C*-algebra of the free group on infinitely many generators, that is the so-called Haagerup states.

math.PR↗
arXiv2013

A Nonconventional Ergodic Theorem for quantum "diagonal measures"

We extend the Nonconventional Ergodic Theorem for generic measures by Furstenberg, to several situations of interest arising from quantum dynamical systems. We deal with the diagonal state canonically associated to the product state (i.e. quantum "diagonal measures"), or to convex combinations of diagonal measures for non ergodic cases. For the sake of completeness, we treat also the Nonconventional Ergodic Theorem for compact dynamical systems, that is when the unitary generating the dynamics in the GNS representation is almost periodic. The Nonconventional Ergodic Theorem allows in a natural way to determine the limit of the three-point correlations, naturally relevant for the knowledge of the ergodic properties of a dynamical system.

math.OA↗
arXiv2012

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.

math-ph↗
arXiv2012

De Finetti theorem on the CAR algebra

The 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 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.

math.OA↗
arXiv2011

Bose Einstein condensation on inhomogeneous amenable graphs

We investigate the Bose-Einstein Condensation on nonhomogeneous amenable networks for the model describing arrays of Josephson junctions. 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. We show that for the nonhomogeneous networks like the comb graphs, particles condensate in momentum and configuration space as well. In this case different properties of the network, of geometric and probabilistic nature, such as the volume growth, the shape of the ground state, and the transience, all play a role in the condensation phenomena. The situation is quite different for homogeneous networks where just one of these parameters, e.g. the volume growth, is enough to determine the appearance of the condensation.

math.OA↗
arXiv2010

Harmonic analysis on perturbed Cayley Trees

We study the mathematical aspects of the Bose Einstein Condensation for the pure hopping model describing arrays of Josephson junctions on non homogeneous networks. The graphs under investigation are obtained by adding density zero perturbations to the homogeneous Cayley Trees. The resulting topological model is described by the (opposite of the) adjacency operator on the graph. In the present paper we investigate some relevant spectral properties of the adjacency of the perturbed network, such the appearance of the hidden spectrum below the norm, the transience character and the Perron Frobenius distribution. All the mentioned properties have a mathematical interest in itself, and have natural applications for the investigation of the BEC of Bardeen Cooper Bosons on the networks under consideration.

math.FA↗
arXiv2010

Disordered fermions on lattices and their spectral properties

We study Fermionic systems on a lattice with random interactions through their dynamics and the associated KMS states. We extend to the disordered CAR algebra, some standard results concerning the spectral properties exhibited by temperature states for disordered quantum spin systems. We discuss the Arveson spectrum and its connection with the Connes and Borchers $\G$-invariants for such W*-dynamical systems. In the case of KMS states exhibiting a natural property of invariance with respect to the spatial translations, some interesting properties, associated with standard spin-glass-like behaviour, emerge naturally. It covers infinite-volume limits of finite-volume Gibbs states, that is the quenched disorder for Fermions living on a standard lattice Z^d. In particular, we show that a temperature state of the systems under consideration can generate only a type III von Neumann algebra (with the type III_0 component excluded). Moreover, in the case of the pure thermodynamic phase, the associated von Neumann algebra is of type III_ł$ for some łin(0,1] independent of the disorder. Such a result is in accordance with the principle of self-averaging which affirms that the physically relevant quantities do not depend on the disorder. The present approach can be viewed as a further step towards fully understanding the very complicated structure of the set of temperature states of quantum spin glasses, and its connection with the breakdown of the symmetry for the replicas.

cond-mat.stat-mech↗
arXiv2010

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.

math.OA↗
arXiv2009

Ergodic properties of Bogoliubov automorphisms in free probability

We show that some $C^*$--dynamical systems obtained by "quantizing" classical ones on the free Fock space, enjoy very strong ergodic properties. Namely, if the classical dynamical system $(X, T, \m)$ is ergodic but not weakly mixing, then the resulting quantized system $(\gg,\a)$ is uniquely ergodic (w.r.t the fixed point algebra) but not uniquely weak mixing. The same happens if we quantize a classical system $(X, T, \m)$ which is weakly mixing but not mixing. In this case, the quantized system is uniquely weak mixing but not uniquely mixing. Finally, a quantized system arising from a classical mixing dynamical system, will be uniquely mixing. In such a way, it is possible to exhibit uniquely weak mixing and uniquely mixing $C^*$--dynamical systems whose GNS representation associated to the unique invariant state generates a von Neuman factor of one of the following types: $I_{\infty}$, $II_{1}$, $III_ł$ where $ł\in(0,1]$. The results listed above are extended to the $q$--commutation relations, provided $|q|<\sqrt2-1$.

math.OA↗
arXiv2009

The entangled ergodic theorem in the almost periodic case

Let $U$ be a unitary operator acting on the Hilbert space $\ch$, and $\a:\{1,..., 2k\}\mapsto\{1,..., k\}$ a pair partition. Then the ergodic average $$ \frac{1}{N^{k}}\sum_{n_{1},...,n_{k}=0}^{N-1} U^{n_{\a(1)}}A_{1}U^{n_{\a(2)}}... U^{n_{\a(2k-1)}}A_{2k-1}U^{n_{\a(2k)}} $$ converges in the strong operator topology provided $U$ is almost periodic, that is when $\ch$ is generated by the eigenvalues of $U$. We apply the present result to obtain the convergence of the Cesaro mean of several multiple correlations.

math.FA↗