1,721,031 research outputs found
Ergodic theorems in Quantum Probability: an application to the monotone stochastic processes
Full text linkWe 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 Monotone and Boolean
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
Invariant Conditional Expectations and Unique Ergodicity for Anzai Skew-Products
Full text linkAnzai skew-products are shown to be uniquely ergodic with
respect to the fixed-point subalgebra if and only if there is a unique conditional expectation onto such a subalgebra which is invariant under the
dynamics. For the particular case of skew-products, this solves a question
raised by B. Abadie and K. Dykema in the wider context of C*-dynamical
systems
Braided endomorphisms of Cuntz algebras
No full textWe discuss sufficient conditions ensuring that certain endomorphisms of infinite factors arising from Cuntz algebras are braided. We analyse some explicit non-trivial examples associated to unitary solutions of quantum Yang-Baxter equations on a Hilbert space of dimension 2. In particular we show the existence of endomorphisms of index 2 associated to representations of Hecke algebras at a primitive fourth root of unity. In this case we compute the associated fusion rules. These fusion rules define a finitely generated *-semiring which is not finite. Such a picture seems to be closely related to the description of (the dual of) a deformation, at a fourth root of unity, of some compact matrix group. This could be of some interest for the investigation of quantum summetries naturally appearing in low-dimensional Quantum Field Theory
Limits of Some Weighted Cesaro Averages
Full text linkWe investigate the existence of the limit of some high order
weighted Cesaro averages
Symmetries and ergodic properties in Quantum Probability
Full text linkWe deal with the general structure of the stochastic processes by using the standard techniques of Operator Algebras. In this context, it appears natural that in the quantum case one can exhibit a huge class of such stochastic processes: each of them is associated to a quotient of the universal object made of the free product -algebra. The quantum (i.e. noncommutative) case describes the most general situation, and the classical (i.e. commutative) probability scheme is seen as a particular case of the quantum one. The ergodic properties of stationary and exchangeable processes are discussed in detail for many interesting cases arising from Quantum Physics and Quantum Probability
On the Thermodynamics of Particles Obeying Monotone Statistics
Full text linkThe aim of the present paper is to provide a preliminary investigation of the thermodynamics of particles obeying monotone statistics. To render the potential physical applications realistic, we propose a modified scheme called block-monotone, based on a partial order arising from the natural one on the spectrum of a positive Hamiltonian with compact resolvent. The block-monotone scheme is never comparable with the weak monotone one and is reduced to the usual monotone scheme whenever all the eigenvalues of the involved Hamiltonian are non-degenerate. Through a detailed analysis of a model based on the quantum harmonic oscillator, we can see that: (a) the computation of the grand-partition function does not require the Gibbs correction factor n! (connected with the indistinguishability of particles) in the various terms of its expansion with respect to the activity; and (b) the decimation of terms contributing to the grand-partition function leads to a kind of “exclusion principle” analogous to the Pauli exclusion principle enjoined by Fermi particles, which is more relevant in the high-density regime and becomes negligible in the low-density regime, as expected
Spectral and ergodic properties of completely positive maps and decoherence
Full text linkIn an attempt to propose more general conditions for decoherence to occur, we study spectral and ergodic properties of unital, completely positive maps on not necessarily unital C⁎-algebras, with a particular focus on gapped maps for which the transient portion of the arising dynamical system can be separated from the persistent one. After some general results, we first devote our attention to the abelian case by investigating the unital ⁎-endomorphisms of, in general non-unital, C⁎-algebras, and their spectral structure. The finite-dimensional case is also investigated in detail, and examples are provided of unital completely positive maps for which the persistent part of the associated dynamical system is equipped with the new product making it into a C⁎-algebra, and the map under consideration restricts to a unital ⁎-automorphism for this new C⁎-structure, thus generating a conservative dynamics on that persistent part
From discrete to continuous monotone C*-algebras via quantum central limit theorems
Full text linkWe prove that all finite joint distributions of creation and annihilation operators in
monotone and anti-monotone Fock spaces can be realised as Quantum Central Limit
of certain operators in 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 weights being related to the above
cited test functions
Wick order, spreadability and exchangeability for monotone commutation relations
Full text linkWe exhibit a Hamel basis for the concrete *-algebra associated to monotone commutation relations realised on the monotone Fock space, mainly composed by Wick ordered words of annihilators and creators. We apply such a result to investigate spreadability and exchangeability of the stochastic processes arising from such commutation relations. In particular, we show that spreadability comes from a monoidal action implementing a dissipative dynamics on the norm closure C*-algebra . Moreover, we determine the structure of spreadable and exchangeable monotone stochastic processes using their correspondence with spreading invariant and symmetric monotone states, respectively
Exchangeable stochastic processes and symmetric states in quantum probability
Full text linkWe 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 ∈ (−1, 1)
(the case q = 0 corresponding to the reduced group C∗-algebra C∗
r (F∞) of the free group F∞
on infinitely many generators), and the Boolean ones. A generalization of de Finetti theorem
to the Fermi CAR algebra (corresponding to the q-deformed Commutation Relations with
q = −1) is proven, by showing that any state is symmetric if and only if it is conditionally
independent and identically distributed with respect to the tail algebra. Moreover, we show
that 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 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 group
C∗-algebra C∗
(F∞), that is the so-called Haagerup states
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