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qGaussian processes: Noncommutative and classical aspects
 Commun. Math. Phys
, 1997
"... Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expec ..."
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Cited by 65 (2 self)
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Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation state. We show that there is a qanalogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on qGaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of qGaussian processes possess a noncommutative kind of Markov property, which ensures that there exist classical versions of these noncommutative processes. This answers an old question of Frisch and Bourret [FB].
OPERATOR SPACES AND ARAKIWOODS FACTORS – A QUANTUM PROBABILISTIC APPROACH –
, 2006
"... Abstract. We show that the operator Hilbert space OH introduced by Pisier embeds into the predual of the hyerfinite III1 factor. The main new tool is a Khintchine type inequality for the generators of the CAR algebra with respect to a quasifree state. Our approach yields a Khintchine type inequalit ..."
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Cited by 5 (2 self)
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Abstract. We show that the operator Hilbert space OH introduced by Pisier embeds into the predual of the hyerfinite III1 factor. The main new tool is a Khintchine type inequality for the generators of the CAR algebra with respect to a quasifree state. Our approach yields a Khintchine type inequality for the qgaussian variables for all values −1 ≤ q ≤ 1. These results are closely related to recent results of Pisier and Shlyakhtenko in the free case. Content: 0. Introduction and notation 1
Derived noncommutative continuous Bernoulli shifts
 In preparation
"... Abstract: We introduce a noncommutative extension of TsirelsonVershik’s noises [TV98, Tsi04], called (noncommutative) continuous Bernoulli shifts. These shifts encode stochastic independence in terms of commuting squares, as they are familiar in subfactor theory [Pop83, GHJ89]. Such shifts are, i ..."
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Cited by 2 (1 self)
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Abstract: We introduce a noncommutative extension of TsirelsonVershik’s noises [TV98, Tsi04], called (noncommutative) continuous Bernoulli shifts. These shifts encode stochastic independence in terms of commuting squares, as they are familiar in subfactor theory [Pop83, GHJ89]. Such shifts are, in particular, capable of producing Arveson’s product system of type I and type II [Arv03]. We investigate the structure of these shifts and prove that the von Neumann algebra of a (scalarexpected) continuous Bernoulli shift is either finite or of type III. The role of (‘classical’) Gstationary flows for TsirelsonVershik’s noises is now played by cocycles of continuous Bernoulli shifts. We show that these cocycles provide an operator algebraic notion for Lévy processes. They lead, in particular, to units and ‘logarithms ’ of units in Arveson’s product systems [Kös04a]. Furthermore, we introduce (noncommutative) white noises, which are operator algebraic versions of Tsirelson’s ‘classical ’ noises. We give examples coming from probability, quantum probability and from Voiculescu’s theory of free probability [VDN92]. Our main result is a bijective correspondence between additive and unital shift cocycles. For the proof of the correspondence we develop tools which are of interest on their own: noncommutative extensions of stochastic Itô integration, stochastic logarithms and exponentials.
Classical dilations à la Quantum Probability of Markov evolutions in discrete time. Quaderno di Dipartimento QDD
"... We study the Classical Probability analogue of the dilations of a quantum dynamical semigroup in Quantum Probability. Given a (not necessarily homogeneous) Markov chain in discrete time in a finite state space E, we introduce a second system, an environment, and a deterministic invertible timehomog ..."
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We study the Classical Probability analogue of the dilations of a quantum dynamical semigroup in Quantum Probability. Given a (not necessarily homogeneous) Markov chain in discrete time in a finite state space E, we introduce a second system, an environment, and a deterministic invertible timehomogeneous global evolution of the system E with this environment such that the original Markov evolution of E can be realized by a proper choice of the initial random state of the environment. We also compare this dilations with the dilations of a quantum dynamical semigroup in Quantum Probability: given a classical Markov semigroup, we show that it can be extended to a quantum dynamical semigroup for which we can find a quantum dilation to a group of ∗automorphisms admitting an invariant abelian subalgebra where this quantum dilation gives just our classical dilation. AMS Subject Classification: 60J10, 81S25. 1
Operator spaces and ArakiWoods factors
"... Abstract. We show that the operator Hilbert space OH introduced by Pisier embeds into the predual of the hyerfinite III1 factor. The main new tool is a Khintchine type inequality for the generators of the CAR algebra with respect to a quasifree state. Our approach yields a Khintchine type inequalit ..."
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Cited by 1 (1 self)
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Abstract. We show that the operator Hilbert space OH introduced by Pisier embeds into the predual of the hyerfinite III1 factor. The main new tool is a Khintchine type inequality for the generators of the CAR algebra with respect to a quasifree state. Our approach yields a Khintchine type inequality for the qgaussian variables for all values −1 ≤ q ≤ 1. These results are closely related to recent results of Pisier and Shlyakhtenko in the free case. 0. Introduction and Notation Probabilistic methods play an important role in the theory of operator algebras and Banach spaces. It is not surprising that a quantized theory of Banach spaces will require tools from quantum probability. This connection between noncommutative probability and the recent theory of operator spaces (sometimes called quantized Banach spaces) is wellestablished through
QHarmonic Oscillator in a Lattice Model
"... We give an explicit proof of the pair partitions formula for the moments of the qharmonic oscillator, and of the claim made by G. Parisi that the qdeformed lattice Laplacian on the ddimensional lattice tends to the qharmonic oscillator in distribution for d !1. 1 ..."
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Cited by 1 (0 self)
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We give an explicit proof of the pair partitions formula for the moments of the qharmonic oscillator, and of the claim made by G. Parisi that the qdeformed lattice Laplacian on the ddimensional lattice tends to the qharmonic oscillator in distribution for d !1. 1
DAMTP/9220 QUANTUM RANDOM WALKS AND TIMEREVERSAL
, 1992
"... ABSTRACT Classical random walks and Markov processes are easily described by Hopf algebras. It is also known that groups and Hopf algebras (quantum groups) lead to classical and quantum diffusions. We study here the more primitive notion of a quantum random walk associated to a general Hopf algebra ..."
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ABSTRACT Classical random walks and Markov processes are easily described by Hopf algebras. It is also known that groups and Hopf algebras (quantum groups) lead to classical and quantum diffusions. We study here the more primitive notion of a quantum random walk associated to a general Hopf algebra and show that it has a simple physical interpretation in quantum mechanics. This is by means of a representation theorem motivated from the theory of Kac algebras: If H is any Hopf algebra, it may be realised in Lin(H) in such a way that ∆h = W(h ⊗ 1)W −1 for an operator W. This W is interpreted as the time evolution operator for the system at time t coupled quantummechanically to the system at time t + δ. Finally, for every Hopf algebra there is a dual one, leading us to a duality operation for quantum random walks and quantum diffusions and a notion of the coentropy of an observable. The dual system has its time reversed with respect to the original system, leading us to a CTPtype theorem. 1
The operators P and Q 1.1 The Hilbert space
, 2011
"... A teljes könyv címe: An invitation to the algebra of the canonical commutation relation ..."
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A teljes könyv címe: An invitation to the algebra of the canonical commutation relation