Quantum Stochastic Calculus on Full Fock Modules

Skeide, Michael

doi:10.1006/jfan.2000.3579

Cited by 23 publications

(20 citation statements)

References 29 publications

(80 reference statements)

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“…(11) According to Theorem 10.1. of [24] (see the end of [24, Chapter 10] to make the link with this particular case), there exists a unique solution to (11) whenever W = (W ij ) 1≤i,j≤n is unitary and R = (R ij ) 1≤i,j≤n is selfadjoint, which is indeed true thanks to Proposition 4.9. Finally, there exists a unique solution (j t (u ij )) n i,j=1 to the coupled stochastic equations (9), and another consequence of [24, Theorem 10.1] is that (j t (u ij )) n i,j=1 is unitary.…”

Section: Proof Of Theorem 47mentioning

confidence: 98%

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Haar states and Lévy processes on the unitary dual group

Cébron

Ulrich

2016

Journal of Functional Analysis

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Abstract. We study states on the universal noncommutative * -algebra generated by the coefficients of a unitary matrix, or equivalently states on the unitary dual group. Its structure of dual group in the sense of Voiculescu allows to define five natural convolutions. We prove that there exists no Haar state for those convolutions. However, we prove that there exists a weaker form of absorbing state, that we call Haar trace, for the free and the tensor convolutions. We show that the free Haar trace is the limit in distribution of the blocks of a Haar unitary matrix when the dimension tends to infinity. Finally, we study a particular class of free Lévy processes on the unitary dual group which are also the limit of the blocks of random matrices on the classical unitary group when the dimension tends to infinity.

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Section: Proof Of Theorem 47mentioning

confidence: 98%

“…The quantum stochastic calculus allows us to write the quantum stochastic differential equation of j t (b * c), thanks to the following result. Theorem 4.11 (Corollary 9.2. of [24]). Let h, h ′ ∈ H and W, W ′ ∈ B(H).…”

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confidence: 95%

Haar states and Lévy processes on the unitary dual group

Cébron

Ulrich

2016

Journal of Functional Analysis

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“…Still today most known examples have been constructed with a calculus for tensor independence based on the fundamental work [HP84]; see the monograph [Par92] and the up-to-date survey [Lin05]. But there are also calculi on other Fock spaces [BSW82, KS92] or modules [GS99,Ske00] or even representation free versions [AFQ92, Kös00]. For all of them a program like the one indicated in the preceding paragraph for the free case still has to be carried out.…”

Section: Commutes For All T ≥ 0 a Dilation Is Unital If I Is Unitamentioning

confidence: 99%

Mini-Workshop: Product Systems and Independence in Quantum Dynamics

Bhat¹,

Franz²,

Skeide³

2009

Oberwolfach Rep.

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Abstract. Quantum dynamics, both reversible (i.e., closed quantum systems) and irreversible (i.e., open quantum systems), gives rise to product systems of Hilbert spaces or, more generally, of Hilbert modules. When we consider reversible dynamics that dilates an irreversible dynamics, then the product system of the latter is equal to the product system of the former (or is contained in a unique way). Whenever the dynamics is on a proper subalgebra of the algebra of all bounded operators on a Hilbert space, in particular, when the open system is classical (commutative) it is indispensable that we use Hilbert modules.The product system of a reversible dynamics is intimately related to a filtration of subalgebras that are independent in a state or conditionally independent in a conditional expectation of the reversible system. This has been illustrated in many concrete dilations that have been obtained with the help of quantum stochastic calculus. Here the underlying Fock space or module determines the sort of quantum independence underlying the reversible system.The mini-workshop brought together experts from quantum dynamics, product systems and quantum independence who have contributed to the general theory or who have studied intriguing examples. As the implications of the tight relationship between product systems and independence had so far been largely neglected, we expect from our mini-workshop a strong innovative impulse to this field.

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“…After these pioneering works, a great number of papers was devoted to develop a theory in non Boson cases (see e.g. [10] for the Fermion case, [15] for universal invariant case, [24] for free, [18] for general quasi-free, [12] for Boolean, [23] for full Fock module). Accardi, Fagnola and Quaegebeur in [3] reached a double result: on the one hand developing a theory independent of the particular representation chosen (as in the classical case) and on the other hand including all the quantum stochastic calculi already appeared (boson and fermion) into a unifying picture.…”

Section: Introductionmentioning

confidence: 99%

Quantum stochastic calculus on interacting Fock spaces: semimartingale estimates and stochastic integral

Crismale¹

2007

COSA

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Quantum Stochastic Calculus on Full Fock Modules

Cited by 23 publications

References 29 publications

Haar states and Lévy processes on the unitary dual group

Haar states and Lévy processes on the unitary dual group

Mini-Workshop: Product Systems and Independence in Quantum Dynamics

Quantum stochastic calculus on interacting Fock spaces: semimartingale estimates and stochastic integral

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