Quantum Markov Semigroups: Product Systems and Subordination

Muhly, Paul S.; Solel, Baruch

doi:10.1142/s0129167x07004254

Cited by 9 publications

(8 citation statements)

References 30 publications

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“…Put E t = C B (B(G, H t )) where H t := F t G with ρ t and ρ t the canonical action of B and of B, repectively, so that F ∼ = E as product system of W * -correspondences. (E is precisely what [19,21] would call theσ 0 -dual of F .) Under these isomorphisms, the covariant representationσ of F induces a normal covariant representation σ of E onG with σ 0 = id B .…”

Section: Proof Of Theorem 11mentioning

confidence: 97%

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Isometric Dilations of Representations of Product Systems via Commutants

Skeide

2008

Int. J. Math.

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We construct a weak dilation of a not necessarily unital CP-semigroup to an E-semigroup acting on the adjointable operators of a Hilbert module with a unit vector. We construct the dilation in such a way that the dilating E-semigroup has a pre-assigned product system. Then, making use of the commutant of von Neumann correspondences, we apply the dilation theorem to prove that covariant representations of product systems admit isometric dilations.

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Section: Proof Of Theorem 11mentioning

confidence: 97%

“…What is still missing is the product system structure of this family. Computations of this type have been detailed also in Muhly and Solel [21] (in the language of σ-duals) so that here we may content ourselves with a sketchy description. G).…”

Section: ) Let T Be the Cp-semigroup Determined By Either Of The Ingrmentioning

confidence: 99%

Isometric Dilations of Representations of Product Systems via Commutants

Skeide

2008

Int. J. Math.

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“…Here one sees for the first time subproduct systems and product systems of Hilbert C * -modules. Muhly and Solel [11] took a dual approach to achieve this, where they have called these Hilbert C * -modules as C * -correspondences.…”

Section: It Is Well-known That a Block Matrixmentioning

confidence: 99%

Structure of block quantum dynamical semigroups and their product systems

Bhat

Kumar

2020

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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Paschke’s version of Stinespring’s theorem associates a Hilbert [Formula: see text]-module along with a generating vector to every completely positive map. Building on this, to every quantum dynamical semigroup (QDS) on a [Formula: see text]-algebra [Formula: see text] one may associate an inclusion system [Formula: see text] of Hilbert [Formula: see text]-[Formula: see text]-modules with a generating unit [Formula: see text]. Suppose [Formula: see text] is a von Neumann algebra, consider [Formula: see text], the von Neumann algebra of [Formula: see text] matrices with entries from [Formula: see text]. Suppose [Formula: see text] with [Formula: see text] is a QDS on [Formula: see text] which acts block-wise and let [Formula: see text] be the inclusion system associated to the diagonal QDS [Formula: see text] with the generating unit [Formula: see text] It is shown that there is a contractive (bilinear) morphism [Formula: see text] from [Formula: see text] to [Formula: see text] such that [Formula: see text] for all [Formula: see text] We also prove that any contractive morphism between inclusion systems of von Neumann [Formula: see text]-[Formula: see text]-modules can be lifted as a morphism between the product systems generated by them. We observe that the [Formula: see text]-dilation of a block quantum Markov semigroup (QMS) on a unital [Formula: see text]-algebra is again a semigroup of block maps.

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“…Θ z (a) := z(I E ⊗ a)z * , a ∈ σ(M ) . We have used this relation between intertwiners and completely positive maps in a number of places in our work (see, e.g., [10,11,13,16]). Its importance lies in the fact that the powers of Θ z can be expressed in terms of z via the formula…”

Section: Weighted Shifts 511mentioning

confidence: 99%

Matricial Function Theory and Weighted Shifts

Muhly

Solel

2016

Integr. Equ. Oper. Theory

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Let T+(E) be the tensor algebra of a W * -correspondence E over a W * -algebra M . In earlier work, we showed that the completely contractive representations of T+(E), whose restrictions to M are normal, are parametrized by certain discs or balls D(E, σ) indexed by the normal * -representations σ of M . Each disc has analytic structure, and each element F ∈ T+(E) gives rise to an operator-valued function Fσ on D(E, σ) that is continuous and analytic on the interior. In this paper, we explore the effect of adding operator-valued weights to the theory. While the statements of many of the results in the weighted theory are anticipated by those in the unweighted setting, substantially different proofs are required. Interesting new connections with the theory of completely positive maps are developed. Our perspective has been inspired by work of Vladimir Müller (J Oper Theory 20(1):3-20, 1988) in which he studied operators that can be modeled by parts of weighted shifts. Our results may be interpreted as providing a description of operator algebras that can be modeled by weighted tensor algebras. Our results also extend work of Gelu Popescu (Mem Am Math Soc 200(941):vi+91, 2009), who investigated similar questions.

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Quantum Markov Semigroups: Product Systems and Subordination

Cited by 9 publications

References 30 publications

Isometric Dilations of Representations of Product Systems via Commutants

Isometric Dilations of Representations of Product Systems via Commutants

Structure of block quantum dynamical semigroups and their product systems

Matricial Function Theory and Weighted Shifts

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