Isometric Dilations of Representations of Product Systems via Commutants

Skeide, Michael

doi:10.1142/s0129167x08004790

Cited by 6 publications

(5 citation statements)

References 32 publications

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“…The dilation result, which we state here, was proved for a fully coisometric covariant representation in [26,Theorem 3.7]. The general case was recently proved in [37,Theorem 1.1]. Theorem 4.9 ([26], [37]) Let {E(t)} t≥0 be a product system over a von Neumann algebra M and let {T t } t≥0 be a fully coisometric covariant representation of the product system on a Hilbert space H. Then there is another Hilbert space K, an isometry u 0 mapping H into K, and fully coisometric, isometric covariant representation {V t } t≥0 of {E(t)} t≥0 on K such that (1) u * 0 V t (ξ)u 0 = T t (ξ) for all ξ ∈ E(t), t ≥ 0.…”

Section: Proofmentioning

confidence: 71%

“…Theorem 4.9 ([26], [37]) Let {E(t)} t≥0 be a product system over a von Neumann algebra M and let {T t } t≥0 be a fully coisometric covariant representation of the product system on a Hilbert space H. Then there is another Hilbert space K, an isometry u 0 mapping H into K, and fully coisometric, isometric covariant representation {V t } t≥0 of {E(t)} t≥0 on K such that…”

Section: This Proves the Measurability Of L(ζ)mentioning

confidence: 99%

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Quantum Markov Semigroups: Product Systems and Subordination

Muhly

Solel

2007

Int. J. Math.

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We show that if a product system comes from a quantum Markov semigroup, then it carries a natural Borel structure with respect to which the semigroup may be realized in terms of a measurable representation. We show, too, that the dual product system of a Borel product system also carries a natural Borel structure. We apply our analysis to study the order interval consisting of all quantum Markov semigroups that are subordinate to a given one.

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Section: Proofmentioning

confidence: 71%

Section: This Proves the Measurability Of L(ζ)mentioning

confidence: 99%

Quantum Markov Semigroups: Product Systems and Subordination

Muhly

Solel

2007

Int. J. Math.

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“…Muhly and Solel [30] constructed from a Markov semigroup on a von Neumann algebra B a product system over the commutant of B. This product system turned out to be the commutant (see Skeide [41,44,43]) of the product system constructed in [14].…”

Section: Product Systems and Subproduct Systemsmentioning

confidence: 99%

Subproduct Systems and Cartesian Systems: New Results on Factorial Languages and their Relations with Other Areas

Gerhold

Skeide

2020

josa

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We point out that a sequence of natural numbers is the dimension sequence of a subproduct system if and only if it is the cardinality sequence of a word system (or factorial language). Determining such sequences is, therefore, reduced to a purely combinatorial problem in the combinatorics of words. A corresponding (and equivalent) result for graded algebras has been known in abstract algebra, but this connection with pure combinatorics has not yet been noticed by the product systems community. We also introduce Cartesian systems, which can be seen either as a set theoretic version of subproduct systems or an abstract version of word systems. Applying this, we provide several new results on the cardinality sequences of word systems and the dimension sequences of subproduct systems.

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“…But it is, in general, impossible to obtain suitable elements ξ s . It is possible to unitalize the kernel to the unitalizations A and B by the unitalization procedure in Skeide [Ske08] or, if all K s,s are strict, to the multiplier algebras.…”

Section: Skeidementioning

confidence: 99%

The Powers sum of spatial CPD-semigroups and CP-semigroups

Skeide

2010

Noncommutative Harmonic Analysis With Applications to Probability II

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Abstract. We define spatial CPD-semigroups and construct their Powers sum. We construct the Powers sum for general spatial CP-semigroups. In both cases, we show that the product system of that Powers sum is the product of the spatial product systems of its factors. We show that on the domain of intersection, pointwise bounded CPD-semigroups on the one side and Schur CP-semigroups on the other, the constructions coincide. This summarizes all known results about Powers sums and generalizes them considerably.1. Introduction. At the 2002 AMS-Workshop on 'Advances in Quantum Dynamics' in Mount Holyoke, Powers described a sum operation for spatial E 0 -semigroups on B(H), the algebra of bounded operators on a Hilbert space H. The result is a Markov semigroup and Powers asked for the product system of that Markov semigroup in the sense of Bhat [Bha96], and if that product system coincides or not with the tensor product of the Arveson systems of the E 0 -semigroups. (By Arveson system we shall refer to product systems of Hilbert spaces as introduced by Arveson [Arv89], while product system refers to the more general situation of Hilbert modules.)Still during the workshop (see Skeide [Ske03a]) we could show that the Arveson system of the Powers sum is our product of spatial product systems introduced in [Ske06] (preprint 2001) immediately for Hilbert modules. In the case of Hilbert spaces, the product is a subsystem of the tensor product. (For modules there is no tensor product of product systems.) Liebscher [Lie03] showed that the product may but need not be all of the tensor product. The question if the subsystem of the tensor product is nevertheless isomorphic to the full tensor product or not, remained open until Powers [Pow04]: It need not.

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Isometric Dilations of Representations of Product Systems via Commutants

Cited by 6 publications

References 32 publications

Quantum Markov Semigroups: Product Systems and Subordination

Quantum Markov Semigroups: Product Systems and Subordination

Subproduct Systems and Cartesian Systems: New Results on Factorial Languages and their Relations with Other Areas

The Powers sum of spatial CPD-semigroups and CP-semigroups

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