Quantum independent increment processes on superalgebras

Accardi, Luigi; Schürmann, Michael; Waldenfels, Wilhelm von

doi:10.1007/bf01162868

Cited by 56 publications

(60 citation statements)

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“…There is a * -mapping ι V : V → T(V) such that for each * -mapping κ : V → A from V into a * -algebra A there exists a unique * -algebra homomorphism T(κ) : T(V) → A with κ = T(κ)•ι V . The * -algebra T(V) becomes a dual group if we extend the mappings v → v (1) Using the procedure described above, we obtain that Lévy processes on T(V) are of the form (5.31) which means that, apart from a multiple of the time t, they are sums of annihilation, preservation and creation processes; cf. [15,16].…”

Section: Definition 42 a Family (ϕ T ) T 0 Is Called A Convolution mentioning

confidence: 99%

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Quantum Lévy processes on dual groups

Ghorbal

Schürmann

2005

Math. Z.

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We present a theory of quantum (non-commutative) Lévy processes on dual groups which generalizes the theory of Lévy processes on bialgebras. It follows from a result of N. Muraki that there exist exactly 5 notions of non-commutative 'positive' stochastic independence. We show that one can associate a commutative bialgebra with each pair consisting of a dual group and one of the 5 notions of independence. This construction is related to a construction of U. Franz. Our construction has the advantage that the important case of free independence is included. We show that Lévy processes are given by their generators which are precisely the conditonally positive linear functionals on the dual group.

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Section: Definition 42 a Family (ϕ T ) T 0 Is Called A Convolution mentioning

confidence: 99%

“…. · b n with n 2 and ψ = D( ).Conversely, for a linear functional ψ on S (B) the convolution exponentials exp (t ψ) be formed (see[1,15]). They satisfy the conditions (4.23) and (4.24).The linear functional D ( ) is an S (0)-derivation with D ( ) = • p B , i.e.…”

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confidence: 99%

Quantum Lévy processes on dual groups

Ghorbal

Schürmann

2005

Math. Z.

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“…One possibility in non-commutative probability, where the state space of a process is replaced by a * -algebra, is to take involutive bialgebras. This approach was followed in [ASW88,Sch93,FS99] and has lead to a rich theory, but it only works for tensor independence. Another possibility is to take the dual (semi-) groups of [Voi87], also called H-algebras [Zha91] or cogroups [BH96].…”

Section: Introductionmentioning

confidence: 99%

Unification of boolean, monotone, anti-monotone, and tensor independence and L�vy processes

Franz

2003

Mathematische Zeitschrift

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The unification of the boolean and the tensor product of quantum probability spaces given by Lenczewski [Len98] is modified to include the monotone product and its mirror image, the anti-monotone product, and used to reduce the theories of boolean, monotone, and anti-monotone Lévy processes on dual semi-groups to the theory of Lévy processes on involutive bialgebras. This leads to a representation theorem for these processes. Further applications are a proof of the Schoenberg correspondence for boolean, monotone, and anti-monotone convolution semi-groups, a realization of boolean, monotone, and anti-monotone creation, annihilation, and conservation operators on boson Fock spaces, and a discussion of the Markov structure of these Lévy processes.

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“…It was first introduced by Accardi, Schürmann and von Waldenfels, in the purely algebraic framework of * -bialgebras [1], and was further developed by Schürmann and others [7,22] who, in particular, extended it to other noncommutative forms of independence (free, boolean and monotone), still in the algebraic context. Inspired by Schürmann's reconstruction theorem, which states that every quantum Lévy process on a * -bialgebra can be equivalently realised on a symmetric Fock space, we first showed how the algebraic theory of quantum Lévy processes can be extended to the natural setting of quantum stochastic convolution cocycles [14].…”

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confidence: 99%

Quantum stochastic convolution cocycles III

Lindsay

Skalski

2011

Math. Ann.

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Every Markov-regular quantum Lévy process on a multiplier C * -bialgebra is shown to be equivalent to one governed by a quantum stochastic differential equation, and the generating functionals of norm-continuous convolution semigroups on a multiplier C * -bialgebra are then completely characterised. These results are achieved by extending the theory of quantum Lévy processes on a compact quantum group, and more generally quantum stochastic convolution cocycles on a C * -bialgebra, to locally compact quantum groups and multiplier C * -bialgebras. Strict extension results obtained by Kustermans, together with automatic strictness properties developed here, are exploited to obtain existence and uniqueness for coalgebraic quantum stochastic differential equations in this setting. Then, working in the universal enveloping von Neumann bialgebra, we characterise the stochastic generators of Markov-regular, *-homomorphic (respectively completely positive and contractive), quantum stochastic convolution cocycles. Mathematics Subject Classification (2000)Primary 46L53 · 81S25; Secondary 22A30 · 47L25 · 16W30 IntroductionLet G be a locally compact quantum semigroup. The main results of this paper are as follows: a concrete realisation of each abstract quantum Lévy process on G

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Quantum independent increment processes on superalgebras

Cited by 56 publications

References 10 publications

Quantum Lévy processes on dual groups

Quantum Lévy processes on dual groups

Unification of boolean, monotone, anti-monotone, and tensor independence and L�vy processes

Quantum stochastic convolution cocycles III

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