Homomorphic Feller Cocycles on a $C^*$-Algebra

Skalski

2011

Math. Ann.

Self Cite

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

Section: In This Case the Convolution Cocycle L Is Nondegenerate If mentioning

confidence: 96%

“…A detailed summary of the relevant results from QS analysis [18][19][20][21]15] is given in [16]. We shall therefore be brief here.…”

Section: Quantum Stochastic Processes Differential Equations and Cocmentioning

confidence: 99%

See 1 more Smart Citation

Quantum stochastic convolution cocycles III

Skalski

2011

Math. Ann.

Self Cite

“…Integration of a sequence is realised by summing the iterated quantum stochastic integrals of its terms (cf. [17,28]). Solutions form a cocycle in the above sense, with respect to the standard shift on Fock space over L 2 (R + ; k), and are strongly 1 2 -Hölder-continuous.…”

Section: Introductionmentioning

confidence: 94%

Quantum stochastic convolution cocycles I

Annales de l'Institut Henri Poincare (B) Probability and Statis

Skalski

2005

Stochastic convolution cocycles on a coalgebra are obtained by solving quantum stochastic differential equations. We describe a direct approach to solving such QSDE's by iterated quantum stochastic integration of matrix-sum kernels. The cocycles arising this way satisfy a Hölder condition, and it is shown that conversely every such Hölder-continuous cocycle is governed by a QSDE. Algebraic structure enjoyed by matrix-sum kernels yields a unital * -algebra of processes which allows easy deduction of homomorphic properties of cocycles on a 'quantum semigroup'. This yields a simple proof that every quantum Lévy process may be realised in Fock space. Finally perturbation of cocycles by Weyl cocycles is shown to be implemented by the action of the corresponding Euclidean group on Schürmann triples.  2005 Elsevier SAS. All rights reserved. RésuméDes cocycles de convolution stochastiques sur une coalgèbre sont obtenus par résolution des équations différentielles stochastiques (EDS) quantiques. Nous décrivons une méthode directe pour résoudre les EDS quantiques par intégration stochastique quantique itérée de noyaux matrice-somme. Les cocycles qui sont obtenus par cette méthode satisfont une condition de Hölder, et nous montrons réciproquement que chaque cocycle Hölder-continu est gouverné par une EDS quantique. La structure algé-brique des noyaux matrice-somme donne une *-algèbre de processus, qui nous permet une déduction facile des propriétés homomorphiques de cocycles sur un groupe quantique. Ce résultat permet d'obtenir un argument simple pour montrer que chaque processus de Lévy quantique peut être realisé dans l'espace de Fock. Finalement, nous montrons que la perturbation des cocycles par des cocycles de Weyl est mise en oeuvre par l'action du groupe euclidien correspondant sur les triplets de Schürmann.  2005 Elsevier SAS. All rights reserved.

“…12 There necessary and sufficient conditions on θ are found by means of a product formula for iterated quantum stochastic integrals (cf. the work of Hudson and co-authors 5 ).…”

Section: Introductionmentioning

confidence: 99%

Multiplicativity via a Hat Trick

Quantum Probability and Infinite-Dimensional Analysis

Wills

2003

A short proof of the homomorphic property for Fock-adapted regular Markovian cocycles is given for two cases. The first is new, and facilitates dilation of quantum dynamical semigroups on a separable C * -algebra. IntroductionLet j be a Fock-adapted regular Markovian cocycle on A with noise dimension space k, where A is a unital C * -algebra acting on a Hilbert space h, and k is a separable Hilbert space. If j is completely bounded (CB) then the cocycle relation it enjoys may be expressed simplywhere σ is the Fock space shift semigroup and  s is the map between matrix spaces M(F s ; A) b and M(F; A) b (defined below) induced by j s . Regularity for the cocycle means that its associated semigroups are norm continuous. If j is completely positive and contractive then it has a CB stochastic generator where k := C ⊕ k and P is the orthogonal projection in B( k) with range k ⊂ k. Conversely, (0.2) entails complete boundedness of θ, which implies that θ generates a regular Markovian cocycle j on A; 10 in turn (0.2) also implies that j is completely positive and contractive (Lemma 2.1 below). The question we seek to answer is when is j multiplicative too, and so * -homomorphic? When k has finite dimension d, Evans 3 proved multiplicativity by showing that the difference J t (a, b, ζ, η) := j t (a)ζ, j t (b)η − ζ, j t (a * b)η satisfies3)