Quantum stochastic convolution cocycles I

Abstract

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.

The Haagerup property for locally compact quantum groups

“…The construction of c follows from a GNS type construction for Lj Ker O , see for example [57,Section 6] (note that a different linearity convention for scalar products is used in [57]). The construction of (7.10) is [53,Theorem 4.6], attributed in that paper to Vergnioux.…”

On quantum stochastic differential equations

“…The following two special cases are relevant for the case of coalgebraic [12] and operator space-theoretic (Section 3 of this paper), quantum stochastic differential equations, respectively. The first applies in particular when V is finite-dimensional.…”

“…Sufficient conditions for the cocycle to satisfy a QSDE weakly, established for the case of C * -algebras in [16], remain valid in the coordinate-free, operator space context of this paper. A new result here, informed by a recent theorem on convolution cocycles [12], is the characterisation of cocycles on finite-dimensional operator spaces which, together with a conjugate process, satisfy a QSDE strongly-namely, they are the locally Hölder-continuous processes with exponent 1/2 whose conjugate process enjoys the same continuity.…”

“…He showed that each quantum Lévy process satisfies a QSDE of a certain type, with initial condition given by the counit of the underlying * -bialgebra (see (5.10) below). The notion of quantum Lévy process was recently generalised to quantum stochastic convolution cocycle on a coalgebra [12]. There it was shown that such objects arise as solutions of coalgebraic quantum stochastic differential equations.…”

Unitary Processes with Independent Increments and Representations of Hilbert Tensor Algebras

“…In [18,19], by a co-algebraic treatment, the second author has proved that any weakly continuous unitary stationary independent increment process on h ⊗ H, h finite dimensional, is unitarily equivalent to a Hudson-Parthasarathy flow with constant operator coefficients; see also [8,9]. In this present paper we treat the case of a unitary stationary independent increment process on h ⊗ H, h not necessarily finite dimensional, with normcontinuous expectation semigroup.…”