Completely positive quantum stochastic convolution cocycles and their dilations

Stochastic generators of completely positive and contractive quantum stochastic convolution cocycles on a C∗-hyperbialgebra are characterised. The characterisation is used to obtain dilations and stochastic forms of Stinespring decomposition for completely positive convolution cocycles on a C∗-bialgebra. Stochastic (or Markovian) cocycles on operator algebras are basic objects of interest in quantum probability ([1]) and have been extensively investigated using quantum stochastic analysis (see [12]). There is also a well-developed theory of quantum Lévy processes, that is stationary, independent-increment, *-homomorphic processes on a *-bialgebra (see [6, 22] and references therein). Close examination of these two directions has naturally led to the notion of quantum stochastic convolution cocycle on a quantum group (or, more generally, on a coalgebra), as introduced and investigated in [14] in an algebraic context, and in [17] in the analytic context of compact quantum groups. The main results have been summarised in [15]. Recent years have also seen an increased interest in the noncommutative generalisation of classical hypergroups ([3]), initiated by Chapovsky and Vainerman ([5]) and continued, for example, in the papers [10] and [11]. Compact quantum hypergroups differ from compact quantum groups in that their coproduct need not be multiplicative. However, it remains completely positive, which makes compact quantum hypergroups, or more generally C∗-hyperbialgebras, an appropriate category for the consideration of completely positive quantum stochastic convolution cocycles in a topological context (for the purely algebraic case see [7]). These cocycles may be viewed as natural counterparts of stationary, independent-increment processes on hypergroups. In [17] it is shown that, under certain regularity conditions, they satisfy coalgebraic quantum stochastic differential equations. The aim of this paper is to prove dilation theorems for quantum stochastic convolution cocycles on a C∗-bialgebra. To this end it is first necessary to establish the detailed structure of the stochastic generators of completely positive and contractive convolution cocycles. We give a direct derivation of this exploiting ideas used in the analysis of standard quantum stochastic cocycles with finite-dimensional noise space ([13]). Once the structure of generators is known, one may consider question of dilating completely positive convolution † Permanent address: Department of Mathematics, University of Łódź, ul. Banacha 22, 90-238 Łódź, Poland. 202 ADAM G. SKALSKI cocycles to ∗-homomorphic ones. In the context of standard quantum stochastic cocycles this problem was treated in [8] and [9] (see also [2]). In the first of these papers it was shown that every Markov-regular completely positive and contractive cocycle arises as the image of a ∗-homomorphic cocycle under a vacuum conditional expectation which averages out some dimensions of the quantum noise. In the second every Markov-regular completely positive and contractive cocycle was shown to be realisable as a composition of a ∗-homomorphic cocycle with conjugation by a contraction operator process. This may be seen as a stochastic Stinespring decomposition. In this paper using the techniques of Goswami, Lindsay, Sinha and Wills we obtain analogous results for convolution cocycles on C∗-bialgebras. Multiplicativity of the coproduct is necessary here for obtaining dilations to ∗-homomorphic cocycles. An alternative approach to the one presented in this paper would be to exploit more directly theorems known for standard quantum stochastic convolution cocycles and properties of the R-map introduced in [17]. In that paper the general form of the stochastic generators of completely positive and contractive convolution cocycles was determined by using a particular representation of the C∗-bialgebra in question and appealing directly to the results of [13], [18] and [20]; similar methods may be further used to obtain the dilation results presented here. One drawback of such an approach is that it involves using the deep Christensen-Evans theorem on quasi-innerness of derivations on represented C∗-algebras. Another is the necessity to reformulate the results of [8] and [9] in coordinate-free quantum stochastic calculus. This is also necessary for overcoming separability assumptions on the noise dimension spaces. Finally the von Neumann algebraic framework used in [8] would require nontrivial modifications. In sum, the method presented here has the advantage of being more elementary. The paper is structured as follows. The first section contains the notation and elements of quantum stochastic analysis and operator space theory needed. In the second section C∗-hyperbialgebras are defined and the well-known technique of obtaining them from C∗bialgebras via a noncommutative conditional expectation is recalled. Basic facts concerning quantum stochastic convolution cocycles and the structure of their stochastic generators in the completely positive and ∗-homomorphic cases are also included here. In the third section a more detailed description of the stochastic generators of Markov-regular, completely positive, contractive convolution cocycles, in terms of a certain tuple of objects, is derived. Dilations of such convolution cocycles on a C∗-bialgebra to *-homomorphic convolution cocycles are given in the fourth section, and the fifth section contains a stochastic Stinespring decomposition.