Completely positive quantum stochastic convolution cocycles and their dilations

Abstract

Abstract. 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 ([Acc]) and have been extensively investigated using quantum stochastic analysis (see [Lin]). There is also a well-developed theory of quantum Lévy processes, that is stationary, independent-increment, *-homomorphic processes on a *-bialgebra (see [Sch], [Fra] 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 [ . 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 [FrS]). These cocycles may be viewed as natural counterparts of stationary, independent-increment processes on hypergroups. In [LS 4 ] 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 ([LiP]). Once the structure of generators is known, one may consider question of dilating completely positive convolution cocycles to * -homomorphic ones. In the context of standard quantum stochastic cocycles this problem was treated in [GLSW] and [GLW] (see also [Bel]). In the first of these papers it was shown

On quantum stochastic differential equations

“…Extension of the results of that paper to the context of compact quantum groups, or more generally operator space coalgebras [14], was our motivation for analysing quantum stochastic differential equations with nontrivial initial conditions on an operator space. Results obtained here have also enabled the development of a dilation theory for completely positive convolution cocycles on a C * -bialgebra [26].…”

“…This is done in the paper [14] which also contains many examples. Dilation of completely positive convolution cocycles on a C * -bialgebra to * -homomorphic convolution cocycles is treated in [26]. The main results, in both the algebraic and analytic cases, are summarized in [13].…”

Quantum Random Walk Approximation on Locally Compact Quantum Groups

“…It follows from Proposition 4.3 and Theorem 4.4 of [26], and their proofs, that there is a Hilbert space h containing k and a χ -structure map θ : A → B( h) such that ϕ is the compression of θ to B( k). The family of *-homomorphisms…”