Random-walk approximation to vacuum cocycles

Quantum random walks are constructed on operator spaces using the concept of matrix-space lifting, a form of ampliation intermediate between those given by spacial and ultraweak tensor products. It is shown that these walks, after suitable scaling, converge in a strong sense to vacuum cocycles, certain vacuum-adapted processes which are Feller cocycles in the sense of Lindsay and Wills. This result is employed to give a new proof of the existence of ∗-homomorphic quantum-stochastic dilations for completely positive contraction semigroups on von Neumann algebras and separable unital C∗ algebras.