Random-walk approximation to vacuum cocycles

Details Random-walk approximation to vacuum cocycles Random-walk approximation to vacuum cocycles

2007-02-23

arxiv.org/abs/math/0702700v4

Journal of the London Mathematical Society (Second Series) 81 (2010), 412-434

Mathematics

Operator Algebras

32 pages; v4: further strengthening of the main theorems; a new example; some re-writing and re-organisation

Scientific paper

10.1112/jlms/jdp075

Quantum random walks are constructed on operator spaces with the aid of matrix-space lifting, a type of ampliation intermediate between those provided by spatial and ultraweak tensor products. Using a form of Wiener-Ito decomposition, a Donsker-type theorem is proved, showing that these walks, after suitable scaling, converge in a strong sense to vacuum cocycles: these are vacuum-adapted processes which are Feller cocycles in the sense of Lindsay and Wills. This 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. The analogous approximation result is also established within the standard quantum stochastic framework, using the link between the two types of adaptedness.

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