Random-walk approximation to vacuum cocycles

Belton, Alexander C. R. (2010) Random-walk approximation to vacuum cocycles. Journal of the London Mathematical Society, 81 (2). pp. 412-434. ISSN 0024-6107

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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 that 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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Journal Article
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Journal of the London Mathematical Society
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26 Apr 2010 10:30
Last Modified:
16 Sep 2023 00:24