Quantum random walk approximation in Banach algebra

Belton's discrete approximation scheme is extended to Banach-algebra-valued sesquilinear quantum stochastic cocycles, through the dyadic discretisation of time. Approximation results for Markov-regular quantum stochastic mapping cocycles are recovered. We also obtain a new random walk approximation for a class of (not necessarily Markov-regular) isometric operator cocycles. Every Lévy process on a compact quantum group is implemented by a unitary cocycle from this class.