ar X iv : m at h / 07 03 33 9 v 1 [ m at h . FA ] 1 2 M ar 2 00 7 APPROXIMATION OF QUANTUM LÉVY PROCESSES BY QUANTUM RANDOM WALKS

Every quantum Lévy process with a bounded stochastic generator is shown to arise as a strong limit of a family of suitably scaled quantum random walks. The note is concerned with investigating convergence of random walks on quantum groups to quantum Lévy processes. The theory of the latter is a natural non-commutative counterpart of the theory of classical Lévy processes on groups ([Hey]). It has been initiated in [ASW] and further extensively developed by Schürmann, Schott and the first named author ([Sch], [FSc], [Fra]). In the series of recent papers ([LS 1−2 ], [Ska]) Lindsay and the second named author introduced and investigated the corresponding notion in the topological context of compact quantum groups (or, more generally, operator space coalgebras). Recent years brought also rapid development of the theory of random walks (discrete time stochastic processes) on discrete quantum groups ([Izu], [NeT], [Col]) initiated by Biane ([Bi 1−3 ]). In the context of quantum stochastic cocycles ([Lin] and references therein) the approximation of continous time evolutions by random walks was first investigated by Lindsay and Parthasarathy ([LiP]). They proved that under suitable assumptions scaled random walks converge weakly to *-homomorphic quantum stochastic cocycles. Recently certain results on the strong convergence have been obtained in papers [Sin] and [Sah] (see also [Bel] for the thorough analysis of the case of the vacuum adapted cocycles). Here we apply the ideas of the latter papers to the approximation of quantum Lévy processes (continuous time processes) on a compact quantum semigroup A by quantum random walks (discrete time processes) on A. Quantum random walks on C *-bialgebras. We start with the discussion of a notion of random walks on compact quantum semigroups. The class contains finite quantum groups, so we are in a natural way generalising the notion of quantum random walks considered in [FGo]. Here, and in everything that follows, ⊗ denotes the spatial tensor product of operator spaces (so in particular, also C *-algebras). Definition 1. A unital C *-algebra A is a C *-bialgebra if it is equipped with two unital *-homomorphisms ∆ : A → A⊗A and ǫ : A → C satisfying the coassociativity and counit conditions: (∆ ⊗ id A)∆ = (id A ⊗ ∆)∆, (ǫ ⊗ id A)∆ = (id A ⊗ ǫ)∆ = id A .