Weak Mixing of Random Walks on Groups

Let G be a locally compact σ-compact group with right Haar measure m and μ a regular probability measure on G. We say that μ is weakly mixing if for all g∈L∞(G) and all f∈L1(G) with ∫fdm=0 we have n−1∑nk=1|〈μk*f,g〉|→0. We show that μ is weakly mixing if and only if μ is ergodic and strictly aperiodic. To prove this we use and prove some results about unimodular eigenvalues for general Markov operators.