AVERAGES OF UNITARY REPRESENTATIONS AND WEAK MIXING OF RANDOM-WALKS

Let S be a locally compact (sigma-compact) group or semigroup, and let T(t) be a continuous representation of S by contractions in a Banach space X. For a regular probability mu on S, we study the convergence o f the powers of the mu-average Ux = integral T(t)x d mu(t). Our main r esults for random walks on a group G are: (i) The following are equiva lent for an adapted regular probability on G: mu is strictly aperiodic ; U-n converges weakly for every continuous unitary representation of G; U is weakly mixing for any ergodic group action in a probability sp ace. (ii) If mu is ergodic on G metrizable, and U-n converges strongly for every unitary representation, then the random walk is weakly mixi ng: n(-1)Sigma(k=1)(n)\[mu(k) f,g]\ --> 0 for g is an element of L(i nfinity)(G) and f is an element of L(1)(G) with integral f d lambda = 0. (iii) Let G be metrizable, and assume that it is nilpotent, or that it has equivalent left and right uniform structures. Then mu is ergod ic and strictly aperiodic if and only if the random walk is weakly mix ing. (iv) Weak mixing is characterized by the asymptotic behaviour of mu(n) on UCBl(G).