20041201
On the asymptotic behaviour of iterates of averages of unitary representations
Publication
Publication
Illinois Journal of Mathematics , Volume 48  Issue 4 p. 1117 1161
Let G be a locally compact group and μ a probability measure on G. Given a unitary representation μ of G, let Pμ, denote the μaverage ∫G π(g) μJ.(dg). μ is called neat if for every unitary representation π and every a in the support of μ, slim n∞ (Pμ n  π(a)nE μ) = 0, where Eμ is a canonically defined orthogonal projection. G is called neat if every almost aperiodic probability measure on G is neat. Previously known results show that every almost aperiodic spread out probability measure is neat, in particular, every discrete group is neat; furthermore, identity excluding groups, in particular, compact groups and nilpotent groups, are neat. In this work neatness of solvable Lie groups, connected algebraic groups, Euclidian motion groups, [SIN] groups, and extensions of abelian groups by discrete groups is established. Neatness of ergodic probability measures on any locally compact group is also proven. The key to these results is the result that when {Xn}n=1 ∞ is the left random walk of law μ on G and π a unitary representation in a separable Hubert space, then for every k = 0,1,..., the sequence π(Xn)1Pμ nk converges almost surely in the strong operator topology.
Additional Metadata  

Illinois Journal of Mathematics  
Organisation  School of Mathematics and Statistics 
Jaworski, W. (2004). On the asymptotic behaviour of iterates of averages of unitary representations. Illinois Journal of Mathematics, 48(4), 1117–1161.
