JOURNAL OF FUNCTIONAL ANALYSIS 85, 86-102 (1989) Convergence of Iterates of Averages of Certain Operator Representations and of Convolution Powers YVES DERRIENNIC UniversitC de Bretagne Occidentale, Dbpartement de Mathbmatiques et Informatique, 6, avenue Le Gorgeu, 29287 Brest, France AND MICHAEL LIN* Department of Mathematics and Computer Science, Ben Gurion University of the Negev, Beer-Sheva, Israel (1) Let G be a locally compact u-compact group and let T(t) be a continuous representation of G by isometries in a uniformly convex Banach space B. Let p be an adapted, strictly aperiodic and spread out probability on G, and let Px = j T( t)x dp( t). Then P” converges strongly. (2) If G is not compact and p is as above, then 11~” *f IIm -+O for every f continuous vanishing at inlinity. 0 1989 Academic Press, Inc. Let G be a locally compact, o-compact group and let T(t) be a con- tinuous representation of G by isometries on a Banach space B. For a probability p on G define the average operator Px = j T(t)x dp(t). We would like to investigate the convergence properties of the sequence P”. The casewhere G is Abelian or compact is easy and known, thus our study is of interest for non compact, non Abelian groups. The motivation for this work is based on two distinct lines of reasoning. By the mean ergodic theorem the Cesaro averages l/n C;;- l P’x con- verge strongly for every x E B, if B is reflexive or more generally “mean ergodic” (that is, B = {x; Vx = x} + (I- I’) B for any linear contraction I’). For any Banach space B the sequence l/n C;f-’ Pi is at least “almost invariant” that is, lim II(Z- P) l/n C;;P1 P’Jl= 0. It is tempting to think that these convergencesshould hold without Cesaro averages,for the sequence P”, when obvious periodic phenomena are excluded. *The research reported here was carried out during the second author’s visit to the Universitt de Bretagne Occidentale. 86 0022-1236189 $3.00 Copyright 0 1989 by Academic Press, Inc. All rights of reproduction in any form reserved. brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by Elsevier - Publisher Connector