Convergence of iterates of averages of group representations. (English) Zbl 0829.47010
Altomare, F. (ed.) et al., Proceedings of the 2nd international conference in functional analysis and approximation theory, Acquafredda di Maratea (Potenza), September 14-19, 1992. Palermo: Circolo Matemático di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 33, 453-461 (1994).
Summary: Let \(G\) be a locally compact \(\sigma\)-compact group, \(\pi\) a continuous representation of \(G\) by invertible isometries on a uniformly convex Banach space \(X\). Then for every adapted and strictly aperiodic probability \(\mu\) on \(G\), the iterates of the average operator \(\Pi_\mu x= \int \pi (g)x d\mu (g)\) converge in s.o.t. iff the sequence \(\Pi_\mu^n x\) is strongly \(\pi\)-equicontinuous.
For groups for which the left and the right uniform structures are equivalent \(\Pi_\mu^n x\) is convergent in \(X\) for every adapted and strictly aperiodic probability \(\mu\).
For the entire collection see [Zbl 0794.00018].
For groups for which the left and the right uniform structures are equivalent \(\Pi_\mu^n x\) is convergent in \(X\) for every adapted and strictly aperiodic probability \(\mu\).
For the entire collection see [Zbl 0794.00018].
