  Let $G$ be a locally compact abelian group, let $\nu$ be a regular
probability measure on $G$, let $X$ be a Banach space, let $\pi\colon G\to
B(X)$ be a bounded strongly continuous representation. Consider the average (or
subordinated) operator $S(\pi,\nu) = \int_{G} \pi(t)\,d\nu(t)\,\colon X\to X$.
We show that if $X$ is a UMD Banach lattice and $\nu$ has bounded angular
ratio, then $S(\pi,\nu)$ is a Ritt operator with a bounded $H^\infty$
functional calculus. Next we show that if $\nu$ is the square of a symmetric
probability measure and $X$ is $K$-convex, then $S(\pi,\nu)$ is a Ritt
operator. We further show that this assertion is false on any non $K$-convex
space $X$.
