I’ve been reading *Quantitative ergodic theorems and their number-theoretic applications* By Gorodnik and Nevo (arXiv:1304.6847). Early on, there is a comment on rates of convergence in the mean ergodic theorem which puzzles me.

Briefly, let $ G$ be some (locally compact second countable) group acting on a measure space $ X$ , let $ F_t \subset G$ , and let $ \beta_t$ denote the uniform probability measure supported on $ F_t$ . Let $ \pi_X (\beta_t)$ denote the operator on $ L^2 (X)$ given by $ \pi_X (\beta_t) f := |F_t|^{-1} \int_{F_t} f(g^{-1} \cdot x) dg$ . On p.8, the authors remark that for a properly ergodic action of a countable amenable group $ G$ on $ X$ , it holds that $ \Vert \pi_X (\beta_t)\Vert = 1$ .

This I believe. But then immediately after, they go on to remark that “it follows that for a properly ergodic action of a countable amenable group, *no uniform rate of convergence* in the mean ergodic theorem can be established, namely that the norm of the averaging operators does not decay at all.” (Emphasis theirs)

But is that really the case? In other words, is the (well known) lack of a uniform rate of convergence in the mean ergodic theorem for amenable groups actually just a direct consequence of the aforementioned fact about the norms of the averaging operators?