Connection between rates of convergence in ergodic theorems and spectral gap property

Question

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?

Answer 1

The "in particular" refers to the previous paragraph. It is the almost invariant sequence of mean 0 functions (which is equivalent to amenability of the acting group) that guarantees the lack of uniform convergence in the ergodic theorem. You can see this because for any Folner set and any $n$, you have a function of norm 1, but integral 0 where $A_nf$ is arbitrarily close to $f$ (and hence far from the limit, 0).