Mean ergodic theorem along Følner sets

Statement

Let $G$ be a countable discrete group with finite nonempty left Følner sets $F_N$ , so $|hF_N\triangle F_N|/|F_N|\to0$ for each $h\in G$ . For a unitary representation $\pi$ and projection $P$ onto its invariant vectors,

\frac1{|F_N|}\sum_{g\in F_N}\pi(g^{-1})v\longrightarrow Pv \quad\text{in Hilbert-space norm}.

The inverse is part of this convention: left Følner sets give the required right translation control for these operators.

Reason and Koopman form

The averages fix invariant vectors. Applied to $\pi(h)w-w$ , their norm is at most $|h^{-1}F_N\triangle F_N|\|w\|/|F_N|$ , which tends to zero. The closed span of such differences is the orthogonal complement of the fixed vectors.

For inverse-pullback Koopman representations, the averages are $|F_N|^{-1}\sum_{g\in F_N}f\circ T_g$ . The result is mean convergence; it does not assert pointwise convergence along arbitrary Følner sequences.

References

Jesse Peterson, Lecture notes on ergodic theory, 2011. §1.6.1, mean ergodic theorem for amenable groups.