The von Neumann ergodic theorem (the Hilbert space version of the mean ergodic theorem) asserts that if $latex {U: H rightarrow H}&fg=000000$ is a unitary operator on a Hilbert space $latex {H}&fg=000000$, and $latex {v in H}&fg=000000$ is a vector in that Hilbert space, then one has

$latex displaystyle lim_{N rightarrow infty} frac{1}{N} sum_{n=1}^N U^n v = pi_{H^U} v&fg=000000$

in the strong topology, where $latex {H^U := { w in H: Uw = w }}&fg=000000$ is the $latex {U}&fg=000000$-invariant subspace of $latex {H}&fg=000000$, and $latex {pi_{H^U}}&fg=000000$ is the orthogonal projection to $latex {H^U}&fg=000000$. (See e.g. these previous lecture notes for a proof.) The same proof extends to more general amenable groups: if $latex {G}&fg=000000$ is a countable amenable group acting on a Hilbert space $latex {H}&fg=000000$ by unitary transformations $latex {T^g: H rightarrow H}&fg=000000$ for $latex {g in G}&fg=000000$, and $latex {v in H}&fg=000000$ is a vector in that…

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