 # The ergodic theorem and Gowers-Host-Kra seminorms without separability or amenability What's new

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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