On the ergodic theorem for affine actions on Hilbert space (Q888711)

A sequence \(\{\mu_n\}\) of probability measures on a countable group \(G\) forms a Reiter sequence if \(\|\mu_n -g\ast \mu_n \| \rightarrow 0 \,\forall g \in G\), and \(g\ast \mu_n (h)=\mu_n (g^-1 h)\). A countable discrete group is amenable if it admits a Reiter sequence. The main result of the paper: Theorem A (Weak Mean Ergodic Theorem). Let \(\pi : G\rightarrow \mathcal{O}(\mathcal{H})\) be an ergodic orthogonal representation of a finitely generated amenable group \(G\), and let \(b : G\rightarrow \mathcal{H}\) be a 1-cocycle associated to \(\pi\). Let \(S\) be a finite symmetric generating set for \(G\), let \(|\cdot|\) denote the word length in \(S\). If \(\{\mu_n \}\) is a Reiter sequence for \(G\), then \[ \int\frac{1}{|g|} b(g)d\mu_n (g) \rightarrow 0 \] in the weak topology on \(\mathcal{H}\). If \(\pi\) is weakly mixing, then \[ \int \frac{1}{|g|} |\langle b(g),\xi \rangle |d\mu_n (g) \rightarrow 0 \] for all \(\xi \in \mathcal{H}\).