The entangled ergodic theorem in the almost periodic case

Let $U$ be a unitary operator acting on the Hilbert space $\mathrm{ch}$, and $\text{a}:\{1,\mathrm{.}\mathrm{.}\mathrm{.},2k\}\mapsto \{1,\mathrm{.}\mathrm{.}\mathrm{.},k\}$ a pair partition. Then the ergodic average $$\frac{1}{N^k}\sum _{n_1,\mathrm{.}\mathrm{.}\mathrm{.},n_k=0}^{N-1}{U}^{n_{\text{a}(1)}}{A}_1{U}^{n_{\text{a}(2)}}\mathrm{.}\mathrm{.}\mathrm{.}{U}^{n_{\text{a}(2k-1)}}{A}_{2k-1}{U}^{n_{\text{a}(2k)}}$$ converges in the strong operator topology provided $U$ is almost periodic, that is when $\mathrm{ch}$ is generated by the eigenvalues of $U$. We apply the present result to obtain the convergence of the Cesaro mean of several multiple correlations.