  Let $U$ be a unitary operator acting on the Hilbert space $H$, $\a:\{1,...,
2k\}\mapsto\{1,..., k\}$ a pair--partition, and finally $A_{1},...,A_{2k-1}\in
B(H)$. We show that the ergodic average $$
\frac{1}{N^{k}}\sum_{n_{1},...,n_{k}=0}^{N-1}
U^{n_{\a(1)}}A_{1}U^{n_{\a(2)}}... U^{n_{\a(2k-1)}}A_{2k-1}U^{n_{\a(2k)}} $$
converges in the strong operator topology when $H$ is generated by the
eigenvectors of $U$, that is when the dynamics induced by the unitary $U$ on
$H$ is almost periodic. This result improves the known ones relative to the
entangled ergodic theorem. We also prove the noncommutative version of the
ergodic result of H. Furstenberg relative to diagonal measures. This implies
that ${\displaystyle \frac{1}{N}\sum_{n=0}^{N-1} U^{n}AU^{n}}$ converges in the
strong operator topology for other interesting situations where the involved
unitary operator does not generate an almost periodic dynamics, and the
operator $A$ is noncompact.
