The entangled ergodic theorem in the almost periodic case

Let UU be a unitary operator acting on the Hilbert space HH, and α:{1,…,2k}↦{1,…,k}α:{1,…,2k}↦{1,…,k} a pair partition. Then the ergodic average1Nk∑n1,…,nk=0N-1Unα(1)A1Unα(2)⋯Unα(2k-1)A2k-1Unα(2k)converges in the strong operator topology provided UU is almost periodic, that is when HH is generated by the eigenvalues of UU. We apply the present result to obtain the convergence of the Cesaro mean of several multiple correlations.