Let $U$ be a unitary operator acting on the Hilbert space $H$, $\a:\{1,...,2k\}\mapsto\{1,..., k\}$ a pair--partition, and finally $A_{1},...,A_{2k-1}\inB(H)$. 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 theeigenvectors 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 theentangled ergodic theorem. We also prove the noncommutative version of theergodic result of H. Furstenberg relative to diagonal measures. This impliesthat ${\displaystyle \frac{1}{N}\sum_{n=0}^{N-1} U^{n}AU^{n}}$ converges in thestrong operator topology for other interesting situations where the involvedunitary operator does not generate an almost periodic dynamics, and theoperator $A$ is noncompact.
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机译：假设$ U $是作用在希尔伯特空间$ H $，$ \ a：\ {1，...，2k \} \ mapsto \ {1，...，k \} $对上的a运算子-分区，最后是$ A_ {1}，...，A_ {2k-1} \ inB（H）$。我们显示遍历平均$$ \ frac {1} {N ^ {k}} \ sum_ {n_ {1}，...，n_ {k} = 0} ^ {N-1} U ^ {n_ { \ a（1）}} A_ {1} U ^ {n _ {\ a（2）}} ... U ^ {n _ {\ a（2k-1）}} A_ {2k-1} U ^ {n_ {\ a（2k）}}当$ U $的特征向量生成$ H $时，即$ H $上由theU $引起的动力学几乎是周期性时，$$收敛于强算子拓扑。相对于纠缠遍历定理，该结果改进了已知的定理。我们还证明了H. Furstenberg相对于对角线测度的遍历结果的非可交换形式。这意味着$ {\ displaystyle \ frac {1} {N} \ sum_ {n = 0} ^ {N-1} U ^ {n} AU ^ {n}} $在其他有趣的情况下会收敛于强算子拓扑卷入式ary元算子不会产生几乎周期性的动力学，并且and元A $是非紧凑的。
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