New topics in ergodic theory

Details New topics in ergodic theory New topics in ergodic theory

2007-02-05

arxiv.org/abs/math/0702103v1

Mathematics

Operator Algebras

18 pages

Scientific paper

The entangled ergodic theorem concerns the study of the convergence in the strong, or merely weak operator topology, of the multiple Cesaro mean $$\frac{1}{N^{k}}\sum_{n_{1},...,n_{k}=0}^{N-1} U^{n_{\a(1)}}A_{1}U^{n_{\a(2)}}... U^{n_{\a(2k-1)}}A_{2k-1}U^{n_{\a(2k)}} ,$$ where $U$ is a unitary operator acting on the Hilbert space $H$, $\a:\{1,..., m\}\mapsto\{1,..., k\}$ is a partition of the set made of $m$ elements in $k$ parts, and finally $A_{1},...,A_{2k-1}$ are bounded operators acting on $H$. While reviewing recent results about the entangled ergodic theorem, we provide some natural applications to dynamical systems based on compact operators. Namely, let $(\mathfrak A,\alpha)$ be a $C^{*}$--dynamical system, where $\mathfrak A=K(H)$, and $\alpha=ad(U)$ is an automorphism implemented by the unitary $U$. We show that $$\lim_{N\to+\infty}\frac{1}{N}\sum_{n=0}^{N-1}\alpha^{n}=E ,$$ pointwise in the weak topology of $\K(H)$. Here, $E$ is a conditional expectation projecting onto the $C^{*}$--subalgebra $$\bigg(\bigoplus_{z\in\sigma_{\mathop{\rm pp}}(U)} E_{z}B(H)E_{z}\bigg)\bigcap K(H) .$$ If in addition $U$ is weakly mixing with $\Omega\in H$ the unique up to a phase, invariant vector under $U$ and $\omega=<\cdot \Omega,\Omega>$, we have the following recurrence result. If $A\in K(H)$ fulfils $\omega(A)>0$, and $00$$ for each $N>N_{0}$.

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