New topics in ergodic theory

2007/02/05 by Francesco Fidaleo, Fidaleo, Francesco
Mathematics · #37A30 #Advanced Banach Space Theory #Advanced Operator Algebra Research #FOS: Mathematics #Operator Algebras (math.OA) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.math/0702103

openalex publication_date 2007/02/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The entangled ergodic theorem concerns the study of the convergence in the strong, or merely weak operator topology, of the multiple Cesaro mean \frac1N^k∑_n₁,...,n_k=0^N-1 U^n_\a(1)A₁U^n_\a(2)... U^n_\a(2k-1)A_2k-1U^n_\a(2k) , where U is a unitary operator acting on the Hilbert space H, \a:\1,..., m\↦\1,..., k\ is a partition of the set made of m elements in k parts, and finally A₁,...,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,α) be a C^*--dynamical system, where \mathfrak A=K(H), and α=ad(U) is an automorphism implemented by the unitary U. We show that lim_N→+∞(1)/(N)∑_n=0^N-1αⁿ=E , pointwise in the weak topology of \K(H). Here, E is a conditional expectation projecting onto the C^*--subalgebra (\bigoplus_z∈σ_\mathop\rm pp(U) E_zB(H)E_z)\bigcap K(H) . If in addition U is weakly mixing with Ω∈ H the unique up to a phase, invariant vector under U and ω=, we have the following recurrence result. If A∈ K(H) fulfils ω(A)>0, and 00 for each N>N₀.

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