Ergodic type theorems for finite von Neumann algebras

Stochastic Banach principle in operator algebras

“…Proof. Corollary 1.1 of [9] implies existence of the projectionẽ 1 in N such that i) there existsα ′ invariant normal state ρ with support supp(ρ) =ẽ 1 and ii) there exists a weakly wandering operator h ∈ N with support I −ẽ 1 . Our goal is to show that similar statements are valid for almost every z ∈ X.…”

“…We reformulate the theorem in a form convenient for applications (Theorem 3.2.4). Based on the principle we give a simplified proof of the stochastic ergodic theorem (compare with [9]). We establish stochastic convergence for Sato's uniform subsequences (Theorem 3.6) and a stochastic ergodic theorem for the Besicovitch Bounded sequences (Theorem 3.7).…”

“…Define α ′ as a linear map on L 1 (M, τ ) satisfying τ (x · α(y)) = τ (α ′ (x)y) for x ∈ L 1 (M, τ ), y ∈ M , and A ′ n (l (x), for x ∈ L 1 (M, τ ). Let us recall some definitions from Grabarnik and Katz[9] and Chilin Litvinov and Skalski[2].…”

A note on non-commutative ergodic theorems for actions of hyperbolic groups

“…Among others we would like to mention work by Oceledetz [28], Tempelman [32], Arnold and Krylov [1], Guivarch [21], Grigorchuk [18,19], Nevo and Stein [25,26], Bufetov [5,6,8], see also [11] . Non commutative analogs are due to work of authors [15,16], Anantharaman-Delaroche [2]. Authors thank Danny Calegari for clarification of proper references and statement of Proposition 1.…”

On weak convergence of iterates in quantum Lp‐spaces(p ≥ 1)

“…The first results in the field of noncommutative ergodic theory were obtained independently by Sinaȋ and Ansělevič [21] and Lance [15]. Developments of the subject are reflected in the monographs of Jajte [13] and Krengel [14] (see also [8,9,10,18]).…”