On ergodic theorems for free group actions on noncommutative spaces

By C. Anantharaman-Delaroche. 2004

92 citations recorded by the source corpus.

We extend in a noncommutative setting the individual ergodic theorem of Nevo and Stein concerning measure preserving actions of free groups and averages on spheres s2n of even radius. Here we study state preserving actions of free groups on a von Neumann algebra A and the behaviour of (s2n(x)) for x in noncommutative spaces Lp(A). For the Cesàro means this problem was solved by Walker. Our approach is based on ideas of Bufetov. We prove a noncommutative version of Rota ``Alternierende Verfahren'' theorem. To this end, we introduce specific dilations of the powers of some noncommutative Markov operators.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ca0ee537ac8ee50ff889a3f8ceb9495bc49c6a37

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In Mathematical Physics

Citations in the corpus

An Asymptotic Property of Factorizable Completely Positive Maps and the Connes Embedding Problem (2014, background): “Anantharaman-Delaroche in [1]) has gained particular significance in quantum information theory in connection with the settling (in the negative) of the asymptotic quantum Birkhoff conjecture.”
$H^{\infty }$ functional calculus and square functions on noncommutative $L^p$-spaces (2006, unknown): “Following [4], we say that T : (M, φ) → (M, φ) is factorizable if there exist a von Neumann algebra (M′, φ′) equipped with a normal faithful state, and two normal unital faithful ∗representations π : M −→ M′ and π̃ : M −→ M′ both satisfying (11.”
Noncommutative Riesz transforms -- a probabilistic approach (2008, background): “te70b]). In the noncommutative situation the condition is satisﬁed if we have either a Markov dilation or a unitary dilation. The general case is open (even for a single completely positive map), see [AD06] for more information. We don’t know whether the condition on 14 gradients are necessary. It i”
BMO spaces associated with semigroups of operators (2011, background): “r a standard dilation or a reversed dilation. We refer to [Ku¨m88] for related questions. A glance at (1.2) shows that a Markov dilation for (Tt) implies that the Pt’s are factorable (in the sense of [AD06]). According to [AD06], we know that a Markov dilation for Tt (standard or reversed) yields a ”