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Article Dans Une Revue Probability Theory and Related Fields Année : 2006

On ergodic theorems for free group actions on noncommutative spaces

Résumé

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 $s_{2n}$ of even radius. Here we study state preserving actions of free groups on a von Neumann algebra $A$ and the behaviour of $(s_{2n}(x))$ for $x$ in noncommutative spaces $L^p(A)$. For the Cesàro means $\frac{1}{n}\sum_{k=0}^{n-1} s_k$ and $p = +\infty$, 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.
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Dates et versions

hal-00003557 , version 1 (13-12-2004)

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Claire Anantharaman-Delaroche. On ergodic theorems for free group actions on noncommutative spaces. Probability Theory and Related Fields, 2006, 135, pp.520-546. ⟨hal-00003557⟩
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