# Nonconventional ergodic averages and multiple recurrence for von Neumann dynamical systems - Mathematics > Operator Algebras

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Abstract: The Furstenberg recurrence theorem or equivalently, Szemer\-edi-s theoremcan be formulated in the language of von Neumann algebras as follows: given aninteger $k \geq 2$, an abelian finite von Neumann algebra $\M,\tau$ with anautomorphism $\alpha: \M \to \M$, and a non-negative $a \in \M$ with$\taua>0$, one has $\liminf {N \to \infty} \frac{1}{N} \sum {n=1}^N \Re\taua \alpha^n a . \alpha^{k-1n} a > 0$; a subsequent result of Hostand Kra shows that this limit exists. In particular, $\Re \taua \alpha^n a> . \alpha^{k-1n} a > 0$ for all $n$ in a set of positive density.From the von Neumann algebra perspective, it is thus natural to ask to whatextent these results remain true when the abelian hypothesis is dropped. Allthree claims hold for $k = 2$, and we show in this paper that all three claimshold for all $k$ when the von Neumann algebra is asymptotically abelian, andthat the last two claims hold for $k=3$ when the von Neumann algebra isergodic. However, we show that the first claim can fail for $k=3$ even withergodicity, the second claim can fail for $k \geq 4$ even assuming ergodicity,and the third claim can fail for $k=3$ without ergodicity, or $k \geq 5$ andodd assuming ergodicity. The second claim remains open for non-ergodic systemswith $k=3$, and the third claim remains open for ergodic systems with $k=4$.

Autor: Tim Austin, Tanja Eisner, Terence Tao

Fuente: https://arxiv.org/