Nonconventional ergodic averages and multiple recurrence for von Neumann dynamical systems

The Furstenberg recurrence theorem (or equivalently Szemerédi’s theorem) can be formulated in the language of von Neumann algebras as follows: given an integer k ≥ 2, an abelian finite von Neumann algebra (M,τ) with an automorphism α : M→M, and a nonnegative a in M with τ(a) > 0, one has liminf N→∞N−1 ∑ n=1N Reτ(aαn(a)⋯α(k−1)n(a)) > 0; a later result of Host and Kra shows this limit exists. In particular, Reτ(aαn(a)⋯α(k−1)n(a)) is positive for all n in a set of positive density. From the von Neumann algebra perspective, it is natural to ask to what remains of these results when the abelian hypothesis is dropped. All three claims hold for k = 2, and we show that all three claims hold for all k when the von Neumann algebra is asymptotically abelian, and that the last two claims hold for k = 3 when the von Neumann algebra is ergodic. However, we show that the first claim can fail for k = 3 even with ergodicity, the second claim can fail for k ≥ 4 even when assuming ergodicity, and the third claim can fail for k = 3 without ergodicity, or k ≥ 5 and odd assuming ergodicity. The second claim remains open for nonergodic systems with k = 3, and the third claim remains open for ergodic systems with k = 4