TY - JOUR
T1 - Nonconventional ergodic averages and multiple recurrence for von neumann dynamical systems
AU - Austin, Tim
AU - Eisner, Tanja
AU - Tao, Terence
PY - 2011
Y1 - 2011
N2 - 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 ∈ M with τ (a) > 0, one has lim inf N→∞ N-1 ∑Nn=1 Re τ(aαn (a) · · · α(K-1)n (a))> 0; a later result of Host and Kra shows this limit exists. In particular, Re τ(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.
AB - 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 ∈ M with τ (a) > 0, one has lim inf N→∞ N-1 ∑Nn=1 Re τ(aαn (a) · · · α(K-1)n (a))> 0; a later result of Host and Kra shows this limit exists. In particular, Re τ(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.
KW - Multiple recurrence
KW - Nonconventional ergodic averages
KW - Szemerédi's theorem
KW - Von Neumann algebras
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U2 - 10.2140/pjm.2011.250.1
DO - 10.2140/pjm.2011.250.1
M3 - Article
AN - SCOPUS:79953792794
SN - 0030-8730
VL - 250
SP - 1
EP - 60
JO - Pacific Journal of Mathematics
JF - Pacific Journal of Mathematics
IS - 1
ER -