E. Szemerédi’s theorem on the existence of arbitrarily long arithmetic progressions in subsets of the integers with positive upper density [Acta Arith. 27, 199–245 (1975; Zbl 0303.10056)] and the equivalent formulation as Furstenberg’s multiple recurrence theorem in ergodic theory [J. Analyse Math. 31, 204–256 (1977; Zbl 0347.28016)] may be formulated as follows. If \((X,\mathcal{B},\mu,T)\) is a measure-preserving system, then \(T\) induces an isomorphism \(\alpha:L^{\infty}(X)\to L^{\infty}(X)\) by \(\alpha f(x)=f(T^{-1}x)\), the measure \(\mu\) induces a trace \(\tau\) defined by \(\tau(f)=\int f\text{d}\mu\), and for any \(k\geq1\) and non-negative \(f\in L^{\infty}(X)\) with positive trace,
\[ \liminf_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\tau\left((\alpha^nf)(\alpha^{2n}f)\cdots (\alpha^{(k-1)n}f)\right)>0. \]
Results of B. Host and B. Kra [Ann. Math. (2) 161, No. 1, 397–488 (2005; Zbl 1077.37002)] refine this further by showing convergence of the multiple ergodic average. Viewing \(L^{\infty}(X)\) as an abelian von Neumann algebra, \(\tau\) as a finite trace on this algebra and \(\alpha\) as an automorphism of the algebra motivates the remarkable results in this paper for the non-abelian setting. Let \(\mathcal M\) be a finite von Neumann algebra with an automorphism \(\alpha\) and a non-negative trace \(\tau\). Then for each \(k\geq1\) three natural manifestations of multiple recurrence arise:
(1) \(\liminf_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N} \text{Re}\,\tau\left((\alpha^nf)(\alpha^{2n}f)\cdots (\alpha^{(k-1)n}f)\right)>0;\)
(2) the limit in (1) exists;
(3) \(\text{Re}\,\tau\left((\alpha^nf)(\alpha^{2n}f)\cdots (\alpha^{(k-1)n}f)\right)>0\) for all \(n\) in a set of positive density.
In contrast to the abelian case, where all three properties hold for any \(k\geq1\), the results here show entirely new phenomena in the non-abelian case. For \(k=2\), all three properties hold (this is the ergodic theorem for von Neumann algebras); (1) can fail for \(k=3\) and (2) can fail for \(k\geq 4\) even in the ergodic case (ergodicity in this context means that the invariant algebra \(\{m\in\mathcal{M}\mid\alpha(m)=m\}\) comprises only the constants \(\mathbb C\times 1\)); (3) can fail for \(k=3\) without ergodicity, or for odd \(k\geq5\) even when assuming ergodicity. Two open problems identified are whether (2) holds for nonergodic systems with \(k=3\), and whether (3) holds for ergodic systems with \(k=4\).
A triple \((\mathcal{M},\tau,\alpha)\) as above is called asymptotically abelian if
\[ \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\| \alpha^n(a)b-b\alpha^n(a) \|=0 \]
for all \(a,b\in\mathcal{M}\), where \(\|\cdot\|\) denotes the \(L^2(\tau)\) norm not the operator norm. Under the hypothesis that the system is asymptotically abelian, some of the abelian machinery used to prove Furstenberg’s multiple recurrence theorem – in particular, a form of the van der Corput lemma – survive, and one of the main results is that all three properties hold for any \(k\geq1\) in this case.