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Recurrence and primitivity for IP systems with polynomial wildcards. (English) Zbl 1337.28025

The paper is devoted to the weak IP-limits of unitary operators in Hilbert space. This investigation is motivated by providing a new joint extension of IP Szemerédi theorem due to H. Furstenberg and Y. Katznelson [J. Anal. Math. 45, 117–168 (1985; Zbl 0605.28012)] and a polynomial Szemerédi theorem due to V. Bergelson and A. Leibman [J. Am. Math. Soc. 9, No. 3, 725–753 (1996; Zbl 0870.11015)].
More precisely, let \({\{U_i^{(s)}\}_{i\in\mathbb{N}}},\) \({1\leq s\leq t}\) be a finite family of sequences of commuting unitary operators on Hilbert space, and \({\mathcal{F}}\) be the collection of non-empty subset of \({\mathbb{N}}.\) By Ramsey’s theorem there exists a sequence \({i_1<i_2<\ldots}\) such that for all \({N\in\mathbb{N}},\) all \({(k_1,k_2,\ldots, k_N)\in\mathbb{N}^N}\) and all \({(s_1,s_2,\ldots, s_N)\in\{1,2,\ldots,t\}^N},\) the limits \[ \lim_{\infty\leftarrow\gamma_1<\gamma_2<\ldots<\gamma_N,\;|\gamma_i|=k_i}V_{\gamma_1}^{(s_1)}V_{\gamma_2}^{(s_2)}\cdots V_{\gamma_N}^{(s)}:=P_{(k_1,k_2,\ldots, k_N)}^{(s_1,s_2,\ldots, s_N)} \] exist in the weak operator topology, where \({V_\gamma^{(s)}=\prod_{j\in\gamma}U_{i_j}^{(s)}}\) for \({\gamma\in\mathcal{F}}\) and \({1\leq s\leq t}.\)
Relying on this fact the authors proved that there exists an IP-ring \({\mathcal{F}^{(1)}}\) such that for all polynomials \({q_s}, {1\leq s\leq t}\) with integer coefficients and zero constant term, the operators \[ P_{(q_1(x),q_2(x),\ldots, q_t(x))}=IP-\lim_{\alpha\in\mathcal{F}^{(1)}}P_{(q_1(n(\alpha)),q_2(n(\alpha)),\ldots, q_t(n(\alpha)))}^{(1,2,\ldots,t)} \] exist in the weak operator topology and are orthogonal projections. As a corollary of this projection theorem an extension of Szemerédi’s theorem is obtained.

MSC:

28D05 Measure-preserving transformations
05D10 Ramsey theory
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
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