Recurrence and primitivity for IP systems with polynomial wildcards (Q2787986)

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 \textit{H. Furstenberg} and \textit{Y. Katznelson} [J. Anal. Math. 45, 117--168 (1985; Zbl 0605.28012)] and a polynomial Szemerédi theorem due to \textit{V. Bergelson} and \textit{A. Leibman} [J. Am. Math. Soc. 9, No. 3, 725--753 (1996; Zbl 0870.11015)].NEWLINENEWLINEMore 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 NEWLINE\[NEWLINE \lim_{\infty\leftarrow\gamma_1<\gamma_2<\ldots<\gamma_N,\;|\gamma_i|=k_i}V_{\gamma_1}^{(s_1)}V_{\gamma_2}^{(s_2)}\cdots V_{\gamma_N}^{(s)}:=P_{(k_1,k_2,\ldots, k_N)}^{(s_1,s_2,\ldots, s_N)} NEWLINE\]NEWLINE exist in the weak operator topology, where \({V_\gamma^{(s)}=\prod_{j\in\gamma}U_{i_j}^{(s)}}\) for \({\gamma\in\mathcal{F}}\) and \({1\leq s\leq t}.\)NEWLINENEWLINERelying 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 NEWLINE\[NEWLINE P_{(q_1(x),q_2(x),\ldots, q_t(x))}=IP-\lim_{\alpha\in\mathcal{F}^{(1)}}P_{(q_1(n(\alpha)),q_2(n(\alpha)),\ldots, q_t(n(\alpha)))}^{(1,2,\ldots,t)} NEWLINE\]NEWLINE 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.