# Journal of Operator Theory

Volume 50, Issue 1, Summer 2003 pp. 3-52.

Non-commutative extensions of classical and multiple recurrence theorems**Authors**: Constantin P. Niculescu, (1) Anton Stroh, (2) and Laszlo Zsido (3)

**Author institution:**(1) University of Craiova, Department of Mathematics, Craiova 200585, Romania

(2) University of Pretoria, Department of Mathematics and Applied Mathematics, Pretoria 0002, South Africa

(3) University of Rome ``Tor Vergata'', Department of Mathematics, Via della Ricerca Scientifica, 00133 Rome, Italy

**Summary:**The aim of this paper is to extend the classical recurrence theorem of A.Y. Khintchine, as well as certain multiple recurrence results of H.~Furstenberg concerning weakly mixing and almost periodic measure preserving transformations, to the framework of $C^{\star }$-algebras $\mathfrak{A}$ and positive linear maps $\Phi :\mathfrak{A}\rightarrow \mathfrak{A}$ preserving a state $\varphi $ on $\mathfrak{A}$. For the proof of the multiple weak mixing results we use a slight extension of a convergence result of Furstenberg in Hilbert spaces, which is derived from a non-commutative generalization of Van der Corput's "Fundamental Inequality" in Theory of uniform distribution modulo 1, proved in Appendix A.

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