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.

Keywords: Poincaré recurrence, $C^{\star}$-dynamical system, almost periodicity, weak mixing, multiple weak mixing

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