NONCOMMUTATIVE EXTENSIONS OF CLASSICAL AND MULTIPLE RECURRENCE THEOREMS

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*-algebras A and positive linear maps Φ : A → A preserving a state φ on 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.