Noncommutative extensions of classical and multiple recurrence theorems

Abstract

The aim of this paper is to extend the classical recurrence the- orem of A.Y. Khintchine, as well as certain multiple recurrence results of H. Furstenberg concerning weakly mixing and almost periodic measure pre- serving transformations, to the framework of C -algebras A and positive linear maps : A ! A preserving a state ' on A. For the proof of the mul- tiple 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.