  Results concerning recurrence and ergodicity are proved in an abstract
Hilbert space setting based on the proof of Khintchine's recurrence theorem for
sets, and on the Hilbert space characterization of ergodicity. These results
are carried over to a non-commutative *-algebraic setting using the
GNS-construction. This generalizes the corresponding measure theoretic results,
in particular a variation of Khintchine's Theorem for ergodic systems, where
the image of one set overlaps with another set, instead of with itself.
