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Almost sure convergence of iterates of contractions in noncommutative $$L_ 2$$-spaces. (English) Zbl 0739.46048
Recently a remarkable progress has been made in the individual ergodic theory of positive contractions in von Neumann algebras. Many pointwise ergodic theorems have been extended to the operator algebra context. The study of such problematics is motivated by the theory of open (irreversible) quantum dynamical systems. From the physical point of view the most important are (semigroups of) completely positive maps on $$C^*$$- or $$W^*$$-algebras but in the context of this paper it seems to be more natural to consider a larger class of positive contractions. We shall discuss the asymptotic behaviour of Schwarz maps in von Neumann algebras. More exactly, we are going to prove some results concerning the iterates of contractions in $$L_ 2$$ over a von Neumann algebra $$M$$ (with respect to a faithful normal state $$\Phi$$ on $$M$$).
##### MSC:
 46L51 Noncommutative measure and integration 46L53 Noncommutative probability and statistics 46L54 Free probability and free operator algebras 47A35 Ergodic theory of linear operators
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