Abstract. We show that the set of those Markov operators on the Schatten class C 1 such that limn→∞ P n − Q = 0, where Q is one-dimensional projection, is norm open and dense. If we require that the limit projections must be on strictly positive states, then such operators P form a norm dense G δ . Surprisingly, for the strong operator topology operators the situation is quite the opposite.

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Abstract. We show that the set of all stochastic strongly continuous semigroups on Ci such that lim t^o c |||T(i) -Qx, III = 0, where Qx. is one-dimensional projection for some state X*, is norm open and dense. Moreover this set forms a norm dense Gs if a state X» is strictly positive. IntroductionMarkov operators P : L 1 (//) -> (P > 0,P*1 = 1) arise as a generalization of the concept of transition probabilities (or stochastic matrices). Their iterates P n are intensively studied as they describe the evolution of classical (commutative) stochastic dynamical systems. Those systems which have a (unique) stationary density /* G L 1 (fi) and such that they are stable (i.e. after perturbations they return asymptotically to their equilibrium) are called mixing. Using mathematical language mixing simply means limn-Kx, P n (/) = /*, independently of the initial distribution /. The reader is referred to the monograph [8] in this regard. In a quantum mechanics the role of densities play states on a Hilbert space H. A natural model of a quantum system, in this context, is a linear operator P which transforms a state into another state. Again one can ask about stability of such systems.In [5] it was shown that the set of those noncommutative Markov operators on C\ (the class of trace operators on 7i) such that lim^oo |||P n -Q||| = 0, where Q is one-dimensional projection, is norm open and dense. Moreover if we request that the limit projections must be on strictly positive states then such operators P form a norm dense Gs-This kinds of questions, about geometric properties of sets of operators, are very important from