On residualities in the set of Markov operators on $\mathcal{C}_1$

We show that the set of those Markov operators on the Schatten class $\mathcal{C}_1$ such that $\lim_{n \to \infty} \Vert P^n - Q \Vert = 0$, 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 $G_{\delta}$. Surprisingly, for the strong operator topology operators the situation is quite the opposite.