  We consider semigroups $\{\alpha_t: \; t\geq 0\}$ of normal, unital,
completely positive maps $\alpha_t$ on a von Neumann algebra ${\mathcal M}$.
The (predual) semigroup $\nu_t (\rho):= \rho \circ \alpha_t$ on normal states
$\rho$ of $\mathcal M$ leaves invariant the face ${\mathcal F}_p:= \{\rho : \;
\rho (p)=1\}$ supported by the projection $p\in {\mathcal M}$, if and only if
$\alpha_t(p)\geq p$ (i.e., $p$ is sub-harmonic). We complete the arguments
showing that the sub-harmonic projections form a complete lattice. We then
consider $r_o$, the smallest projection which is larger than each support of a
minimal invariant face; then $r_o$ is subharmonic. In finite dimensional cases
$\sup \alpha_t(r_o)={\bf 1}$ and $r_o$ is also the smallest projection $p$ for
which $\alpha_t(p)\to {\bf 1}$. If $\{\nu_t: \; t\geq 0\}$ admits a faithful
family of normal stationary states then $r_o={\bf 1}$ is useless; if not, it
helps to reduce the problem of the asymptotic behaviour of the semigroup for
large times.
