  The eigenvectors of an ergodic semigroup of linear normal positive unital
maps on a von Neumann algebra are described. Moreover, it is shown by means of
examples, that mere positivity of the maps in question is not sufficient for
Frobenius theory as in S. Albeverio and R. H{\o}egh-Krohn, \emph{Frobenius
theory of positive maps of von Neumann algebras}, Comm. Math. Phys.
\bf{64}(1978), 83--94, to hold.
