  Consider the set $\mathcal{S}=\lbrace\rho_{SE}\rbrace$ of possible initial
states of the system-environment, steered from a tripartite reference state
$\omega_{RSE}$. Buscemi [F. Buscemi, Phys. Rev. Lett. 113, 140502 (2014)]
showed that the reduced dynamics of the system, for each $\rho_{S}\in
\mathrm{Tr}_{E}\mathcal{S}$, is always completely positive if and only if
$\omega_{RSE}$ is a Markov state. There, during the proof, it has been assumed
that the dimensions of the system and the environment can vary through the
evolution. Here, we show that this assumption is necessary: we give an example
for which, though $\omega_{RSE}$ is not a Markov state, the reduced dynamics of
the system is completely positive, for any evolution of the system-environment
during which the dimensions of the system and the environment remain unchanged.
As our next result, we show that the result of Muller-Hermes and Reeb [A.
Muller-Hermes and D. Reeb, Ann. Henri Poincare 18, 1777 (2017)], of
monotonicity of the quantum relative entropy under positive maps, cannot be
generalized to the Hermitian maps, even within their physical domains.
