  In an attempt to propose more general conditions for decoherence to occur, we
study spectral and ergodic properties of unital, completely positive maps on
not necessarily unital $C^*$-algebras, with a particular focus on gapped maps
for which the transient portion of the arising dynamical system can be
separated from the persistent one. After some general results, we first devote
our attention to the abelian case by investigating the unital $*$-endomorphisms
of, in general non-unital, $C^*$-algebras, and their spectral structure. The
finite dimensional case is also investigated in detail, and examples are
provided of unital completely positive maps for which the persistent part of
the associated dynamical system is equipped with the new product making it a
$C^*$-algebra, and the map under consideration restricts to a unital
$*$-automorphism for this new $C^*$-structure, thus generating a conservative
dynamics on that persistent part.
