  We study a class of quantum channels describing a quantum system, split into
the direct sum of an excited and a ground sector, undergoing a one-way transfer
of population from the former to the latter; this construction, which provides
a generalization of the amplitude-damping qubit channel, can be regarded as a
way to upgrade a trace non-increasing quantum operation, defined on the excited
sector, to a possibly trace preserving operation on a larger Hilbert space. We
provide necessary and sufficient conditions for the complete positivity of such
channels, and we also show that complete positivity is equivalent to simple
positivity whenever the ground sector is one-dimensional. Finally, we examine
the time-dependent scenario and characterize all CP-divisible channels and
Markovian semigroups belonging to this class.
