  We consider the additivity of the minimal output entropy and the classical
information capacity of a class of quantum channels. For this class of channels
the norm of the output is maximized for the output being a normalized
projection. We prove the additivity of the minimal output Renyi entropies with
entropic parameters contained in [0, 2], generalizing an argument by Alicki and
Fannes, and present a number of examples in detail. In order to relate these
results to the classical information capacity, we introduce a weak form of
covariance of a channel. We then identify several instances of weakly covariant
channels for which we can infer the additivity of the classical information
capacity. Both additivity results apply to the case of an arbitrary number of
different channels. Finally, we relate the obtained results to instances of
bi-partite quantum states for which the entanglement cost can be calculated.
