  The continuity properties of the convex closure of the output entropy of
infinite dimensional channels and their applications to the additivity problem
are considered.
  The main result of this paper is the statement that the superadditivity of
the convex closure of the output entropy for all finite dimensional channels
implies the superadditivity of the convex closure of the output entropy for all
infinite dimensional channels, which provides the analogous statements for the
strong superadditivity of the EoF and for the additivity of the minimal output
entropy.
  The above result also provides infinite dimensional generalization of Shor's
theorem stated equivalence of different additivity properties.
  The superadditivity of the convex closure of the output entropy (and hence
the additivity of the minimal output entropy) for two infinite dimensional
channels with one of them a direct sum of noiseless and entanglement-breaking
channels are derived from the corresponding finite dimensional results.
  In the context of the additivity problem some observations concerning
complementary infinite dimensional channels are considered.
