  The channels, and more generally superoperators acting on the trace class
operators of a quantum system naturally form a Banach space under the
completely bounded trace norm (aka diamond norm). However, it is well-known
that in infinite dimension, the norm topology is often "too strong" for
reasonable applications. Here, we explore a recently introduced
energy-constrained diamond norm on superoperators (subject to an energy bound
on the input states). Our main motivation is the continuity of capacities and
other entropic quantities of quantum channels, but we also present an
application to the continuity of one-parameter unitary groups and certain
one-parameter semigroups of quantum channels.
