  We develop an approximation approach to infinite dimensional quantum channels
based on detailed investigation of the continuity properties of entropic
characteristics of quantum channels and operations (trace-nonincreasing
completely positive maps) as functions of a pair ``channel, input state''. The
obtained results are then applied to the following problems: continuity of the
$\chi$-capacity as function of a channel; strong additivity of the
$\chi$-capacity for infinite dimensional channels; the analytical expression
for the convex closure of the output entropy of arbitrary quantum channel.
