  We develop the notions of hypercontractivity (HC) and the log-Sobolev (LS)
inequality for completely bounded norms of one-parameter semigroups of
super-operators acting on matrix algebras. We prove the equivalence of the
completely bounded versions of HC and LS under suitable hypotheses. We also
prove a version of the Gross Lemma which allows LS at general $q$ to be deduced
from LS at $q=2$.
