Definition 41.3.4 (Completely Bounded Map).label Let $A, B$ be $C^{*}$-algebras with $A$ being unital, $S \subset A$ be an operator system, and $\phi: S \to B$ be a linear map, then
\[\norm{\phi}_{\text{cb}}= \sup_{n \in \natp}\norm{\phi_n}_{L(M_n(S); M_n(B))}\]
is the completely bounded norm of $\phi$.
The map $\phi$ is completely bounded if $\norm{\phi}_{\text{cb}}< \infty$, and completely contractive if $\norm{\phi}_{\text{cb}}\le 1$.
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