Definition 38.1.4 (Completely).label Let $A, B$ be $C^{*}$-algebras and $\phi: A \to B$ be a function, then $\phi$ induces mappings $\phi_{n} = \text{Id}_{n} \otimes \phi: M_{n}(A) \to M_{n}(B)$. A property of $\phi$ holds completely if it holds for $\phi_{n}$ for all $n \in \natp$.