Example 41.3.13.label Let $A$ be a $C^{*}$-algebra, $y, z \in A$, $\phi: A \to A$ be defined by $x \mapsto yxz$, then
- (1)
$\phi$ is completely bounded.
- (2)
If $y = z^{*}$, then $\phi$ is completely positive.
Proof. (1): For each $n \in \natp$ and $T \in M_{n}(A)$,
\[\phi_{n}(T) = (1_{M_n(\complex)}\otimes y)T(1_{M_n(\complex)}\otimes z)\]
$\square$
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