Theorem 41.3.15.label Let $A$ be a unital $C^{*}$-algebra, $S \subset A$ be a subspace, $X$ be a compact Hausdorff space, and $\phi \in L(S; C(X; \complex))$, then:
- (1)
$\norm{\phi}_{\text{cb}}= \norm{\phi}_{L(S; C(X; \complex))}$.
- (2)
If $S$ is an operator system and $\phi$ is positive, then $\phi$ is completely positive.
Proof, [Proposition 3.8, Po02]. By Proposition 41.3.14, the norm and order relation on $C(X; M_{n}(\complex)) \iso M_{n}(C(X; \complex))$ are pointwise. As such, assume without loss of generality that $X$ is a single point and $C(X; \complex) = \complex$.
Let $a = (a_{ij}) \in M_{n}(S)$. For each $\xi, \eta \in \complex^{n}$,
\begin{align*}\dpn{\phi_n(a)\xi, \eta}{\complex^n}&= \sum_{i, j = 1}^{n} \xi_{j} \ol{\eta_i}\dpn{a_{ij}, \phi}{S}= \angles{\sum_{i, j = 1}^n a_{ij}\xi_j\ol{\eta_i}, \phi}_{S}\end{align*}
where
\begin{align*}\sum_{i, j = 1}^{n} a_{ij}\xi_{j}\ol{\eta_i}&= (\one \otimes \ol{\eta}) \cdot a \cdot (\xi \otimes \one)\end{align*}
(1):
\begin{align*}|\dpn{\phi_n(a)\xi, \eta}{\complex^n}|&\le \norm{\phi}_{S^*}\cdot \norm{\sum_{i, j = 1}^n a_{ij}\xi_j\ol{\eta_i}}_{S}\\ \norm{\sum_{i, j = 1}^n a_{ij}\xi_j\ol{\eta_i}}_{S}&\le \norm{\xi}_{\complex^n}\norm{\eta}_{\complex^n}\norm{a}_{M_n(S)}\\ \norm{\phi_n(a)}_{M_n(\complex)}&\le \norm{\phi}_{S^*}\norm{a}_{M_n(S)}\\ \norm{\phi}_{\text{cb}}&\le \norm{\phi}_{S^*}\end{align*}
(2): If $\xi = \eta$, then
\[\sum_{i, j = 1}^{n} a_{ij}\eta_{j}\ol{\eta_i}= (\xi \otimes \one)^{*} \cdot a \cdot (\xi \otimes \one) \ge 0\]
so
\[\dpn{\phi_n(a)\xi, \eta}{\complex^n}= \dpn{(\xi \otimes \one)^* \cdot a \cdot (\xi \otimes \one), \phi}{S}\ge 0\]
$\square$
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