Proposition 41.2.11.label Let $A$ be a unital $C^{*}$-algebra, $S \subset A$ be an operator space, $H$ be a complex Hilbert space, and $\phi: S \to B(H)$ be a unital positive map, then for each $a \in S$, $w(\phi(a)) \le \norm{a}_{A}$.
Proof. Let $a \in S$ with $\norm{a}_{A} \le 1$, then for each $\lambda \in \complex$, $2 + (\lambda a) + (\lambda a)^{*} \ge 0$. Since $\phi$ is positive, $2 + (\lambda \phi(a)) + (\lambda \phi(a))^{*} \ge 0$ as well. Therefore $w(\phi(a)) \le 1$ by Proposition 41.2.10.$\square$
Post a Comment