Proposition 34.8.2.label Let $A$ be a unital $C^{*}$-algebra and $x \in A$ be normal, then $x$ is positive if and only if $\sigma_{A}(x) \subset [0, \infty)$.
Proof. Using the continuous functional calculus, $x$ is positive if and only if $\Gamma_{A[x]}(x) = \text{Id}$ is positive in $C(\sigma_{A}(x); \complex)$, if and only if $\sigma_{A}(x) = \Gamma_{A[x]}(x)(\Omega(A[x])) \subset [0, \infty)$ by Proposition 33.8.2.$\square$
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