Proposition 34.8.3.label Let $A$ be a unital $C^{*}$-algebra and $x \in A_{sa}$, then the following are equivalent:
- (1)
$x$ is positive.
- (2)
$\sigma_{A}(x) \subset [0, \infty)$.
- (3)
There exists $\lambda \ge \norm{x}_{A}$ such that $\norm{\lambda - x}_{A} \le \lambda$.
Proof, [Lemma II.11.3, Zhu93]. (1) $\Leftrightarrow$ (2): Proposition 34.8.2.
(2) $\Leftrightarrow$ (3): By assumption, $\sigma_{A}(x) \subset \real$, so Theorem 34.4.3 implies that
\[\norm{\lambda - x}_{A} = [\lambda - x]_{sp}= \sup\bracsn{\lambda - \mu|\mu\in \sigma_A(x)}\]
which is bounded above by $\lambda$ if and only if $\sigma_{A}(x) \subset [0, \infty)$.$\square$
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