Proposition 34.3.6.label Let $A$ be a unital $C^{*}$-algebra and $x \in A$ be self-adjoint, then $\sigma_{A}(x) \subset \real$.
Proof. Let
\[y = \exp(ix) = \sum_{n = 0}^{\infty} \frac{i^{n}x^{n}}{n!}\]
then
\[y^{*}= \sum_{n = 0}^{\infty} \frac{(-i)^{n} (x^{*})^{n}}{n!}= \exp(-ix^{*})\]
Since $x$ is normal, $y$ is also normal. By Proposition 33.6.3,,
\[y^{*}y = \exp(-ix^{*} + ix) = \exp(-ix + ix) = 1\]
so $y$ is unitary. By Proposition 34.2.5 and the Spectral Mapping Theorem, $\exp(i\sigma_{A}(x)) = \sigma_{A}(y) \subset \partial B_{\complex}(0, 1)$. Thus $i\sigma_{A}(x) \subset \bracs{\text{Re} = 0}$, and $\sigma_{A}(x) \subset \real$.$\square$
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