Proposition 34.2.5.label Let $A$ be a unital $C^{*}$-algebra and $x \in A$ be unitary, then $\sigma_{A}(x) \subset \partial B_{\complex}(0, 1)$.
Proof, [Proposition II.8.2, Zhu93]. By Lemma 34.2.2, $\norm{x}_{A} = 1$, so $\sigma_{A}(x) \subset \ol{B_\complex(0, 1)}$. Thus
\[\bracsn{\ol{\lambda}|\lambda \in \sigma_A(x)}= \sigma_{A}(x^{*}) = \sigma_{A}(x^{-1}) \subset \ol{\complex \setminus B_\complex(0, 1)}\]
by (4) of Proposition 34.1.3. For any $\lambda \in \complex$, $\lambda \in \ol{B_\complex(0, 1)}$ and $\ol \lambda \in \ol{\complex \setminus B_\complex(0, 1)}$ if and only if $|\lambda| = 1$. Therefore $\sigma_{A}(x) \subset \partial B_{\complex}(0, 1)$.$\square$
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