Removed parts from Zhu citations.
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\label{proposition:unitary-spectrum}
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Let $A$ be a unital $C^*$-algebra and $x \in A$ be unitary, then $\sigma_A(x) \subset \partial B_\complex(0, 1)$.
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\end{proposition}
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\begin{proof}[Proof, {{\cite[Proposition II.8.2]{Zhu}}}. ]
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\begin{proof}[Proof, {{\cite[Proposition 8.2]{Zhu}}}. ]
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By \autoref{lemma:unitary-unit}, $\norm{x}_A = 1$, so $\sigma_A(x) \subset \ol{B_\complex(0, 1)}$. Thus
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\[
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\bracsn{\ol{\lambda}|\lambda \in \sigma_A(x)} = \sigma_A(x^*) = \sigma_A(x^{-1}) \subset \ol{\complex \setminus B_\complex(0, 1)}
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