Removed parts from Zhu citations.

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Bokuan Li
2026-07-07 11:39:54 -04:00
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is a unital $C^*$-isomorphism.
\end{theorem}
\begin{proof}[Proof, {{\cite[Theorem II.9.4]{Zhu}}}. ]
\begin{proof}[Proof, {{\cite[Theorem 9.4]{Zhu}}}. ]
By construction $\Gamma_A$ is a unital algebra homomorphism.
To see that $\Gamma_A$ preserves involutions, let $y \in A$ be self-adjoint. By \autoref{proposition:gelfand-transform-gymnastics} and \autoref{proposition:self-adjoint-spectrum}, $\Gamma_A(y)(\Omega(A)) = \sigma_A(y) \subset \real$, so $\Gamma_A(y) \in C(\Omega(A); \real)$. For any $x \in A$, write $x = \text{Re}(x) + i\text{Im}(x)$, where $\text{Re}(x)$ and $\text{Im}(x)$ are both self-adjoint, then