Proof, [Theorem II.9.4, Zhu93]. By construction $\Gamma_{A}$ is a unital algebra homomorphism.

To see that $\Gamma_{A}$ preserves involutions, let $y \in A$ be self-adjoint. By Proposition 33.8.2 and Proposition 34.3.6, $\Gamma_{A}(y)(\Omega(A)) = \sigma_{A}(y) \subset \real$, 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

so $\Gamma_{A}(x^{*}) = \Gamma_{A}(x)^{*}$.

Now, for each $x \in A$, Corollary 34.3.5 and Proposition 33.8.2 imply that

Thus $\Gamma_{A}$ is an isometry, and $\Gamma_{A}(A)$ is a closed subalgebra of $C(\Omega(A))$.

Since $\Gamma_{A}(1_{A}) = 1$, $\Gamma_{A}(A)$ contains constants. As $\Gamma_{A}(A)$ separates points and is closed under complex conjugation, $\Gamma_{A}(A) = C(\Omega(A))$ by the Stone-Weierstrass Theorem.$\square$