Theorem 34.7.1 (Spectral Theorem for $C^{*}$-Algebras).label Let $A$ be a unital $C^{*}$-algebra and $x \in A$ be normal, then the mapping
is a homeomorphism.
Proof. Firstly, $A[x]$ is commutative by Proposition 34.2.2. Thus Corollary 34.2.4 and (3) of Proposition 33.8.2 imply that
and $\Phi$ is a surjection onto $\sigma_{A}(x)$.
On the other hand, the Gelfand-Naimark Theorem implies that $\Phi(x^{*}) = \ol{\Phi(x)}$, so since $A[x]$ is the smallest $C^{*}$-algebra containing $x$, any element $\psi \in \Omega(A[x])$ is uniquely determined by $\psi(x)$. Therefore $\Phi$ is injective.
Finally, since $\Omega(A[x])$ is equipped with the weak* topology and $\Phi$ is the evaluation map at $x$, it is continuous.
By Proposition 5.16.5, $\Phi$ is a homeomorphism.$\square$
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