Theorem: Continuous Functional Calculus Commutes with Character on Generated C*-Subalgebra

Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $x \in A$ be normal. 
Let $\map {\sigma_A} x$ denote the spectrum of $x$ in $A$.
Let $\map \CC {\map {\sigma_A} x}$ be the space of continuous functions on $\map {\sigma_A} x$.
Let $f \in \map \CC {\map {\sigma_A} x}$.
Let $\map f x$ be the continuous functional calculus of $x$ applied to $f$.
Let $B$ be the $\text C^\ast$-algebra generated by $\set { {\mathbf 1}_A, x, x^\ast}$. 
Let $\Phi_B$ be the spectrum of $B$. 
Let $\phi \in \Phi_B$.
Then $\map \phi {\map f x} = \map f {\map \phi x}$.

Proof:
From Spectrum of Element of Unital C*-Subalgebra of Unital C*-Algebra, we have:
:$\map {\sigma_A} {\map f x} = \map {\sigma_B} {\map f x}$
Let $\norm {\, \cdot \,}_\infty$ be the supremum norm on $\map {\sigma_B} x$. 
From Subalgebra Generated by Inclusion and Conjugate is Everywhere Dense in Space of Continuous Functions on Compact Subset of Complex Numbers, there exists a sequence $\sequence {p_k}_{k \in \N}$ such that:
:$\ds \map {p_k} z = \sum_{j \mathop = 1}^{N_k} \alpha_{j, k} z^{n_{j, k} } \overline z^{m_{j, k} }$
for each $z \in \map {\sigma_B} x$ and some $N_k \in \N$, $\alpha_{j, k} \in \C$, $n_{j, k}, m_{j, k} \ge 0$, such that:
:$\norm {p_k - f}_\infty \to 0$ as $n \to \infty$
From Existence and Uniqueness of Continuous Functional Calculus, we have that $\Theta_x$ is a linear isometry and so:
:$\norm {\map {p_k} x - \map f x} \to 0$ as $n \to \infty$
Since $\Theta_x$ is an algebra homomorphism, we note that:
{{begin-eqn}}
{{eqn	| l = \map {\Theta_x} {p_k}
		| r = \sum_{j \mathop = 1}^{N_k} \alpha_{j, k} \map {\Theta_x} {z^{n_{j, k} } } \map {\Theta_x} {\overline z^{m_{j, k} } }
		| c = {{Defof|Algebra Homomorphism}}
}}
{{eqn	| r = \sum_{j \mathop = 1}^{N_k} \alpha_{j, k} \paren {\map {\Theta_x} z}^{n_{j, k} } \paren {\map {\Theta_x} {\overline z} }^{m_{j, k} }
}}
{{eqn	| r = \sum_{j \mathop = 1}^{N_k} \alpha_{j, k} x^{n_{j, k} } \paren {x^\ast}^{m_{j, k} }
		| c = {{Defof|Continuous Functional Calculus}}, $\Theta_x$ is a $\ast$-algebra homomorphism
}}
{{end-eqn}}
From Character on Banach Algebra is Continuous, we have that $\phi$ is bounded, giving:
:$\cmod {\map \phi {\map {p_k} x} - \map \phi {\map f x} } \to 0$ as $n \to \infty$
From Character on C*-Algebra is *-Algebra Homomorphism, $\phi$ is a $\ast$-algebra homomorphism. 
Hence we have:
:$\ds \map \phi {\map {p_k} x} = \sum_{j \mathop = 1}^{N_k} \alpha_{j, k} \map \phi { x^{n_{j, k} } \paren {x^\ast}^{m_{j, k} } } = \sum_{j \mathop = 1}^{N_k} \alpha_{j, k} \paren {\map \phi x}^{n_{j, k} } \overline {\map \phi x}^{m_{j, k} } = \map {p_k} {\map \phi x}$
Moreover since $\norm {p_k - f}_\infty \to 0$ as $n \to \infty$, we obtain $\map {p_k} {\map \phi x} \to \map f {\map \phi x}$ as $n \to \infty$.
We have already established that $\map {p_k} {\map \phi x} = \map \phi {\map {p_k} x} \to \map \phi {\map f x}$ as $n \to \infty$.
Hence we conclude that $\map f {\map \phi x} = \map \phi {\map f x}$.
{{qed}}
Category:Continuous Functional Calculus
Category:Characters (Banach Algebras)
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