\section{The Borel Functional Calculus}
\label{section:borel-functional-calculus}

\begin{definition}[Borel Functional Calculus]
\label{definition:borel-functional-calculus}
    Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, and $x \in A$ be normal, then there exists a unique continuous unital *-homomorphism
    \[
    C(\sigma_A(x); \complex)^{**} \to A[x] \quad f \mapsto f(x)
    \]

    such that:
    \begin{enumerate}
        \item $\one(x) = 1_A$, $\text{Id}(x) = x$, and $\overline{\text{Id}}(x) = x^*$.
        \item The mapping $f \mapsto f(x)$ is continuous from the weak* topology on $C(\sigma_A(x); \complex)^{**}$ to the strong operator topology on $B(H)$. 
    \end{enumerate}
\end{definition}
\begin{proof}
    Since the \autoref{definition:continuous-functional-calculus}
\end{proof}