Let $A$ be a unital complex $C^*$-algebra, and let $a\in A$ be normal. Let $\sigma_A(a)$ denote the spectrum of $a$ in $A$. For $f,g\in C(\sigma_A(a))$, let $f(a)$ and $g(a)$ denote the elements of $A$ obtained from the continuous functional calculus for $a$. Then
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\begin{align*}
(fg)(a)=f(a)g(a).
\end{align*}
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If $\bar f:\sigma_A(a)\to\mathbb C$ is the function $\bar f(\lambda)=\overline{f(\lambda)}$, then