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When $N$ is a normal operator on $H$ with spectral measure $E$,let $B(\sigma(N))$ be the $C^*$ algebra of bounded Borel functions on $\sigma(N)$, we have the map $\psi\mapsto \psi(N)$, which is a representation of the $C^*$ algebra $B(\sigma(N))$.

But we also have the map from $C(\sigma(N))\to B(H)$, $\psi \mapsto \psi (N)$. Is the Borel functional calculus an extension of continuous functional calculus?

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Yes, in both cases they are faithful representations that map the identity function to $N$. So they agree on polynomials. Being continuous, they agree on continuous functions.

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