Let $T\in B(H)$ is normal, and $B(\sigma(T))$ denote the $C^*$ algebra of all bounded Borel functions on $\sigma(T)$. Then is it true that $B(\sigma(T))$ is a closed $C^*$ algebra under the sup. norm ?
Actually I try to prove the Spectral representation Theorem of a normal operator in case the representation of $C^*(T)$ over $H$ is cyclic, i.e there exists $x\in H$ such that $\{\phi(T)x:T \in C^*(T)\}$ is dense in $H$. We have $E$ is the the spectral measure of $T$, and $\mu$ is the positive regular Borel measure $\langle E(\cdot)x,x\rangle$ . So I define the following map $$ U\colon \{F(T)x: F\in B(\sigma(T))\}\rightarrow L_{2}(\mu) $$ by $U(F(T)x)=F$. I proved that the map $U$ is well defined and isometric. I want to prove that it has a dense range and can be extended to a unitary map on $H$. To prove the fist assertion I said that the $S$:the set of all simple functions on $\sigma(T)$ is contained in $B(\sigma(T))$, and hence the closure of $S$ under the sup. norm is contained in the closure of $B(\sigma(T))$. Hence the $C(\sigma(T))\subset B(\sigma(T))$, but $C(\sigma(T))$ is dense in $L_{2}(\mu)$, which proves that $U$ has a dense range? Is that correct?
I'm stuck with the second assertion , so can you help me please! Thank you.