Let $A$ be an unital commutative C*-subalgebra of $B(H)$, and $\Omega$ be its character space. By spectral theorem $$\phi: B_\infty(\Omega)\to B(H);~~~~~f\to \int f \, dP$$ is a $*-$ homomorphism where $B_\infty(\Omega)$ is the space of bounded Borel-measurable functions and $P$ is the spectral measure correspondence $A$.
If $x\in B(H)$ is normal, put $A=C^*(x,1)$ (C*-algebra generated by $x,1$). Could we show that $\phi(B_\infty (\sigma(x))) = vn(x,1)$ (von Neumann algebra generated by $x,1$)? Thanks in advance.