I am reading John B. Conway. A Course in Functional Analysis. Springer, 1997 (second edition). Theorem X.4.10c states:
If $(X,\Omega)$ is a measurable space, $H$ is a Hilbert space and $E$ is a spectral measure on $(X,\Omega,H)$, let $\Phi(X,\Omega)$ be the algebra of all $\Omega$-measurable functions $\phi:X\to\mathbb{C}$ and define $\rho:\Phi(X,\Omega)\to C(H)$ by $\rho(\phi)=\int\phi\mbox{d}E$. Then for $\phi,\psi$ in $\Phi(X,\Omega)$, if $\psi$ is bounded, $\rho(\phi)\rho(\psi)=\rho(\psi)\rho(\phi)=\rho(\phi\psi)$.
I doubt this claim. If $\psi=0$ and $\mbox{dom }\rho(\phi)\neq H$, then $\mbox{dom }(\rho(\psi)\rho(\phi))\neq H$, but $\mbox{dom }\rho(\phi\psi)=\mbox{dom }\rho(0)=H$.
Am I missing something?