Let $A$ be a closed subalgebra of bounded normal operators on a Hilbert space $H$. the spectral theorem then implies the existence of a compact topological space $\Delta$, and a resolution of the identity $E$ on $\Delta$ such that the map
$$ f \mapsto \int_\Delta f(x) dE(x) $$
is an isometry between $L^\infty(\Delta,E)$ and $A$. Is it possible to generalize this theorem in some form to more general 'subalgebras' of unbounded operators on a Hilbert space? For a practical example, the family of all bounded operators from $\mathcal{S}(\mathbf{R}^d)$ to $\mathcal{S}(\mathbf{R}^d)$ form an algebra $B$, and each of these operators act as unbounded operators from $L^2(\mathbf{R}^d)$, so can we talk about spectral theorems for closed normal subalgebras $A$ of $B$?