One version of the spectral theorem can be phrased as (cp. "Canonical Quantum General Relativity" by Thiemann):
Let $\mathfrak{P}$ be a $*$-Algebra and $\mathfrak{U} \subset \mathfrak{P}$ be an abelian, unital $C^*$-subalgebra. Let $(\mathcal{H}, \pi)$ be a representation of $\mathfrak{P}$. Then there exists a unitary map $U:\mathcal{H} \rightarrow L^2(\sigma, d\mu)$ where $\sigma$ is some disjoint union of the spectra $\Delta(\mathfrak{U})$ and $d \mu$ are regular Borel measures thereon, such that $\mathfrak{U}$ is represented via multiplication operators on $L^2(\sigma, d\mu)$.
What I need is the compatibility of this theorem with inductive limits in the following sense:
I would like to have this theorem:
Let $\mathfrak{P}$ be a $*$-algebra which is given by the inductive limit of a directed family of $*$-algebras $\mathfrak{P}_\gamma$. Let $\mathfrak{U} \subset \mathfrak{P}$ be an abelian, unital $C^*$-subalgebra which is given by the inductive limit of a directed family of abelian, unital $C^*$-subalgebras. Let $(\mathcal{H}, \pi)$ be a representation of $\mathfrak{P}$. Then there exists a unitary map $U: \mathcal{H} \rightarrow L^2(\sigma, d\mu)$ where $\sigma$ is a disjoint union of the projective limits of spectra $\lim_\leftarrow \Delta(\mathfrak{U}_\gamma)$ and $d \mu$ are regular Borel measures thereon, such that $\mathfrak{U}$ is represented via multiplication operators on $L^2(\sigma, d \mu)$.
Proving this theorem would require the compatibility of representations, cyclic representations, subalgebras, etc. with inductive limits.
My question is: Do you know (some analogue of) this theorem? If so, could you give me a reference? Are there reasons, why a proof of this theorem could be impossible or hard?