I have some (possibly basic) questions about $C^*$-algebras, the Stinespring Theorem (Theorem 3.6 in Takesaki's book), and unbounded operators. This is motivated by quantum mechanics, where unbounded operators are common. For example, the momentum operator $\hat{p}$ (which is related to $-i \partial^{\,}_x$), or the boson number operator $\hat{n}$, whose eigenvalues are $n \in \mathbb{N}^{\,}_0$. For reference, I have come across these two works from 2008 and from 2020, which seem relevant, though I lack the background to understand them fully in finite time.
Do the two references above generically extend the Stinespring Theorem to unbounded operators? In other words, if a Hilbert space $\mathcal{H}$ admits unbounded operators, and I replace the usual set of bounded operators $\mathcal{B}(\mathcal{H})$ with the set End($\mathcal{H}$) of all (possibly self-adjoint) linear operators on $\mathcal{H}$, does the following aspect of the Stinespring Theorem still hold:
Let $\mathcal{H}$ and $\mathcal{K}$ be Hilbert spaces, with corresponding operator sets ${\rm End}(\mathcal{H})$ and ${\rm End}(\mathcal{K})$. If the adjoint map $\psi^* = \phi: \mathcal{A} \to {\rm End}(\mathcal{H})$ is completely positive and unital, then there exists a unital *-homomorphism $\pi : \mathcal{A} \to {\rm End} (\mathcal{K})$ and an isometry $V : \mathcal{H} \to \mathcal{K}$ such that, $ \forall A \in \mathcal{A}$, $\phi (A) = V^* \pi (A) V$, up to multiplication by a unitary.
The statement above is the part of the Theorem relevant to me, and please correct me if I've been sloppy. For quantum mechanics, the $C^*$-algebra $\mathcal{A}$ is unital, the map $\psi$ and its adjoint $\phi$ are generally completely positive and trace preserving (CPTP), we can restrict to self-adjoint operators in End($\mathcal{H}$), and/or require that dim($\mathcal{K}$) ≥ dim($\mathcal{H}$), if helpful.
If these papers are not familiar—and because I am new to *-algebras—it would also be helpful to me to know, e.g., what goes wrong in trying to construct $C^*$-algebras with unbounded operators, if anything? If there are issues, do they persist if we restrict to unital algebras / CPTP maps / self-adjoint operators? Or, are things like "completely positive" or other aspects of $C^*$-algebras difficult to define with unbounded operators? Lastly, do unbounded operators pose a particular obstacle to proving the Stinespring Theorem beyond the considerations above?
If this is a bad question in any respect, please let me know how I could improve it / what is missing. I did not find any similar questions while searching Stack, and my more general search came up with the two papers referenced above. Still, if you think another answer might be relevant, please comment with a link.