GNS construction for the Schrödinger's hydrogen atom Does anybody know if the H-atom (no spin, no special relativity) has been treated in the literature starting from the axioms of QM by I.E. Segal?  These axioms are:
We call a structure $\frak{A}$ which statisfies the following postulates a closed system of observables, or for short, a system.
POSTULATES
I. Algebraic postulates.


*

*$\frak{A}$ is a real linear space.

*There exists in $\frak{A}$ and identity element I and for every $U\in\frak{A}$ and positive integer n an element $U^n \in \frak{A}$, these being such that the usual rules for operating with polynomials in a single variable are valid: if f, g, and h are polynomials with real coefficients, and if $f(g(\alpha)) = h(\alpha)$ for all real $\alpha$, then $f(g(U)) = h(U)$. Here: 
$$f(U) = \beta_0 I + \sum_{k=1}^m \beta_k U^k$$, if $$f(\alpha) = \sum_{k=1}^m \beta_k \alpha^k$$


II. Metric postulates.
There is defined for each observable U a non-negative real number $||U||$ such that:


*

*If $\alpha$ is an arbitrary real number and U and V are arbitrary elements of $\frak{A}$, then $||\alpha U|| = |\alpha|\cdot||U||$, $||U+V||\leq ||U|| + ||V||$. The vanishing of $||U||$ implies $U=0$. And $\frak{A}$ is topologically complete when regarded as a metric space with the distance between U and V defined as $||U-V||$. (In other words, $\frak{A}$ is a real Banach space relative to $||U||$ as a norm.) 

*$||U^2 - V^2 || \leq \mbox{Max}~[~||U^2||, ||V^2||~]$.

*$||U^2|| = ||U||^2 $.

*$ ||\sum_{U\in\frak{R}} U^2|| \leq ||\sum_{U\in\frak{S}} U^2||$, if $\frak{R}\subset\frak{S}$, $\frak{R}$ and $\frak{S}$ being finite subsets of $\frak{A}$.

*$U^2$ is a continuous function of $U$. 
I know that the spinless non-specially relativistic H-atom has been dealt with at a satisfactory mathematical level from very many PoV, such as the direct functional analytical approach by Tosio Kato, the path integral by Hagen Kleinert, the group representation theory of $SO(4)$ up to $SO(4,2)$ starting from the disguised form by Pauli (1926) and Fock (1934) up to Barut, Itzykson and others in the 1960s. 
But from the PoV of algebraic QM, built on the axioms of I.E. Segal, I've not seen this seemingly simple model dealt with anywhere. 
By my limited knowledge, I see that $\frak{A}$, this time as a complex linear space, can be further endowed with an involution operation (Hermitean adjoint) which would turn it into an involution Banach algebra, i.e. a $C^{*}$-algebra (https://ncatlab.org/nlab/show/C-star-algebra). The trouble is that the operators from such a topological algebra must be bounded, but the fundamental observables in the H-atom are unbounded. 
 A: When you have an unbounded selfadjoint operator $A$ (as those relevant for the theory of Hydrogen atom)  you can always consider the class  of bounded selfadjoint operators $A_n = \int_{[-n,n]} \lambda dP(\lambda)$, where  $P$ is the spectral  measure of $A$ and $n\in \mathbb N$. This class includes the whole information of $A$ and has the physical meaning of an approximation of $A$ by an instrument not capable to make measurements outside the range $[-n,n]$. The unital $C^{*}$ algebra of a physical system can always be viewed as the von Neumann algebra generated by all families $\{A_n\}_{n\in \mathbb N}$ for every observable $A$ taking int account the fact that von Neumann algebras are $C^{*}$ algebras. This way, Segal's approach is matched. There are subtleties however, since the standard approach deals with the strong operator topology, while algebraic, $C^{*}$-algebra one,   uses uniform topology. This is still an open issue and there exist several different viewpoints on the subject. Also the use of more flexible unital $^{*}$ algebras has some problems.
