quantum mechanics - 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.

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


2. 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:

1. 
   

   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.)

   
   


2. 
   

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

   
   


3. 
   

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

   
   



4. 
   

   $ ||\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 $\frak{A}$.

   
   


5. 
   

   $U^2$ 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.