The introduction you outlined is basically reminiscent of the old quantum mechanics, anyway in the approach you depicted it is the culmination and not the premise of the construction, and the comment was surely intended to be explanatory of the far origin of these choice. I now try to resume the history.
There are basically two approaches to mathematical quantum mechanics. The first one very complex and stratified in its development, but simple in the premise was discussed by John Von Neumann in a lot of papers after the "Foundation of quantum mechanics", the second one is basically conceveid to be an extension for the first, and is this second approach you are referring to: the GNS approach.
Anyway both of them are surely derived after an abstraction process very far beginning on the methods of classical mechanics, joint to the newest evidence from atomic and particle physics of the first quantum mechanics.
Just as in classical mechanics we define functions of observables dynamical quantity so the founders of quantum mechanics conceived it is possible in quantum mechanics, anyway we need to clarify in which sense this is possible and explanation isn't fully depleted from the naive extension of classical theory of the measure, based on real numbers, but it need of a clear axiomatic and this was furnished from John Von Neumann (and in some way from Heisenberg, Dirac, and Schroedinger before him formulated this axiomatic)
Anyway, just as in classical mechanics there is a notion of repeatibility and regularity, so there is in quantum mechanics. The true difference is in the outcome of the measures, deterministic in classical, probabilistic in quantum mechanics. So that measure processes are conceived deterministic in a statistical sense, and, for example, the component energies of the isotropic harmonic oscillator sums exactly in mean value, but the variance is zero if the considered states are eigenstates. Old quantum mechanics can be founded on few axioms about the measures and led Von Neumann, in a natural way to linear operators acting, like a non commutative algebra, on Hilbert spaces.
In order to grant correspondence principle we, following the founders of quantum mechanics, need to hypothesize the existence of intrinsically deterministically evolving observable, and just the measure process make the difference, because these dynamical "quantities" with respect to the measures doesn't appear as real numbers, this point was the first time realized some time after the Copenaghen interpretation was developed.
So they are assumed, after Heisenberg (speaking of non commutative numbers) and Jordan (speaking of matrices), and Schroedinger (speaking of operators acting on functional space of probability) all these three point of view were showed to be in a certain strict framework to be equivalent, from Dirac assuming they are algebraic elements obeying to canonical commutation relation generalizing the Poisson algebra.
In brief the Dirac point can be summarized in assuming an Hilbert space structure for the states, and in developing step by step a theory of observables compatible with the Copenaghen interpretation spirit and with the correspondence principle.
Anyway Von Neumann felt the need to obtain an axiomatic foundation based on more general operators algebras, and an axiomatic of measure, unifying from scratch the theoretical
framework, in fact obtaining a more general theory with respect to the Heisenberg and Dirac theoretical "prejudices". The Von Neumann point was in fact based on the general representation theory in the geometrical framework of Banach operator algebras of operators in Hilbert space, and in particular on the CCR irreducible representation theory, but from this point the research of Von Neumann continued in search of an intrinsic point of view based on the geometry of observable.
After time and time was in fact recognized that part of quantum theory of measure is nothing else then a generalized probabilistic theory in a Banach algebra and the general setting of Gelfand Najmark Segal construction rebuild intrinsically the Hilbert spaces. Anyway the field extension of this setting is very problematic and a hierarchy of Hilbert spaces appears. Anyway in this way a circle is closed and a new loop is opened: in the GNS approach to quantum mechanics we postulate that operators are living in an abstract algebra, obeying familiar rules for an algebra with an involution (the * operation). Via Gelfand theorem the commutative case led to the algebra of complex valued continuous functions in an Hausdorf space, the spectrum of the algebra (which will led the ordinary numerical set of coordinates of classical mechanics), and more in general to a spectral theory, culminating in the GNS construction, which associate to a given linear form an Hilbert space and a representation for the algebra.
Anyway the true achievement of this approach is the net of algebras, that is very more general with respect to the Hilbert space interpretation of quantum mechanics,this achievement is useful in relativistic field theory and leads to very far reaching results firstly partially discovered by Von Neumann in some papers, and after then developed from Araky, Haag, Kastler. In this full setting is now possible to address in more precise terms the question of the cluster decomposition principle implicit in the deterministic evolutionary scheme, and the question of repeatability principle of classical and quantum mechanics, and to understand quantitatively something about the limitation, arising from the change of the state of the universe, to this principle, which can be espressed, for example, in term of a change of representation, becaused from the change of the linear form representing the thermokinetic state of "universe", without any change in the postulates of quantum field theory and the derived quantum mechanics. This is perhaps the perspective of the search about KMS theorem.
I'm not very satisfied from this resume, anyway I think you can correct and integrate it, and I hope to read and write something else more precise and delimited.
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