An Axiomatic Approach to the Formalism of Quantum Mechanics. II.

Abstract

Another axiom has to be introduced to make the formalism given in Part I physically equivalent to the conventional Hilbert‐space formalism. Then it is shown that, given a certain fundamental set of observables, a B*‐algebra $\mathfrak{A}$ can be built into which the set O_b of all bounded observables can be mapped injectively. The closure of the algebra generated by the image of O_b into $\mathfrak{A}$ is $\mathfrak{A}$ itself. A Hilbert space H exists into which the set O₀ of all pure physical states can be mapped injectively. The closure of the subset of H which is the image of this mapping is H itself. The algebra $\mathfrak{A}$ can be mapped in such a fashion into the C*‐algebra $\mathfrak{B}$ (H) of all bounded observables that these mappings provide, essentially, just a translation of the original formalism in a Hilbert‐space formalism.