From this viewpoint, the move from a classical to a quantum mechanical system is not a move from a comutative to a non-commutative algebra of a real-valued observables, but, instead, a move from a commutative algebra to a partial commutative algebra of observables.
Of course, every non-commutative algebra determines an underlying partial commutative algebra and also its diagram of commutative subalgebras.
That fact that assuming the structure of a non-commutative algebra is the wrong assumption has already been observed in the literature (see, for example, ),
but it is often replaced by another wrong assumption, namely that of assuming the structure of a Jordan algebra.
These differing assumptions on the structure of affect the size of its automorphisum group and, hence, of the allowable symmetries of the system (the weaker the assumed structure on , the larger is its automorphism group).
— The Mathematical Foundations of Quantum Mechanics
— David A. Edwards
2019.06.18 Tuesday ACHK