I am not able to find a proper explanation of how classical mechanics can be expressed as C*-algebra. In the brief description that I found (please see here below few references), it is said that classical mechanics is described as abelian $C^{*}$-algebra, but I missed how important steps as the representation of the phase space or the description of Poisson brakets are introduced. I asked this question also on Physics StackExchange, but no answer so far.
References
"Deterministic Chaos in Infinite Quantum Systems" by F. Benatti;
"Operator Algebras and Quantum Statistical Mechanics" by O. Bratteli, D. Robinson.