/sci/ - Science & Math

In pic related my professor told us that if you have a classical system with hamiltonian [math]H(p_i,q_i)[/math] then the relative quantum system is given if the Schrödinger eqn is satisfied and also the observables become self adjoint operators. Is this notation implying that there exist a map, matching every classical observable to its corresponding operator? (would that be a map from [math]\left( C^{\infty}(M),+, * \right) \rightarrow A[/math] , where A is an algebra of operators)