  Koopman and von Neumann (KvN) extended the Liouville equation by introducing
a phase space function $S^{(K)}(q,p,t)$ whose physical meaning is unknown. We
show that a different $S(q,p,t)$, with well-defined physical meaning, may be
introduced without destroying the attractive "quantum-like" mathematical
features of the KvN theory. This new $S(q,p,t)$ is the classical action
expressed in phase space coordinates. It defines a mapping between observables
and operators which preserves the Lie bracket structure. The new evolution
equation reduces to Schr\"odinger's equation if functions on phase space are
reduced to functions on configuration space. This new kind of "quantization"
does not only establish a correspondence between observables and operators, but
provides in addition a derivation of quantum operators and evolution equations
from corresponding classical entities. It is performed by replacing
$\frac{\partial}{\partial p}$ by $0$ and $p$ by $\frac{\hbar}{\imath}
\frac{\partial}{\partial q}$, thus providing an explanation for the common
quantization rules.
