  This work concerns some issues about the interplay of standard and geometric
(Hamiltonian) approaches to finite-dimensional quantum mechanics, formulated in
the projective space. Our analysis relies upon the notion and the properties of
so-called frame functions, introduced by A.M. Gleason to prove his celebrated
theorem. In particular, the problem of associating quantum state with positive
Liouville densities is tackled from an axiomatic point of view, proving a
theorem classifying all possible correspondences. A similar result is
established for classical observables representing quantum ones. These
correspondences turn out to be encoded in a one-parameter class and, in both
cases, the classical objects representing quantum ones result to be frame
functions. The requirements of $U(n)$ covariance and (convex) linearity play a
central r\^ole in the proof of those theorems. A new characterization of
classical observables describing quantum observables is presented, together
with a geometric description of the $C^*$-algebra structure of the set of
quantum observables in terms of classical ones.
