  The common structure of the space of pure states $P$ of a classical or a
quantum mechanical system is that of a Poisson space with a transition
probability. This is a topological space equipped with a Poisson structure, as
well as with a function $p:P\times P-> [0,1]$, with certain properties. The
Poisson structure is connected with the transition probabilities through
unitarity (in a specific formulation intrinsic to the given context).
  In classical mechanics, where $p(\rho,\sigma)=\dl_{\rho\sigma}$, unitarity
  poses no restriction on the Poisson structure. Quantum mechanics is
characterized by a specific (complex Hilbert space) form of $p$, and by the
property that the irreducible components of $P$ as a transition probability
space coincide with the symplectic leaves of $P$ as a Poisson space. In
conjunction, these stipulations determine the Poisson structure of quantum
mechanics up to a multiplicative constant (identified with Planck's constant).
  Motivated by E.M. Alfsen, H. Hanche-Olsen and F.W. Shultz ({\em Acta Math.}
{\bf 144} (1980) 267-305) and F.W. Shultz ({\em Commun.\ Math.\ Phys.} {\bf 82}
(1982) 497-509), we give axioms guaranteeing that $P$ is the space of pure
states of a unital $C^*$-algebra. We give an explicit construction of this
algebra from $P$.
