  In the study of open quantum systems modeled by a unitary evolution of a
bipartite Hilbert space, we address the question of which parts of the
environment can be said to have a "classical action" on the system, in the
sense of acting as a classical stochastic process. Our method relies on the
definition of the Environment Algebra, a relevant von Neumann algebra of the
environment. With this algebra we define the classical parts of the environment
and prove a decomposition between a maximal classical part and a quantum part.
Then we investigate what other information can be obtained via this algebra,
which leads us to define a more pertinent algebra: the Environment Action
Algebra. This second algebra is linked to the minimal Stinespring
representations induced by the unitary evolution on the system. Finally in
finite dimension we give a characterization of both algebras in terms of the
spectrum of a certain completely positive map acting on the states of the
environment.
