  An extension of the conditional expectations (those under a given subalgebra
of events and not the simple ones under a single event) from the classical to
the quantum case is presented. In the classical case, the conditional
expectations always exist; in the quantum case, however, they exist only if a
certain weak compatibility criterion is satisfied. This compatibility criterion
was introduced among others in a recent paper by the author. Then,
state-independent conditional expectations and quantum Markov processes are
studied. A classical Markov process is a probability measure, together with a
system of random variables, satisfying the Markov property and can equivalently
be described by a system of Markovian kernels (often forming a semigroup). This
equivalence is partly extended to quantum probabilities. It is shown that a
dynamical (semi)group can be derived from a given system of quantum observables
satisfying the Markov property, and the group generators are studied. The
results are presented in the framework of Jordan operator algebras, and a very
general type of observables (including the usual real-valued observables or
self-adjoint operators) is considered.
