  We develop the theory of quasi--invariant (resp. strongly quasi--invariant)
states under the action of a group $G$ of normal $*$--automorphisms of a
$*$--algebra (or von Neumann alegbra) $\mathcal{A}$. We prove that these states
are naturally associated to left--$G$--$1$--cocycles. If $G$ is compact, the
structure of strongly $G$--quasi--invariant states is determined. For any
$G$--strongly quasi--invariant state $\varphi$, we construct a unitary
representation associated to the triple $(\mathcal{A},G,\varphi)$. We prove,
under some conditions, that any quantum Markov chain with commuting, invertible
and hermitean conditional density amplitudes on a countable tensor product of
type I factors is strongly quasi--invariant with respect to the natural action
of the group $\mathcal{S}_{\infty}$ of local permutations and we give the
explicit form of the associated cocycle. This provides a family of non--trivial
examples of strongly quasi--invariant states for locally compact groups
obtained as inductive limit of an increasing sequence of compact groups.
