Quasi-invariant states

We develop the theory of quasi–invariant (resp. strongly quasi–invariant) states under the action of a group $G$ of normal $\ast$–automorphisms of a $\ast$–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 $\phi$, we construct a unitary representation associated to the triple $(\mathcal{A},G,\phi )$. 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 ${\mathcal{S}}_{\mathrm{\infty }}$ 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.