Quasi-invariant states

In this paper, we develop the theory of quasi-invariant (respectively, strongly quasi-invariant) states under the action of a group [Formula: see text] of normal ∗-automorphisms of a ∗-algebra (or von Neumann algebra) [Formula: see text]. We prove that these states are naturally associated to left-[Formula: see text]-[Formula: see text]-cocycles. If [Formula: see text] is compact, the structure of strongly [Formula: see text]-quasi-invariant states is determined. For any [Formula: see text]-strongly quasi-invariant state [Formula: see text], we construct a unitary representation associated to the triple [Formula: see text]. We prove, under some conditions, that any quantum Markov chain with commuting, invertible and Hermitian 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 [Formula: see text] of local permutations and we give the explicit form of the associated cocycle. This provides a family of nontrivial examples of strongly quasi-invariant states for locally compact groups obtained as inductive limit of an increasing sequence of compact groups.