Quasi-invariant states

In this paper, we develop the theory of quasi-invariant (respectively, strongly quasi-invariant) states under the action of a group G of normal & lowast;-automorphisms of a & lowast;-algebra (or von Neumann algebra) 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 (A,G,phi). 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 S-infinity 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.