Quasi-invariant states

TL;DR

This paper develops the theory of quasi-invariant states under group actions on algebras, characterizes their structure, and provides examples related to quantum Markov chains and locally compact groups.

Contribution

It introduces the concept of quasi-invariant states under automorphism groups, characterizes strongly quasi-invariant states for compact groups, and constructs associated unitary representations.

Findings

Strongly quasi-invariant states are associated with unitary representations.

Quantum Markov chains with certain properties are strongly quasi-invariant under local permutations.

Explicit cocycles are constructed for examples involving inductive limits of compact groups.

Abstract

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.